ON APERIODIC RELATIONAL MORPHISMS
BENJAMIN STEINBERG
Abstract. We characterize aperiodic relational morphisms as those
that are injective on regular H-classes. This result is applied to obtain simple proofs and generalizations of McAlister’s results on joins of
aperiodic semigroups and groups. Also, we show that if H is a proper,
non-trivial pseudovariety of groups, then
A ∗ H ( (A ∗ G) ∩ H.
We provide coordinate-free formulations and proofs of Rhodes’s Presentation Lemma and generalizations. As an application, we give simpler
proofs of Tilson’s theorem on the complexity of semigroups with at most
2 non-zero J -classes and Rhodes’s theorem that complexity is not local.
1. Introduction and the Aperiodicity Lemma
In this paper all semigroups are taken to be finite. A relational morphism [8] ϕ : S −→
◦ T of semigroups is a relation ϕ such that:
• sϕ 6= ∅, all s ∈ S;
• sϕs0 ϕ ⊆ (ss0 )ϕ, all s, s0 ∈ S.
Recall that if V is a pseudovariety of semigroups, a relational morphism
ϕ : S −→
◦ T is a V-relational morphism [46, 24] if, T ≥ T 0 ∈ V implies
T 0 ϕ−1 ∈ V. The class RV of V-relational morphisms is clearly closed under
composition and is a (positive) pseudovariety of relational morphisms in the
sense of [28] and a closed class in the sense of Tilson [46].
Of particular interest are the aperiodic relational morphisms. They play
a key role in the Krohn-Rhodes complexity theory [14, 15, 16] via the Fundamental Lemma of Complexity [25, 46]. They are also important in connection with the concatenation operator in formal language theory [42, 24].
Recall that a semigroup is aperiodic if all its subgroups are trivial. Then by
an aperiodic relational morphism, we mean a member of RA . The following
well-known lemma [16] characterizes aperiodic homomorphisms.
Lemma 1.1. Let ϕ : S → T be a homomorphism. Then the following are
equivalent:
(1) ϕ is aperiodic;
Date: September 19, 2003.
1991 Mathematics Subject Classification. 20M07.
Key words and phrases. Aperiodic relational morphisms, the presentation lemma, joins.
The author was supported in part by NSERC and by POCTI approved project
POCTI/32817/MAT/2000 in participation with the European Community Fund FEDER.
1
2
BENJAMIN STEINBERG
(2) for all idempotents e ∈ T , eϕ−1 is aperiodic;
(3) ϕ is injective on subgroups (in the terminology of [16], ϕ is a γ-map);
(4) ϕ is injective on regular H-classes.
The principal tool in this paper is an extension of this lemma to relational
morphisms. Although it might not seem surprising, this extension (which
the author believes to be new) allows for simple proofs (and extensions) of
McAlister’s results [20, 21] on (regular) direct product decompositions of a
regular semigroup via aperiodic (or more generally fundamental) semgroups
and groups. In the process we answer a question of Karnofsky and Rhodes
from 1982 [13] as to whether there exists a semigroup in A ∗ G whose subgroups are Abelian (in Ab) but does not belong to A ∗ Ab. In fact, we show
more generally that, for any proper non-trivial pseudovariety H of groups,
there is a semigroup with commuting idempotents and subgroups in H that
does not belong to A ∗ H.
Our extension of Lemma 1.1 is also used to obtain a coordinate-free proof
of a version of Rhodes’s Presentation Lemma [5]. We give a new (and we
hope simplified) proof of Tilson’s results on the complexity of semigroups
with at most 2 non-zero J -classes [44]. Finally, we give an easier proof of
Rhodes’s result [27, 5] that complexity is not local.
Let us introduce some terminology so that we may formulate our result.
Let ϕ : S −→
◦ T be a relational morphism and let X ⊆ S. Then ϕ is said to
be injective on X if, for all x1 , x2 ∈ X,
(1.1)
x1 ϕ ∩ x2 ϕ 6= ∅ =⇒ x1 = x2 .
Notice that a relational morphism is injective on S if and only if it is a
division. Our main technical lemma is then:
Lemma 1.2 (Aperiodicity Lemma). Let ϕ : S −→
◦ T be a relational morphism. Then the following are equivalent:
(1)
(2)
(3)
(4)
ϕ is aperiodic;
for all idempotents e ∈ T , eϕ−1 is aperiodic;
ϕ is injective on subgroups;
ϕ is injective on regular H-classes.
Recall that a subset A of a semigroup S is said to be V-pointlike for a
pseudovariety V if, for all relational morphisms ϕ : S −→
◦ T with T ∈ V,
−1
there exists t ∈ T with A ⊆ tϕ . (The notion of pointlike sets is due to
Rhodes). One says that V has decidable pointlikes if there is an algorithm
to compute the V-pointlike subsets of finite semigroups. Such algorithms
exist for many pseudovarieties [2, 3, 33, 35, 36]. It is well known (see [10] for
instance) that there is always a relational morphism ϕ : S −→
◦ T with T ∈ V
such that A ⊆ S is V-pointlike if and only if A ⊆ tϕ−1 for some t ∈ T . By
considering such a relational morphism, we immediately obtain the following
corollary of the Aperiodicity Lemma. We recall that the Mal’cev product
m V is the pseudovariety of all semigroups with an aperiodic relational
A
ON APERIODIC RELATIONAL MORPHISMS
3
morphism to an element of V (see below for the general definition of Mal’cev
products).
Corollary 1.3. Let V be a pseudovariety for which it is decidable whether
a pair of H-equivalent regular elements of a finite semigroup is V-pointlike.
m V is decidable. In particular, if V has decidable pointlikes,
Then A m V is decidable.
A
Thus, by [42], under the conditions of the above corollary, one can determine the variety of languages obtained from the V-languages [8] by closing
under concatentation; see also [24].
We point out that it is often easier to compute V-pointlikes for Hequivalent regular elements than the V-pointlikes in general. For instance,
the case of G-pointlike R-equivalent regular elements is already handled by
the techniques of Rhodes and Tilson [30], while the general case requires
the deep results of Ash [3]. The case A is particularly enlightening since
H-equivalent regular elements are always A-pointlike and so there is nothing
to decide.
By the Fundamental Lemma of Complexity [25, 46], the pseudovariety of
m (G ∗ A). Hence deciding complexity
semigroups of complexity one is A one may be reduced to computing G ∗ A-pointlike pairs of H-equivalent
regular elements; the decidability of G ∗ A is established in [13]. We believe
this approach could lead to an easier proof of the decidability of complexity
one than that announced recently by Rhodes. See Section 4 below on the
Presentation Lemma [5] for more.
We mention that the two-sided complexity one-half pseudovarieties
m A and A m LG
LG m is the
are decidable (here LG is the pseudovariety of local groups and Mal’cev product operator; see below for definitions). The first follows since
m ( ) preserves decidability [26, 43]. The second follows by
the operator LG Corollary 1.3 and the decidability of pointlikes for LG, which was shown by
the author in [36].
This paper can really be viewed as two separate papers that have the
same theoretical underpinnings (the Aperiodicity Lemma and the Division
Lemma), but are otherwise independent of each other. Section 2 introduces
our technical tools; Section 3 is concerned with applying these tools to join
decompositions involving aperiodic semigroups and groups; Sections 4, 5 and
6 are concerned with the Presentation Lemma, Tilson’s 2J -class theorem
and the non-locality of complexity; these two parts are independent of each
other.
2. Technical Lemmata
We begin by proving the Aperiodicity Lemma. Let Reg(S) denote the set
of regular elements of a semigroup S; recall that s ∈ S is said to be regular if
s ∈ sSs for all s ∈ S. Also recall that s, s0 are said to be inverses if ss0 s = s
4
BENJAMIN STEINBERG
and s0 ss0 = s0 ; every regular element has an inverse [6, 16]. One says S is
regular if Reg(S) = S.
If s ∈ S, we use sω to denote the unique idempotent power of s and sω−1
to denote the unique inverse of sω s in the minimal ideal of hsi.
2.1. Proof of the Aperiodicity Lemma. The implications (4) =⇒
(3) =⇒ (2) are clear.
It is well known that (2) =⇒ (1), but we include the proof for completeness. Suppose T 0 ≤ T is aperiodic. Let S 0 = T 0 ϕ−1 and ϕ0 : S 0 −→
◦ T0
be the restriction. Then ϕ0 = α−1 β where α : R S 0 is a surjective homomorphism and β : R → T 0 is a homomorphism. Let G ≤ T 0 ϕ−1 be a
subgroup. Then there is a subgroup [8, 16] G0 of R with G0 α = G. Since T 0
is aperiodic, G0 β is an idempotent e and hence G ≤ eϕ−1 , it follows that G
is trivial by (2). So T 0 ϕ−1 is aperiodic.
The interesting verification is (1) =⇒ (4). Suppose x, y ∈ Reg(S), x H y
and t ∈ xϕ ∩ yϕ. Let x0 be an inverse of x and choose t0 ∈ x0 ϕ. Then xx0 ,
x0 x are idempotents. Since xx0 R x R y and x0 x L x L y, xx0 y = y and
yx0 x = y.
Then (using relation notation) we have:
u = (tt0 )ω t ϕ (xx0 )ω x = x
u = (tt0 )ω t ϕ (xx0 )ω y = y
v = (t0 t)ω−1 t0 ϕ (x0 x)ω−1 x0 = x0
uv = (tt0 )ω , vu = (t0 t)ω
uvu = u, vuv = v
Thus xx0 and yx0 ϕ-relate to the idempotent uv. Moreover yx0 H xx0 (they
belong to Rx ∩Lx0 ). Since (uv)ϕ−1 is aperiodic and xx0 is idempotent (and so
hyx0 i is a subgroup in uvϕ−1 with identity xx0 ), we conclude that yx0 = xx0 .
Hence
y = yx0 x = xx0 x = x,
as desired.
2.2. The Division Lemma. Let ϕ : S → T be a homomorphism and X a
subset of S. We say that a relational morphism ψ : S −→
◦ U is injective on
ϕ-classes of X if ψ is injective on equivalence classes of ϕ|X . The following
is a special case of the Slice Theorem [33, 35] that we prove for completeness.
If ϕi : S −→
◦ Ti are relational morphisms, i = 1, 2, then set ϕ1 ⊕ ϕ2 =
∆(ϕ1 × ϕ2 ) where ∆ : S → S × S is the diagonal map. We say ϕ1 ⊕ ϕ2 is
the direct sum of ϕ1 and ϕ2 .
Lemma 2.1 (Division Lemma). Suppose ϕ : S → T is a homomorphism
and ψ : S −→
◦ U is injective on ϕ-classes of X ⊆ S. Then ϕ ⊕ ψ is injective
on X. In particular, if ψ is injective on ϕ-classes of S, then ϕ ⊕ ψ is a
division.
ON APERIODIC RELATIONAL MORPHISMS
5
Proof. Suppose s1 , s2 ∈ (t, u)(ϕ ⊕ ψ)−1 ∩ X. Then s1 ϕ = t = s2 ϕ and
u ∈ s1 ψ ∩ s2 ψ. It follows, since ψ is injective ϕ-classes on X, that s1 = s2
and so ϕ ⊕ ψ is injective on X.
2.3. Lifting J -classes. The following is a relational morphism analogue
of a classical lemma [16] about lifting J -classes under surjective homomorphisms. We use ≤J for the J -preorder [6, 16]. The notation S 1 will always
mean S with a new identity adjoined. This is the object part of a functor
from the category of semigroups with relational morphisms to the category
of monoids with relational morphisms (the arrow part is written ϕ 7→ ϕ1 ).
Lemma 2.2 (Lifting J -classes). Let ϕ : S −→
◦ T be a relational morphism
and J ⊆ S a J -class. Suppose J 0 ⊆ T is ≤J -minimal such that J 0 ϕ−1 ∩ J 6=
∅. Then J ⊆ J 0 ϕ−1 . Moreover, if J is regular, than J 0 is regular and unique.
Proof. Let x ∈ J be such that xϕ ∩ J 0 contains an element, say y. Let r ∈ J.
Choose s, t ∈ S 1 with sxt = r. Let se, e
t ∈ T 0 be elements ϕ1 -relating to s
e
and t, respectively. Then r ϕ sey t ≤J y, so by minimality, seye
t ∈ J 0 proving
0
−1
that J ⊆ J ϕ .
Suppose J is regular and let e ∈ J be an idempotent. Choose z ∈ eϕ ∩ J 0 .
Then e = eω ϕ z ω ≤J z and so z ω ∈ J 0 by minimality, whence J 0 is regular.
Let J 00 be another such minimal J -class. Suppose w ∈ J 00 ϕ-relates to
e. Then e ϕ zw ≤J z, w and so, by minimality, z J zw J w, proving
J 0 = J 00 .
We remark that a similar lemma holds for L and R, but not for H
(see [16]).
3. Applications to Joins
We begin with some notions from [16]. Let K be any of Green’s relations
R, L, H or J . Recall [16] that a homomorphism ϕ : S → T of semigroups
is called an K0 -morphism if
(3.1)
s1 , s2 ∈ Reg(S), s1 ϕ = s2 ϕ =⇒ s1 K s2
It is called a K-morphism if (3.1) holds without assuming regularity [16].
Let N be the pseudovariety of nilpotent semigroups; LS be the pseudovariety of left simple semigroups; RS be the pseudovariety of right simple
semigroups; and G be the pseudovariety of groups. Let LG be the pseudovariety of local groups (semigroups S such that eSe ∈ G for all idempotents
m W be the Mal’cev product of V and W; that is, V mW
e ∈ S). Let V consists of all semigroups S with a relational morphism ϕ : S −→
◦ T ∈W
such that eϕ−1 ∈ V for all idempotents e ∈ T .
One can verify that, for a homomorphism ϕ : S → T , the following holds:
m N-morphism;
(1) ϕ is an L0 -morphism if and only if ϕ is an LS m N-morphism;
(2) ϕ is an R0 -morphism if and only if ϕ is an RS m N-morphism;
(3) ϕ is a H0 -morphism if and only if ϕ is a G (4) ϕ is a J 0 -morphism if and only if ϕ is an LG-morphism.
6
BENJAMIN STEINBERG
m N consists precisely of those semigroups with a unique idemSince G potent, it follows easily from (4) that an H0 -morphism is the same thing as
an idempotent-separating homomorphism.
It is shown in [16, 26, 43] that each semigroup S has a maximal congru0
ence ≡K0 such that ϕK0 : S S K = S/≡K0 is a K0 -morphism; one says that
0
0
S K is the minimal K0 -image of S. Moreover, the assignment S 7→ S K is a
(functorial) endomorphism of the category of finite semigroups with onto
homomorphisms that preserves the division relation [16, 26, 43]. In particular, every semigroup S has a functorially minimal idempotent-separating
0
image S H .
One can verify easily that ≡H0 = (≡R0 ∩ ≡L0 ). Since the cases of R0
and L0 are dual, we just recall the construction of ≡L0 for the reader. We
shall not be too interested in J 0 -morphisms in this paper. Define ≡ on S
by s1 ≡ s2 if and only if for all s ∈ Reg(S)
(3.2)
ss1 J s ⇐⇒ ss2 J s and in this case ss1 L ss2
(Of course, in this case ss1 H ss2 by stability of finite semigroups). One can
0
show [16] that S/≡ = S L .
Notice that if S is an inverse semigroup (i.e. is regular with commuting
idempotents), then ≡L0 (=≡H0 ) is the largest idempotent separating congruence and identifying L-classes with idempotents turns (3.2) into the Munn
representation [22].
Here is our first application of the Aperiodicity and Division Lemmas.
Recall that a division ψ : S −→
◦ T of regular semigroups is called regular if
its graph #ψ is a regular subsemigroup of S × T .
Corollary 3.1. Let ϕ : S −→
◦ T be an aperiodic relational morphism. Then
0
H
0
ϕH ⊕ ϕ : S → S × T is injective on Reg(S). In particular, if S is regular,
ϕH0 ⊕ ϕ is a division. If S is regular and T is a group, then ϕ H0 ⊕ ϕ is a
regular division.
Proof. By the Aperiodicity Lemma, ϕ is injective on regular H-classes and
hence on ϕH0 -classes of Reg(S). Everything except the last sentence then
follows from the Division Lemma.
Let ψ = ϕH0 ⊕ ϕ. For the last statement, we show that #ψ is a regular
semigroup. Indeed, suppose ([s], t) ∈ #ψ. Let s0 be an inverse of s. It is
well known [11] that ([s0 ], t−1 ) ∈ #ψ. Clearly ([s], t), ([s0 ], t−1 ) are inverses
in #ψ. So, indeed, #ψ is regular, as required.
We now wish to generalize some results of McAlister [20, 21]. Let H be
m H = A ∗ H (where the latter is the
a pseudovariety of groups. Recall A semidirect product of pseudovarieties) [11].
Recall that a regular semigroup is called fundamental if its maximal
idempotent-separating congruence is trivial. A regular semigroup S is called
orthodox if E(S) is a semigroup, where E(S) is the set of idempotents of S;
if E(S) is a semilattice, S is called inverse.
ON APERIODIC RELATIONAL MORPHISMS
7
Theorem 3.2. Let S be a regular semigroup and H be a pseudovariety of
groups. If S ∈ A ∗ H, then S has a regular division to a direct product of
a member of H and a fundamental regular semigroup. In particular, every
orthodox (inverse) semigroup has a regular division to a direct product of a
group and a fundamental orthodox (inverse) semigroup.
Proof. The first part is immediate from Corollary 3.1. The last part follows
from the results of Rhodes and Tilson characterizing regular Type II elements [30] or from independent work of McAlister [21] showing that every
orthodox semigroup has an E-unitary cover and hence belongs to A∗G. The last statement was first proved by McAlister [20]. It could also be
deduced from the results of [30], where Rhodes and Tilson show that a
group mapping semigroup in A ∗ G divides a direct product of a group
with its minimal L0 = H0 -image and the fact that a regular semigroup is a
direct sum of its minimal H0 -image and its group mapping images. A nontrivial semigroup is called group mapping if contains a (necessarily unique)
0-simple, non-aperiodic 0-minimal ideal I on which it acts faithfully on both
the left and right.
We now generalize and refine McAlister’s theorem describing regular semigroups in A ∨ G (the pseudovariety join of A and G) [20, 21]. As in the
above case, one could also deduce McAlister’s results from [30]. Recall that
V ∗r W denotes the reverse semidirect product of V and W (here reverse
semidirect products project to the left hand factor). Recall that the selfconjugate core (i.e. Type II subsemigroup [30, 11]) of a semigroup S is the
smallest subsemigroup containing E(S) and closed under weak conjugation
(x is a weak conjugate of y if there exist a, b ∈ S such that aba = a and
x = ayb or x = bya). We write S ≺ T if S divides T (meaning S has a
division to T ).
Theorem 3.3. Let S be a regular semigroup and H be a pseudovariety of
groups. Then the following are equivalent:
(1) S admits a regular division to direct product A × G with A ∈ A
regular and G ∈ H;
(2) S ∈ A ∨ H;
(3) S ∈ (A ∗ H) ∩ (G ∗ A) ∩ (A ∗r G);
(4) S ∈ A ∗ H and H is a congruence.
In particular, a regular semigroup belongs to A ∨ G (as a pseudovariety or
as an e-pseudovariety) if and only if H is a congruence and its self-conjugate
core is aperiodic. In particular, an orthodox semigroup belongs to A ∨ G if
and only if H is a congruence.
Proof. Clearly (1) =⇒ (2) =⇒ (3).
For (3) =⇒ (4), we follow [13]. Suppose ϕ : S −→
◦ G o A is a division
with A ∈ A and G ∈ G. Without loss of generality, we may assume A
is a monoid. Consider the projection π : G o A → A; this is clearly an
0
0
L-morphism [16]. Hence (G o A)L = AL . Since, as mentioned earlier, the
8
BENJAMIN STEINBERG
0
0
0
assignment T 7→ T L preserves the division relation, S L ≺ AL and is hence
0
0
aperiodic [13]; dually S R is aperiodic. Since S H is a subdirect product of
0
0
0
S L and S R , it follows that S H is aperiodic. For a regular semigroup, ≡H0
0
is the largest congruence contained in H. However since S H is aperiodic and
S is regular, H-equivalent elements of S are identified by ≡H0 ; so ≡H0 = H.
We conclude that H is a congruence.
For (4) =⇒ (1), first observe that if H is a congruence on a regular
0
semigroup S, then S/H is aperiodic and equals S H . An application of
Corollary 3.1 completes the equivalence.
The remaining statements follow since it is shown in [30] that the selfconjugate core of S is aperiodic if and only if S ∈ A ∗ G and the selfconjugate core of an orthodox semigroup is always aperiodic – in fact it is
E(S) (c.f. [11]).
We mention that in [13] the following was established:
0
S ∈ G ∗ A ⇐⇒ S L ∈ A.
In [37, 38], it is shown that the decidability of A∗H can be reduced to the
decidability of Sl∗H for inverse. Margolis, Sapir and Weil reduced this latter
problem to computing the pro-H closure of a finitely generated subgroup of
a free group [19]. In particular, this can be done for the pseudovarieties
of p-groups [32, 19], nilpotent groups [19] and decidable pseudovarieties of
Abelian groups [7, 34]. Hence if H is any of these pseudovarieties of groups,
the membership problem for A ∨ H is decidable for regular semigroups by
Theorem 3.3.
Auinger and the author have recently constructed a decidable pseudovariety of (solvable) groups U for which membership in Sl∗U is undecidable [4].
It is also shown in this paper that A ∨ U is not decidable, but the construction uses non-regular semigroups. We ask whether A ∨ U is undecidable for
regular semigroups.
3.1. Wreath products and A ∗ H. Margolis asked (private communication) whether membership in A ∨ H for inverse semigroups reduces to H
being a congruence and subgroups belonging to H. We show that this is
not the case. In the process we answer an old question of Karnofsky and
Rhodes [13] concerning the existence of a semigroup in A ∗ G with Abelian
subgroups that does not belong to A ∗ Ab.
In what follows, we make use of the H-kernel of a semigroup. If S is a
semigroup, then
\
KH (S) =
1ϕ−1 .
ϕ:S −→
◦ G∈H
For instance, KG (S) is the self-conjugate core by Ash’s theorem [3]. It is
well known [30, 47, 11, 38] that
(3.3)
Reg(KH (S)) = Reg(S) ∩ KH (S).
ON APERIODIC RELATIONAL MORPHISMS
9
m H if and only if KH (S) ∈ V; see [11, 38]. It is also known
Also S ∈ V that KH (S) is an endofunctor of the category of semigroups that preserves
m H.
surjective homomorphisms [30, 38]. Recall that Sl ∗ H = Sl In what follows, we shall use the wreath product of partial transformation
semigroups. For the sake of readability, we provide the definition. Our
definition is equivalent to that in [8], but we state things in what we hope
is a more transparent way.
Let (Y, T ), (X, S) be partial transformation semigroups. Their wreath
product (Y, T ) o (X, S) = (Y × X, W ) consists of all partial transformations
g of Y × X that can be represented by a pair (f, s) with s ∈ S, f ∈ T X such
that
(
(y(xf ), xs) if y(xf ) and xs are defined,
(y, x)g =
∅
else.
In what follows, we shall need the following straightforward proposition,
whose proof we leave as an exercise. Recall that H denotes the pseudovariety
of all semigroups whose subgroups belong to H.
Proposition 3.4. Suppose (X, M ) is a partial transformation monoid and
(Y, G) is a transformation group. Then the wreath product projection
π : (Y × X, W ) = (Y, G) o (X, M ) → (X, M )
induces a surjective L-morphism ψ : W M . Moreover, if M is aperiodic
and G ∈ H, a pseudovariety of groups, then W ∈ H.
In the sequel we shall use the fact [6, 16] that if S ≤ T and a, b ∈ S are
regular then a K b in S if and only if a K b in T where K is any of R,
L or H. If X is a set, I(X) will denote the symmetric inverse monoid on
X [6, 16, 18].
Proposition 3.5. Let H be a non-trivial pseudovariety of groups and 1 6=
G ∈ H. Suppose S is an inverse monoid not belonging to Sl ∗ H; assume S
acts faithfully by partial permutations on a set X. Then
(G × X, W ) = (G, G) o (X, S) ∈
/ A∗H
and is an inverse monoid of partial permutations. If, in addition, S is
aperiodic, then H is a congruence on W and W ∈ H.
Proof. It is well known that (G × X, W ) is a transformation inverse monoid
of partial permutations. By Proposition 3.4, the projection ψ : W S given
by (f, s) 7→ s is a well-defined L-morphism and hence is an H-morphism,
since W and S are inverse semigroups. If S is aperiodic, then W ∈ H and,
since ψ is an H-morphism, we must have that the congruence associated to
ψ is H – that is, H is a congruence.
We now aim to show that W ∈
/ A ∗ H. To do this, we exhibit distinct Hequivalent elements in the inverse subsemigroup (c.f. (3.3)) KH (W ); hence
KH (W ) ∈
/ A.
10
BENJAMIN STEINBERG
Recall that our hypotheses imply that KH (S) contains an element s that
is not idempotent. Since the functor KH preserves surjective morphisms,
there exists an element w ∈ KH (W ) with wψ = s; let (h, s) ∈ GX × S
represent w. Choose x ∈ X such that xs is defined and xs 6= x. Consider
v ∈ W represented by (f, 1) where
(
g 6= 1 y = x,
yf =
1
otherwise.
Clearly v belongs to the group of units of W and has inverse represented by
(f −1 , 1). A standard argument [30, 11, 38] then shows that u = vwv −1 ∈
KH (W ). Also uψ = s = wψ. Hence, since ψ is an H-morphism, u and w
are H-equivalent in W and hence in then inverse subsemigroup KH (W ). It
remains to show that u 6= w. Observe
(1, x)w = (xh, xs).
On the other hand
(1, x)u = (1, x)vwv −1 = (g, x)wv −1 = (gxh, xs)v −1 = (gxh, xs)
since xsf −1 = 1 (using xs 6= x). But gxh 6= xh, so u 6= w, as desired.
Corollary 3.6. Let H be a non-trivial pseudovariety of groups such that
there is an aperiodic inverse monoid S ∈
/ Sl ∗ H. Then there is an inverse
monoid W ∈ H for which H is a congruence, but that does not belong to
A ∗ H and hence does not belong to A ∨ H.
For the above corollary to be of use, we need examples of pseudovarieties
H for which the hypotheses apply. Suppose H satisfies a non-trivial group
identity of the form w = 1. Let M (w) be the Munn tree of w [23, 18]. Then
the transition inverse monoid S of M (w) is aperiodic. However, since M (w)
cannot be embedded in a permutation automaton with transition group in
H, we see that S ∈
/ Sl ∗ H. Hence the above corollary applies to show there
are inverse monoids in (A ∨ G) ∩ H that do not belong to A ∗ H. D. Cowan
constructed for Gnil , the pseudovariety of nilpotent groups, an aperiodic
inverse monoid not in Sl ∗ Gnil . His example is the following1: take
1 2 3
1 2 3
a=
and b =
;
2
3
3 2
then one can show that S = ha, b, a−1 , b−1 i fails to satisfy w = w 2 for each
word w = [x1 , x2 , ..., xn ] and so S ∈
/ Sl∗Gnil (alternatively, the results of [19]
apply to easily give this); clearly S ∈ A. The question as to whether every
aperiodic inverse monoid belongs to Sl∗Gsol is open for the pseudovariety of
solvable groups. The author conjectures that if H is a proper pseudovariety
of groups, then there is an aperiodic inverse monoid that does not belong to
Sl ∗ H.
1We thank D. Cowan for allowing us to include his example.
ON APERIODIC RELATIONAL MORPHISMS
11
Notice, since the pseudovariety of Abelian groups is equational, that we
have established the existence of an inverse monoid with Abelian subgroups
that does not belong to A∗Ab. Since all inverse monoids belong to Sl∗G ⊆
A ∗ G, we have settled an old question of Karnofsky and Rhodes [13] about
the existence of such semigroups. In fact, using a result of Higgins and
Margolis [12] we can prove the following stronger result.
Theorem 3.7. Let 1 ( H ( G. Then
A ∗ H 6= (A ∗ G) ∩ H.
More specifically there is a monoid (Q, W ) of partial permutations such
that W ∈
/ A ∗ H, W ∈ H, and W has an idempotent separating homomorphism to an aperiodic monoid of partial permutations.
Proof. In [12], an aperiodic transformation monoid (X, S) of partial permutations is constructed such that:
(1) The inverse hull [6] of (X, S) does not belong to Sl ∗ H;
(2) S contains all the rank 1 partial permutations of X;
(3) S ∈
/ Sl ∗ H.
Note that (3) follows from (1) and (2) and the results of [38], which show
that a transitive monoid of partial permutations belongs to Sl ∗ H if and
only if its inverse hull does. (A different argument for (3) is given in [12]).
Consider again the wreath product (G × X, W ) = (G, G) o (X, S) where
1 6= G ∈ H. As in the proof of Proposition 3.5: (G × X, W ) is a monoid of
partial permutations with subgroups in H; the projection to (X, S) induces
an L-morphism ψ : W S, in fact, the projection is idempotent separating
since if e ∈ E(S), the unique idempotent mapping to e is represented by
(1, e); there are distinct elements u, w of W with u, w ∈ KH (W ) such that
0 6= uψ = wψ. However, in the current context u, w need not be regular and
we do not know whether u H w holds in either W or KH (W ). So we need
to do some more work to show that W ∈
/ A ∗ H.
For each x ∈ X, let 1x denote the partial identity on {x}. By (2), 1x ∈ S
for all x ∈ X. Since uψ = wψ, u can be represented by (h, s) and w by (f, s)
such that, for some x ∈ X with xs defined, xh 6= xf . Let e be represented
by (1, 1x ) and set r = eu and z = ew. Then
(1, x)r = (xh, xs) 6= (xf, xs) = (1, x)z
so r 6= z. Since e is idempotent and u, w ∈ KH (W ), we obtain r, z ∈ KH (W ).
We now aim to prove r, z are H-equivalent regular elements of KH (W ),
thus showing that KH (W ) ∈
/ A, as desired. This will be done in two steps.
Our first step is to show that r, z are regular elements of W ; it will then
follow, using (3.3), that r, z are regular in KH (W ).
Our second goal is to show that r H z in I(G × X). Since we will have
shown that r and z are regular in KH (W ) ≤ I(G × X), it will follow that
r H z in KH (W ).
12
BENJAMIN STEINBERG
We first show r R e R z in W . We may then conclude that r and z are
regular in W . Clearly r ≤R e. For the converse, let k ∈ S be the rank one
map taking xs to x and h0 : X → G be such that xsh0 = (xh)−1 . Then
(
(gxh, xs)(h0 , k) = (g, x) y = x,
(g, y)r(h0 , k) =
∅
otherwise
(g, y)e =
(
(g, x) y = x,
∅
otherwise
and so r R e in W . A similar argument, with f in the role of h, shows that
z R e in W .
Since r, z both have domain G × {x} and range G × {xs}, they are Hequivalent in I(G × X) [6]. We conclude r H z in KH (W ), completing the
proof that KH (W ) ∈
/ A.
One last application of this construction is the following result. Recall
that Gp denotes the pseudovariety of p-groups.
Theorem 3.8. Let p be a prime and H a pseudovariety of groups such that
Gp ∗ H = H. Then the following are equivalent:
(1) The decidability of membership in Sl ∗ H for inverse monoids;
(2) The decidability of membership in A ∗ H.
Proof. In [37, 38], it is shown that (1) implies (2). For the converse, suppose
M is an inverse monoid; we may assume that M acts faithfully on a set X
by partial permutations. Consider the wreath product
(Zp , Zp ) o (X, M ) = (U, W );
we saw earlier that W is an inverse monoid. The claim is that W ∈ A ∗ H
if and only if M ∈ Sl ∗ H. Proposition 3.5 shows that if M ∈
/ Sl ∗ H, then
W ∈
/ A ∗ H. Conversely, if M ∈ Sl ∗ H, then there is a set Y containing
X and permutations N extending those of M , such that G = hN i ∈ H
(c.f. [19, 38]). Consider
(Zp , Zp ) o (Y, G) = (Z, W 0 ).
Clearly U ⊆ Z; we show that each element of W can be extended to an
element of W 0 . Let (f, m) represent an element of W and n ∈ N extend m.
Choose g : Y → Zp such that xg = xf for all x ∈ X. Clearly (g, n) ∈ W 0
extends (f, m), so W ∈ Sl ∗ Gp ∗ H ⊆ A ∗ H, as required.
4. The Presentation Lemma
We begin this section with some historical remarks and discussion.
ON APERIODIC RELATIONAL MORPHISMS
13
4.1. Brief history and discussion. There are several versions of the Presentation Lemma in existence. There is the original, due to Rhodes [5] and
a more general, coordinate-free version due to Tilson [50, 49]. The version given here is even more general. All “Presentation Lemmas” have the
common feature of creating, seemingly out of nowhere, a join decomposition. Our version, like the results of the previous section, uses the Division
Lemma (i.e. the Slice Theorem) and the Aperiodicity Lemma to obtain the
result. Rhodes told the author back in 1997 that the Slice Theorem should
imply some sort of generalization of the Presentation Lemma; sadly, it took
the author nearly 5 years to discover it. Tilson [49] uses a transformation
semigroup version of the Division Lemma
The Presentation Lemma arose out of the work in [30], where the special case of presentations over the trivial pseudovariety is dealt with (in a
different language). In particular, the result that if a group mapping semigroup is in A ∗ G, then it divides a direct product of a group with its right
letter mapping image is the immediate precursor of both the Presentation
Lemma and the results of the previous section. Further predecessors of the
Presentation Lemma can be found in [44] and [27].
Common to all versions of the Presentation Lemma is the notion of a
cross-section. Rhodes cross-sections [5], which are in some sense a coordinate notion, are the least general, but the most useful for computations. In
some sense his Presentation Lemma has a stronger statement than the other
versions because he guarantees this more restrictive notion of a cross-section.
Rhodes cross-sections are officially defined for group mapping semigroups,
but make sense for any right mapping semigroup. Recall [16] that a semigroup S is right (left) mapping if it has a distinguished (necessarily unique)
regular 0-minimal I such that S acts faithfully on the right (left) of I and
that S is group mapping if it is right and left mapping and I contains a nontrivial group (or S is trivial). Rhodes’s definition has built into it the notion
of sets and partitions and the Rhodes approach is aimed at pseudovarieties
of the form A ∗ G ∗ V (usually he assumes V is a complexity pseudovariety).
A presentation in the Rhodes sense is then a mixture of sets, partitions and
cross-sections. The paper [5] makes heavy use of coordinates, rendering it
difficult conceptually.
Tilson cross-sections are also defined for group mapping semigroups (again
his definition can be extended to right mapping semigroups). His notion is
coordinate-free, as are his proofs. Let R be a non-zero R-class of the distinguished ideal of group mapping semigroup S. Roughly speaking, Tilson
shows that if (R, S) divides a wreath product (Q, A)o(P, T ) with A aperiodic,
then (R, S) has a Tilson cross-section over a subtransformation semigroup
of (P, T ).
We are now ready to introduce our notion of a cross-section.
4.2. Cross-sections. Let S be a semigroup and let X ⊆ S. Then a crosssection for X is a relational morphism ϕ : S −→
◦ T that is injective on
14
BENJAMIN STEINBERG
H-classes of X. By injective on H-classes of X, we mean
(4.1)
s, s0 ∈ X, s H s, sϕ ∩ s0 ϕ 6= ∅ =⇒ s = s0
If T ∈ V, then we say S has a cross-section for X over V.
The following elementary proposition will be of use to us.
Proposition 4.1. Suppose that X is an R-class (dually L-class) of a semigroup S contained in a J -class J. Then ϕ : S −→
◦ T is a cross-section for
X if and only if it is a cross-section for J.
Proof. We just handle the case where X is an R-class. Clearly ϕ is a crosssection for J if and only if it is a cross-section for every R-class of J. Hence
if ϕ is a cross-section for J, it is a cross-section for X.
Suppose ϕ is a cross-section for X and let X 0 be an R-class of J. Suppose
s1 H s2 are elements of X 0 with t ∈ s1 ϕ ∩ s2 ϕ. Let y ∈ Ls1 ∩ X. Then
there exist s, s0 ∈ S 1 with sy = s1 and s0 s1 = y. Green’s lemma [6] then
says that s· : X → X 0 and s0 · : X 0 → X are inverse bijections preserving H.
Hence s0 s1 H s0 s2 and s0 s1 , s0 s2 ∈ X. Moreover, if t0 ∈ s0 ϕ1 , then s0 si ϕ t0 t,
i = 1, 2. We conclude that s0 s1 = s0 s2 , whence s1 = s2 since s0 · : X 0 → X is
a bijection.
The Aperiodicity Lemma has the following restatement.
Proposition 4.2 (Aperiodicity Lemma – Cross-section version). A relational morphism ϕ : S −→
◦ T is a cross-section for Reg(S) if and only if it
is aperiodic.
Theorem 4.3 (Cross-section Theorem). Let V be a pseudovariety. Then
m V if and only if every regular R-class has a cross-section over V.
S ∈ A
Proof. If ϕ : S −→
◦ T is an aperiodic relational morphism with T ∈ V, then
ϕ is a cross-section for Reg(S) and hence for every regular R-class.
Conversely, if, for each regular
L R-class R, there
Q is a cross-section ϕR :
S −→
◦ TR with TR ∈ V, then
ϕ
:
S
−→
◦
TR is a cross-section for
R R
Reg(S) and hence an aperiodic relational morphism.
By a generalized transformation semigroup (gts) (X, S) we mean a semigroup S acting by partial transformations on X (but we do not assume faithfulness). Our definition of a relational morphism follows that of [50] and is
essentially that of Eilenberg [8] with the arrows turned around. So a relational morphism ϕ : (X, S) −→
◦ (Y, T ) is a fully defined relation ϕ : X −→
◦ Y
such that for each s ∈ S, there exists ŝ ∈ T such that, for all y ∈ Y ,
yϕ−1 s ⊆ yŝϕ−1 . One says that ŝ covers s.
Let R be a regular R-class of S with associated J -class J. Then there
results a gts (R, S) via the Schützenberger representation [6, 16] given, for
s ∈ S, r ∈ R, by
(
rs
rs ∈ R,
(4.2)
rs =
undefined otherwise.
ON APERIODIC RELATIONAL MORPHISMS
15
where on the left hand side of (4.2) rs is interpreted in (R, S). The associated faithful partial transformation semigroup is denoted (R, RMJ (S)). We
remark that RMJ (S) is the right mapping semigroup associated to J [16]
and only depends on J.
There is also an action of S on the right of the L-classes B of J given by
(
Lrs
Lrs ⊆ J,
(4.3)
Lr s =
undefined otherwise.
The associated faithful partial transformation semigroup (called the right
letter mapping semigroup associated to J [16]) is denoted (B, RLM J (S)).
There is a morphism (R, RMJ (S)) → (B, RLMJ (S)) given by r 7→ Lr . The
map corresponds to modding out by the action of the maximal subgroup G
of J on the left of R. Hence [16]
L0
(R, RMJ (S)) ≤ (G, G) o (B, RLMJ (S)).
Notice that S (see Section 3) is just the direct sum of the various right
letter mapping representations; see [16] and (3.2). Left mapping and left
letter mapping are defined dually.
Theorem 4.4 (Cross-section Lemma – Semigroup Version). Let S be a
right mapping semigroup with distinguished J -class J. Let ϕ : S −→
◦ T be
a cross-section for J. Then
S ≺ T × RLMJ (S).
m V, then S ∈ W
Moreover, if V, W are pseudovarieties with V ⊆ W ⊆ A if and only if S has a cross-section for J over V and RLM J (S) ∈ W.
Proof. By the Division Lemma, it suffices to show that if s1 , s2 have the
same image in RLMJ (S) and s1 ϕ ∩ s2 ϕ 6= ∅, then s1 = s2 . First observe
that, for r ∈ R, rsi is defined if and only if Lr si is defined in RLMJ (S),
i = 1, 2. Thus s1 and s2 are defined at the same elements. Suppose rsi is
defined, i = 1, 2. Then Lrs1 = Lrs2 by our assumption. Since rs1 , rs2 ∈ R,
we see that rs1 H rs2 . Let t ∈ s1 ϕ ∩ s2 ϕ and let t0 ∈ rϕ. Then rsi ϕ t0 t,
i = 1, 2. Hence, since ϕ is a cross-section for J, rs1 = rs2 . We conclude
that s1 = s2 since S is right mapping.
m V, then S has a cross-section for
For the final statement, if S ∈ W ⊆ A J over V by Theorem 4.3 and trivially RLMJ (S) ∈ W. For the converse, we
have by the first part of the theorem that if ϕ : S −→
◦ T is a cross-section
for J, then S ≺ T × RLMJ (S). So if T ∈ V and RLMJ (S) ∈ W, then
S ∈ W.
The main reason the above result is of use is that if S is a non-trivial group
mapping semigroup with distinguished J -class J, then |RLMJ (S)| < |S|.
Also the direct sum of the projections of an arbitrary semigroup S to its varmV
ious group mapping images is aperiodic [16]. Hence membership in A reduces to the group mapping case. An induction argument and the above
16
BENJAMIN STEINBERG
m V reduces to detertheorem then show that deciding membership in A mining whether a non-trivial group mapping semigroup has a cross-section
over V for its distinguished J -class. It is not hard to deduce Theorem 3.3
from this result and its dual applied with V = G and W = A ∨ G.
We now wish to relate our notion of a cross-section with other notions in
the literature. Let S be a semigroup and R be a regular R-class contained
in a J -class J. We define a Tilson cross-section for R to be a relational
morphism ϕ : (R, S) −→
◦ (Q, T ) such that ϕ : R −→
◦ Q is injective on Hclasses. Actually Tilson defines this just for the case of group mapping
semigroups [49]. If G is the maximal subgroup of J and B is the set of
L-classes, then R can be (non-canonically) identified with G × B. The
condition that ϕ is injective on H-classes then says that for each q ∈ Q,
the subset qϕ−1 of G × B, when viewed as a relation B −→
◦ G, is in fact a
partial function. In [49] the definition of a cross-section is formulated in this
manner, although Tilson tells me that more recent versions use the former
formulation. If there is a Tilson cross-section ϕ : (R, S) −→
◦ (Q, T ) with
T ∈ V, then we say that S has a Tilson cross-section over V for R.
For those wondering what a Rhodes cross-section is, roughly speaking it
is a relational morphism ϕ : (B, RLMJ (S)) −→
◦ (Q, T ) that extends to a
relational morphism of G-transformation semigroups
(G × B, S) −→
◦ (G, G) o (Q, T )
that is a Tilson cross-section. Rhodes’s actual definition [5] presupposes
that (Q, T ) divides a wreath product of a group with a different transformation semigroup and this is where his sets and partitions come from; see
Subsection 4.3 below for more.
Rhodes cross-sections stand in the same relation to Tilson cross-sections
as the proof of the Type II theorem for regular elements in [30] relates to
that in [47]. In both cases, the latter is more elegant, but the former gives
additional information that is useful for computations.
Now we aim to show how Tilson cross-sections relate to our notion of a
cross-section. Recall that if ϕ : (R, S) −→
◦ (Q, T ) is a relational morphism,
then one has the companion relation fϕ : S −→
◦ T given by
(4.4)
sfϕ = {t ∈ T | t covers s}.
Proposition 4.5. Let R be a regular R-class of a semigroup S and let
ϕ : (R, S) −→
◦ (Q, T ) be a Tilson cross-section. Then fϕ : S −→
◦ T is a
cross-section for R.
Proof. Suppose s1 H s2 are elements of R with t ∈ s1 fϕ ∩ s2 fϕ . Let e be an
idempotent in R and let q ∈ eϕ. Then
si = esi ∈ qϕ−1 si ⊆ (qt)ϕ−1 ,
i = 1, 2. Since ϕ is a Tilson cross-section, we deduce s1 = s2 proving that
the companion relation fϕ is a cross-section.
ON APERIODIC RELATIONAL MORPHISMS
17
Suppose now that ϕ : S −→
◦ T is a relational morphism and let R be
a regular R-class of S. Let R0 be an ≤R -minimal R-class of T with R ∩
R0 ϕ−1 6= ∅. By the remark after Lemma 2.2, R0 is regular and R ⊆ R0 ϕ−1 .
Let J 0 be the J -class of R0 . We can define a relational morphism ϕR :
(R, S) −→
◦ (R0 , RMJ 0 (T )) by restricting ϕ to R. Given s ∈ S to obtain
a cover ŝ, simply take the image in RMJ 0 (T ) of any element of sϕ. The
following proposition is then immediate.
Proposition 4.6. Let ϕ : S −→
◦ T be a cross-section for R. Then
ϕR : (R, S) −→
◦ (R0 , RMJ0 (T ))
is a Tilson cross-section for R.
The previous two propositions can be summarized as follows.
Proposition 4.7. Let S be a semigroup and R a regular R-class of S.
Then S has a cross-section for R over V if and only if (R, S) has a Tilson
cross-section over V.
As a consequence we obtain a transformation semigroup version of the
presentation lemma. This is a strengthened version of Tilson’s presentation
lemma since he deals with semidirect instead of Mal’cev products (see the
historical remarks above).
Theorem 4.8 (Cross-section Lemma – Partial Transformation Semigroup
Version). Let S be a right mapping semigroup with distinguished J -class J.
Let B denote the set of L-classes of J and let R ⊆ J be an R-class. Let
ϕ : (R, S) −→
◦ (Q, T ) be a Tilson cross-section for R. Then
(R, S) ≺ (Q, T ) × (B, RLMJ (S)).
m V, then S ∈ W
Moreover, if V, W are pseudovarieties with V ⊆ W ⊆ A if and only if S has a Tilson cross-section over V for R and RLM J (S) ∈ W.
Proof. For the first statement, note that the projection
π : (R, S) (B, RLMJ (S))
identifies two elements if and only if they are L and hence H equivalent.
Since ϕ separates H-equivalent elements, the obvious transformation semigroup version of the Division Lemma shows that ϕ ⊕ π is a division.
The second statement follows from Theorem 4.4 and Proposition 4.7. m V reduces to the decidability of the
So we see that membership in A existence of Tilson cross-sections for non-trivial group mapping semigroups.
Rhodes’s Presentation Lemma [5] is about complexity. Recall from [16, 8]
that one defines inductively the pseudovariety Cn of semigroups of complexity at most n by C0 = A and Cn = A ∗ (G ∗ Cn−1 ). Since
m V,
V ⊆ A ∗ V ⊆ A
both Cross-section Lemmas given above apply to computing the complexity
of a group mapping semigroup. The Fundamental Lemma of Complexity [25]
18
BENJAMIN STEINBERG
m
states that the leftmost ∗ in the definition of Cn can be replaced with and so we can reduce to the group mapping case.
We would thus like to give a detailed study of what it means to have a
Tilson cross-section for a pseudovariety of the form G∗V. This will allow us
to formulate a Presentation Lemma that is exactly a coordinate-free version
of the Rhodes Presentation Lemma as formulated in [5]. This leads us to
presentations.
4.3. Presentations. In this section, we recall the definition of the derived
gts of a relational morphism of gts [8]. First we need the notion of a parameterized relational morphism of gts: A parameterized relational morphism
of transformation semigroups Φ : (Q, S) −→
◦ (P, T ) consists of a relational
morphism ϕ1 : (Q, S) −→
◦ (P, T ) and a relational morphism ϕ2 : S −→
◦ T
such that ϕ2 ⊆ fϕ1 where fϕ1 is the companion relation (4.4).
The derived gts of Φ is DΦ = (#ϕ1 , DΦ ) where
(4.5)
DΦ = {(p, (s, t), p0 ) ∈ P × #ϕ2 × P | pt ⊆ p0 }
The action is given by
(4.6)
(q, p)(p0 , (s, t), p1 ) =
(
(qs, p1 )
p0 = p and qs defined,
undefined otherwise.
In this paper, an automaton is a pair (Q, X) with Q a set and X a set
acting on Q via a transition function δ : Q × X → Q. That is, (Q, X) is an
automaton in the usual sense without initial or terminal states; these are
called modules in [50]. Any gts is an automaton in an obvious way. An
automaton is called injective if X acts via partial permutations.
An automaton congruence on (Q, X) is an equivalence relation ≡ on Q
such that q ≡ q 0 and qx, q 0 x defined imply qx ≡ q 0 x. That is, ≡ is an
equivalence relation such that the quotient automaton
(Q, X)/≡ = (Q/≡, X)
is well defined. An automaton congruence ≡ is called injective if the quotient
automaton is injective.
In what follows, we do not distinguish between a partition and its associated equivalence relation.
Definition 4.9 (Presentation). Let S be a semigroup and R a regular Rclass. Then a presentation for R is a pair (Φ, P) where:
(1) Φ : (R, S) −→
◦ (Q, T ) is a parameterized relational morphism;
(2) P is an injective automaton congruence on DΦ such that
(x, q) P (x0 , q 0 ) =⇒ q = q 0 ;
(3) (x, q) P (x0 , q 0 ) and x H x0 implies (x, q) = (x0 , q 0 ).
The qϕ−1
2 , q ∈ Q, are called the sets of the presentation; P is called the
partition of the presentation. These notions are coordinate-free versions of
those considered in [5].
ON APERIODIC RELATIONAL MORPHISMS
19
In general, we call a partition P on DΦ admissible if it satisfies condition
(2) of Definition 4.9. It is easy to verify that any injective automaton congruence which refines an admissible partition is again admissible. Since it is well
known that every automaton has a (polynomial time constructible) minimal
injective congruence [41, 47, 38, 50], it follows that Φ : (R, S) −→
◦ (Q, T ) is
part of a presentation if and only if the minimal injective congruence on DΦ
satisfies (2) and (3) of Definition 4.9 and that this is decidable.
If T from (1) of Definition 4.9 above belongs to the pseudovariety V, then
we say S has a presentation for R over V.
We remark that one can always take (Q, T ) to be faithful by composing
Φ with the natural map from (Q, T ) to the associated faithful partial transformation semigroup. One can verify that conditions (2) and (3) are still
satisfied.
Our next goal is to show that S has a presentation over V if and only if
it has a cross-section over G ∗ V. We actually prove a more general result.
The proof mixes ideas from [47, 5, 49] with ideas from [36, 33]. To do
this, we shall need to make use of the semigroupoid De r(ψ) associated to a
relational morphism ψ : S −→
◦ T [40, 28] of semigroups. If T is a semigroup,
we remind the reader that T 1 means T with a new identity 1. Then
Obj(De r(ψ)) = T 1
Arr(De r(ψ)) = T 1 × #ψ
(tL , (s, t)) : tL → tL t
(tL , (s, t))(tL t, (s0 , t0 )) = (tL , (ss0 , tt0 ))
There is also a faithful functor De r(ψ) → #ψ given by
(4.7)
(tL , (s, t)) 7−→ (s, t).
Theorem 4.10 (Pointlikes via Presentations). Let R be a regular R-class
of a semigroup S and let V be a pseudovariety. Then X ⊆ R is G ∗
V-pointlike if and only if, for all parameterized relational morphisms Φ :
(R, S) −→
◦ (Q, T ) with T ∈ V and for all admissible partitions P on D Φ ,
there exists q ∈ Q such that X ⊆ qϕ−1
1 and X × {q} is contained in a single
block of P.
Proof. Suppose X is G ∗ V-pointlike, Φ : (R, S) −→
◦ (Q, T ) is a parameterized relational morphism, and P is an admissible partition on DΦ . We write
[(r, q)] for the partition block of (r, q) in P.
Since any partial permutation of the finite set P = #ϕ1 /P can be extended to a permutation of P , for each (p, (s, t), p0 ) ∈ DΦ , choose a permutation f(p,(s,t),p0 ) that extends the action of (p, (s, t), p0 ) on DΦ /P. Let (P, G)
be the permutation group obtained by setting
G = hf(p,(s,t),p0 ) | (p, (s, t), p0 ) ∈ DΦ i.
20
BENJAMIN STEINBERG
Define a relational morphism of gts
α : (R, S) −→
◦ (P, G) o (Q, T ) = (P × Q, W )
rα = {(p, q) | q ∈ rϕ1 , p = [(r, q)]}.
To see that α is indeed a relational morphism, first we observe that if
r ∈ R and q ∈ rϕ1 , then ([(r, q)], q) ∈ rα, and so rα 6= ∅. Suppose s ∈ S.
Choose t ∈ sϕ2 . Define f ∈ GQ as follows:
(
f(q,(s,t),qt) if qt is defined,
(4.8)
qf =
arbitrary otherwise.
and let ŝ be represented by (f, t). Let (p, q) ∈ P × Q. We must show
(p, q)α−1 s ⊆ (p, q)ŝα−1 . If (p, q)α−1 = ∅, we are done. Else, let r ∈ (p, q)α−1
be arbitrary. Then q ∈ rϕ1 and p = [(r, q)]. If rs is undefined, we are done.
So assume rs is defined. Then rs ⊆ qtϕ−1
1 , from which we may deduce that
qt is defined. Now
(r, q)(q, (s, t), qt) = (rs, qt),
whence
[(r, q)]f(q,(s,t),qt) = [(rs, qt)].
Thus (4.8) gives
(p, q)(f, t) = (p(qf ), qt) = ([(r, q)]f(q,(s,t),qt) , qt) = ([(rs, qt)], qt)
with qt ∈ rsϕ1 and so (p, q)(f, t) ∈ rsα. We conclude
(p, q)α−1 s ⊆ (p, q)ŝα−1 ,
establishing that α is a relational morphism.
Let fα : S −→
◦ W be the companion relation (4.4). Since (P, G) is comc
plete, W divides GQ oT (where Qc is Q with an adjoined sink state; see [8])
and so belongs to G ∗ V. Since X is G ∗ V-pointlike, X ⊆ wfα−1 for some
w ∈ W . Let e ∈ R be an idempotent and q ∈ eϕ1 . Set p = [(e, q)]. Then
(p, q) ∈ eα. Let s ∈ X. Then es = s. Since (p, q)α−1 s ⊆ (p, q)wα−1 , we
see that s ∈ (p, q)wα−1 . We may conclude that (p, q)w is defined – say
it is equal to (p0 , q 0 ); so s ∈ (p0 , q 0 )α−1 . By definition of α, we conclude
p0 = [(s, q 0 )]. Since p0 and q 0 depend only on (p, q) and w, and not on s, we
may conclude X × {q 0 } is contained in a single block of P, as desired.
For the converse, suppose ρ : S −→
◦ G o T is a relational morphism with
T ∈ V. Let π : G o T T denote the projection and let ψ = ρπ.
We use α : #ψ S and β : #ψ → T for the projections. Let R be an
≤R -minimal R-class of #ψ mapping under α into R. Then, by the R-class
version of Lemma 2.2 or [16], Rα = R and R is regular. Let R0 be the
R-class of T defined by Rβπ ⊆ R0 .
We then have a parameterized relational morphism Φ : (R, S) −→
◦ (R 0 , T )
given by
β = {r 0 | (r, r0 ) ∈ R}
rϕ1 = α|−1
R
ON APERIODIC RELATIONAL MORPHISMS
21
and ϕ2 = ψ. To see that this is a parameterized relational morphism,
0
consider r 0 ∈ R0 and let r ∈ r 0 ϕ−1
1 . Note that (r, r ) ∈ R. Suppose rs ∈ R
and t ∈ sψ. Then
(r, r0 )(s, t) = (rs, r 0 t) ≤R (r, r0 )
so by minimality of R, we see (rs, r 0 t) ∈ R and so rs ∈ (r 0 t)ϕ−1
1 . Therefore,
t covers s.
Let e be an idempotent of R. Then (e, t) ∈ R for some t ∈ T . Hence
(e, f ) = (e, t)ω ≤R (e, t) and so by minimality of R, (e, f ) ∈ R ⊆ #ψ. Thus
e of (f, (e, f )) in De r(ψ). Notice
f ∈ eϕ1 ; also f = f 2 . Consider the R-class R
e and so R
e is a regular R-class.
that (f, (e, f )) : f → f is an idempotent of R
We claim
e = {(f, (s, t)) | t ∈ sϕ1 } = {(f, (s, t)) | (s, t) ∈ R}.
(4.9)
R
To see this, suppose first that (f, (e, f )) R (f, (s, t)). Then, by considering
our faithful functor (4.7), we obtain (e, f ) R (s, t) in #ψ. We may deduce
that (s, t) ∈ R, i.e. t ∈ sϕ1 .
Conversely, if t ∈ sϕ1 , then (s, t) ∈ R and so the arrow (f, (s, t)) is well
defined. Since (e, f ) is an idempotent, (e, f )(s, t) = (s, t) and so
Suppose
and
(f, (e, f ))(f, (s, t)) = (f, (s, t)) : f → t.
(s, t)(s0 , t0 )
= (e, f ) with (s0 , t0 ) ∈ #ψ. Then (t, (s0 , t0 )) ∈ De r(ψ)
(f, (s, t))(t, (s0 , t0 )) = (f, (ss0 , tt0 )) = (f, (e, f ))
e and establishing (4.9).
proving (f, (s, t)) ∈ R
Consider the relational morphism ζ : De r(ψ) −→
◦ G defined by
(4.10)
(tL , (s, t))ζ = {tLg | (g, t) ∈ sρ}.
It is well known [40, 48] that ζ is a relational morphism.
e by
The idea in what follows is that #ϕ1 = R can be identified with R
(4.9). Moreover, each element (tL , (s, t), tL t) ∈ DΦ can be identified with
e in
the element (tL , (s, t)) ∈ De r(ψ). If we let De r(ψ) act on the right of R
the natural way, then this identification preserves actions.
Define a relation P on R = #ϕ1 by
(4.11)
(s, t) P (s0 , t0 ) ⇐⇒ t = t0 and (f, (s, t))ζ ∩ (f, (s0 , t0 ))ζ 6= ∅.
It is clear that P is symmetric and reflexive. Transitivity is part of
Lemma 4.11 below, whose proof will be deferred until after the proof of
Theorem 4.10.
To see that P is an automaton congruence, let
(s, t) P (s0 , t) and (tL , (s1 , t1 ), tR ) ∈ DΦ .
Then (s, t)(tL , (s1 , t1 ), tR ) and (s0 , t0 )(tL , (s1 , t1 ), tR ) defined implies t = tL =
t0 , tR = tL t1 , ss1 , s0 s1 ∈ R and
(4.12)
(4.13)
(s, t)(tL , (s1 , t1 ), tL t1 ) = (ss1 , tt1 )
(s0 , t0 )(tL , (s1 , t1 ), tL t1 ) = (s0 s1 , tt1 )
22
BENJAMIN STEINBERG
Suppose g ∈ (f, (s, t))ζ ∩ (f, (s0 , t))ζ. Choose g 0 ∈ (t, (s1 , t1 ))ζ. Then
clearly
gg 0 ∈ (f, (ss1 , tt1 ))ζ ∩ (f, (s0 s1 , tt1 ))ζ
and so (ss1 , tt1 ) P (s0 s1 , tt1 ), as desired. We conclude P is an automaton
congruence.
To see that P is injective, suppose that
(s, t)(tL , (s1 , t1 ), tR ) and (s0 , t0 )(tL , (s1 , t1 ), tR )
are defined and in the same P-block. Then t = tL = t0 , tR = tL t1 and (4.12)
and (4.13) hold. Since (ss1 , tt1 ) R (s, t) in #ψ, there exists (u, v) ∈ #ψ
with (ss1 , tt1 )(u, v) = (s, t). Let
(e0 , f0 ) = (s1 u, t1 v)ω ∈ #ψ, (w, z) = (u, v)(s1 u, t1 v)ω−1 ∈ #ψ.
Then straightforward calculations show
(4.14)
(s, t)(e0 , f0 ) = (s, t)
(4.15)
(s1 , t1 )(w, z) = (e0 , f0 )
(4.16)
(4.17)
(ss1 , tt1 )(w, z) = (s, t)
0
(s s1 , tt1 )(w, z) = (s0 e0 , tf0 ) = (s0 e0 , t).
By (4.11), there exists g ∈ (f, (ss1 , tt1 ))ζ ∩ (f, (s0 s1 , tt1 ))ζ. Choose g 0 ∈
(tt1 , (w, z))ζ. Multiplying (f, (ss1 , tt1 )) and (f, (s0 s1 , tt1 )) by (tt1 , (w, z))
and taking into account (4.16), (4.17), we see that
gg 0 ∈ (f, (s, t))ζ ∩ (f, (s0 e0 , t))ζ.
Thus (s, t) P (s0 e0 , t) by (4.11).
Observing that tf0 = t, say by (4.15), we have that (t, (e0 , f0 )) : t → t is an
idempotent of De r(ψ). Hence 1 ∈ (t, (e0 , f0 ))ζ. Since (f, (s0 , t))(t, (e0 , f0 )) =
(f, (s0 e0 , t)), if h ∈ (f, (s0 , t))ζ, then h ∈ (f, (s0 e0 , t))ζ. So (s0 , t) P (s0 e0 , t)
by (4.11). By transitivity of P, we obtain (s, t) P (s0 , t) = (s0 , t0 ).
Since (s, t) P (s0 , t0 ) implies t = t0 by (4.11), we conclude that P is an
admissible partition.
By assumption, there exists t ∈ R0 with X ⊆ tϕ−1
1 and X × {t} contained in a single P-block. By Lemma 4.11 below, there exists g ∈ G with
(f, (x, t)) ⊆ gζ −1 for all x ∈ X. Choose g 0 ∈ (1, (e, f ))ζ. Then, for all x ∈ X
g 0 g ∈ (1, (x, t))ζ. Hence, by (4.10), (g 0 g, t) ∈ xρ for all x ∈ X. We conclude
X is G ∗ V-pointlike.
In this lemma we retain the notation of the above proof.
Lemma 4.11. Suppose g ∈ (f, (s, t))ζ and (s0 , t0 ) P (s, t). Then g ∈
(f, (s0 , t0 ))ζ. Hence P is transitive. Moreover, if Y ⊆ tϕ−1
1 , then Y × {t}
is contained in a single P-block if and only if there exists g ∈ G with
g ∈ (f, (y, t))ζ, all y ∈ Y .
ON APERIODIC RELATIONAL MORPHISMS
23
Proof. The last two statements are straightforward consequences of the first
e
statement. For the first statement, let a = (f, (s, t)), b = (f, (s0 , t0 )). Since R
0
is regular, a has an inverse a = (t, (s0 , t0 )). A standard argument [11, 3, 38]
shows that if h ∈ aζ, then h−1 ∈ a0 ζ. Suppose h ∈ aζ ∩ bζ. Then
a0 a ζ 1 ζ a0 b.
Since a R b, aa0 b = b. So
b = aa0 b ζ g1 = g,
as desired.
Corollary 4.12 (Presentation Lemma – Semigroup Version). Let R be a
regular R-class of S and V be a pseudovariety. Then S has a cross-section
for R over G ∗ V if and only if S has a presentation for R over V.
Proof. Suppose S has a presentation over V. Then Theorem 4.10 and Definition 4.9 show that no H-equivalent elements of R are G ∗ V-pointlike.
Hence S has a cross-section over G ∗ V for R.
Conversely, suppose that S has a cross-section over G ∗ V for R. Then
no pair {x, y} of distinct H-equivalent elements of R is G ∗ V-pointlike, so
for each such pair we can find a parameterized relational morphism Φx,y :
(R, S) −→
◦ (Qx,y , Tx.y ) with Tx,y ∈ V and an admissible partition Px,y on
DΦx,y such that for no q ∈ Qx,y are (x, q) and (y, q) in the same block of P.
Let Φ be the direct sum of the Φx,y . Then there are natural projections from
DΦ to the DΦx,y and hence natural projections ψx,y : DΦ → DΦx,y /Px,y . Let
P be the automaton congruence on DΦ associated to the direct sum of the
ψx,y . Clearly P is an admissible partition satisfying (3) of Definition 4.9.
Hence (Φ, P) is a presentation for R over V.
We mention briefly an alternate proof of Corollary 4.12 that is more constructive. If (Φ, P) is a presentation for R, then the relational morphism
constructed in the proof of Theorem 4.10 is easily seen to be a Tilson crosssection for R and hence, by Proposition 4.5, its companion relation is a
cross-section.
Conversely, if S has a cross-section over G ∗ V, we may assume it is of
the form ψ : S −→
◦ G o T with T ∈ V. Then the pair (Φ, P) constructed in
the proof of Theorem 4.10 is easily seen to be a presentation.
Taking into account the above discussion and Theorem 4.8, we can now
state our version of the Presentation Lemma; see [5] for the original version
due to Rhodes. Here SX denotes the symmetric group on a set X.
Theorem 4.13 (Presentation Lemma – Partial Transformation Semigroup
Version). Let S be a right mapping semigroup with distinguished J -class J.
Let B denote the set of L-classes of J and let R ⊆ J be an R-class. Suppose
(Φ, P) is a presentation for R, where Φ : (R, S) −→
◦ (Q, T ). Then
(R, S) ≺ (#ϕ1 /P, S#ϕ1 /P ) o (Q, T ) × (B, RLMJ (S)).
24
BENJAMIN STEINBERG
m (G ∗ V),
Moreover, if V, W are pseudovarieties with G ∗ V ⊆ W ⊆ A then S ∈ W if and only if RLMJ (S) ∈ W and S has a presentation for R
over V.
Applying this theorem to the case of complexity, we obtain the following
result of Rhodes [5].
Corollary 4.14. If S is a group mapping semigroup with distinguished J class J and R-class R, then S ∈ Cn if and only if RLMJ (S) ∈ Cn and S
has a presentation for R over Cn−1 .
It follows by an induction argument that deciding membership in Cn
reduces to determining whether a non-trivial group mapping semigroup has
a presentation for an R-class of its distinguished J -class over Cn−1 .
5. Tilson’s 2J -class Theorem
Tilson proved [44] that complexity is decidable for semigroups with at
most two non-zero J -classes; we call such a semigroup a 2J -semigroup.
Our aim is to provide a proof of this theorem using the machinery of presentations and cross-sections that we have developed – we hope the reader finds
this proof conceptually and technically simpler than that of [44]. A proof
using the Presentation Lemma is given in [5], but therein they only prove
the decidability of complexity for such semigroups – they do not provide
the elegant characterization of complexity obtained for such semigroups by
Tilson in [44].
First we point out some standard reductions [44, 5]. Recall that if S is a
semigroup, then the depth d(S) is the size of the longest chain of J -classes
containing non-trivial groups [45]. The Depth Decomposition Theorem [45]
shows that d is an upper bound for complexity. So the complexity of a 2J semigroup is at most 2. Since complexity 0 is clearly decidable, we are left
with distinguishing complexity 2 from 1.
Using the Depth Decomposition Theorem, we see that in order to have a
chance of having complexity 2, our 2J -class semigroup S must have 2 nonzero J -classes, each containing a non-trivial group, and forming a chain. So
the J -structure is of the form: J1 >J J2 >J 0.
Let e ∈ J1 . Then the Reduction Theorem [31, 45] shows that S divides
a Rees matrix semigroup over the monoid eSe and hence the complexity
of S and eSe are the same. It is a straightforward exercise to verify that
eSe is a 2J -semigroup. We are thus reduced to the case of a 2J -semigroup
consisting of a group of units and a (completely) 0-simple 0-minimal ideal.
Such monoids are called sometimes called small monoids [17]. Notice that
by adjoining a 0, we may always assume that our small monoid has a 0.
If V is a pseudovariety, EV denotes the pseudovariety of semigroups S
such that hE(S)i ∈ V. We now state Tilson’s theorem.
ON APERIODIC RELATIONAL MORPHISMS
25
Theorem 5.1 (Tilson). Let M ∈
/ A be a small monoid with group of units
H and 0-simple minimal ideal I. Then M has complexity 1 if and only if:
For each L -class L ⊆ I \ 0, LH ∪ 0 ∈ EA.
Otherwise, M has complexity 2.
Corollary 5.2 (Tilson). Complexity is decidable for semigroups with at
most 2 non-zero J -classes.
Note that LH ∪ 0 is indeed a subsemigroup of M by stability. The proof
of Theorem 5.1 will require some preparation.
5.1. Proof of Necessity. We retain the notation of Theorem 5.1. Let G
be the maximal subgroup of I and set J = I \ 0. First we prove that if there
is an L-class L ⊆ J with LH ∪ 0 ∈
/ EA, then M does not have complexity
1. To do this, we need the notion of a V-stabilizer pair, which is essentially
due to Rhodes and Tilson; compare with earlier results in [29, 30].
We begin by recalling that if T is a semigroup, t ∈ T and A ⊆ T , then
t−1 A = {y ∈ T | ty ∈ A}.
One defines At−1 dually. Notice that t−1 {t} is just the right stabilizer of t.
The definition of a V-stabilizer pair is then as follows. Fix a semigroup
S and a pseudovariety V. Then (s, A) ⊆ S × 2S is a V-stabilizer pair if,
for all ϕ : S −→
◦ T with T ∈ V, there exists t ∈ T such that s ∈ tϕ−1
−1
and A ⊆ t {t}ϕ−1 . Notice that if (x, A) is a V-stabilizer pair, then so is
(x, hAi) and so one often assumes that A is a subsemigroup. Equivalently,
(s, A) is a V-stabilizer pair if and only if the graph with a single vertrex
labelled by s and |A| loops labelled by A is V-inevitable in the sense of
Almeida [1]; see also Ash [3].
The usual motivation for studying V-stabilizer pairs is to obtain the decidability of semidirect products of the form W ∗ V, where W is local in
the sense of Tilson [48]. Our intention is to use them to obtain a sufficient
condition for a set to be W ∗ V-pointlike. The reader is referred to the
introduction for the notion of pointlike sets.
Lemma 5.3. Suppose (s, A) is a V-stabilizer pair for S with A ≤ S a
subsemigroup. Let B ⊆ A be W-pointlike. Then sB is W ∗ V-pointlike.
Proof. We may assume A 6= ∅. A standard argument shows that it suffices
to consider relational morphisms ϕ : S −→
◦ W oV with W ∈ W and V ∈ V.
Let ψ = ϕπ where π : W o V → V is the semidirect product projection.
Choose t ∈ T such that t ∈ sψ and A ≤ t−1 {t}ψ −1 .
Define ρ : A −→
◦ W by
aρ = {tw | (w, v) ∈ aϕ, v ∈ t−1 {t}}.
First note that aρ 6= ∅ since A ≤ t−1 {t}ψ −1 .
Suppose ai ρ twi and ai ϕ (wi , vi ) with vi ∈ t−1 {t}, i = 1, 2. Then
(w1 v1w2 , v1 v2 ) = (w1 , v1 )(w2 , v2 ) ∈ (a1 a2 )ϕ,
26
BENJAMIN STEINBERG
and v1 v2 ∈ t−1 {t}. Moreover,
t
(w1 v1w2 ) = tw1 tv1w2 = tw1 tw2
since tv1 = t. Thus tw1 tw2 ∈ (a1 a2 )ρ, finishing the proof that ρ is a relational
morphism. In the monoidal setting, it is straightforward to verify 1 ∈ 1ρ.
Since B is W-pointlike, there exists w ∈ W such that B ⊆ wρ−1 . Since
t ∈ sψ, there exists y ∈ W such that (y, t) ∈ sϕ. We claim that sB ⊆
(yw, t)ϕ−1 . Indeed, if b ∈ B, then w ∈ bρ implies there exists w 0 ∈ W and
v ∈ t−1 {t} so that w = tw0 and (w0 , v) ∈ bϕ. Then
(yw, t) = (y, t)(w 0 , v) ∈ sϕbϕ ⊆ (sb)ϕ,
as desired.
We retain the notation above concerning our small monoid.
Lemma 5.4. Let L ⊆ I \ 0 be an L-class and e ∈ L be an idempotent.
Then (e, L0 H ∪ H) is an A-stabilizer pair where L0 is the L-class of e in
LH ∪ H ∪ 0.
Proof. Let ϕ : M −→
◦ T be a relational morphism with T ∈ A. Let J 0
be the (unique) ≤J -minimal J -class of T relating to J; then J ⊆ J 0 ϕ−1
by Lemma 2.2. As in the proof of the Aperiodicity Lemma, there is an
idempotent f ∈ T with H ≤ f ϕ−1 . Choose ee ∈ J 0 ∩ eϕ. Since
e = e1 ϕ eef ≤J ee,
we see that eef ∈ J 0 ∩ eϕ by minimality. So without loss of generality, we
may assume eef = ee. Moreover, since
e = eω ϕ eeω ≤J ee,
eeω ∈ J 0 and so we may also assume that ee is an idempotent.
Let T 0 = ee−1 {e
e}. We show that L0 H ∪H ⊆ T 0 ϕ−1 . Since T 0 is a semigroup
containing f and H ⊆ f ϕ−1 , we see that T 0 ϕ−1 is a subsemigroup containing
H. Therefore it suffices to show L0 ⊆ T 0 ϕ−1 .
Clearly e ∈ T 0 ϕ−1 . Suppose e 6= z ∈ L0 . Since L0 is regular, z is regular
in LH ∪ H ∪ 0. Hence the R-class of z in LH ∪ H ∪ 0 contains an idempotent
e0 . Suppose that x belongs to Re ∩ Le0 , where we take R and L-classes in
LH ∪ H ∪ 0 throughout this proof. Then we have the eggbox picture in
Figure 1.
e ···
.. . .
.
.
z ···
0
e x
..
.
Figure 1. Eggbox picture for LH ∪ H ∪ 0.
ON APERIODIC RELATIONAL MORPHISMS
27
So, Green’s Lemma [6, 16] tells us that u 7→ uz gives a bijection between
Re ∩ Le0 and He . In particular, there exists y ∈ Re ∩ Le0 with yz = e. Since
y ∈ LH, we may write y = wh with w ∈ L and h ∈ H. Also note that
y = ey since y R e.
Since w ∈ L, w = we. Let t ∈ wϕ. Then
w = we ϕ te
e ≤J ee.
Setting w
e = te
e, we have, by minimality of J 0 , that w
e ∈ J 0 ∩ wϕ. Moreover,
by stability of T , we must have w
e L ee. Since ee is an idempotent,
we
ee = w.
e
(5.1)
Since z ∈ L0 , z = ze. Let ẑ ∈ zϕ. Then
J0
z = ze ϕ ẑe
e ≤J ee,
so if ze = ẑe
e, then ze ∈
∩ zϕ by minimality. Moreover, by stability of T ,
we must have ze L ee.
Summing things up, we have
e = yz = eyz = ewhz ϕ eew
efeze ≤J ee,
So, by minimality, eew
efeze ∈ J 0 . By stability of T , we conclude eewf
e ze ∈
Ree ∩ Lze = Hee. Since T is aperiodic, we deduce
(5.2)
However, by (5.1), we have
(5.3)
J0
ee = eewf
e ze.
ee = eewe
eef ze = eewe
eeze
and so eewe
ee ∈
and hence, by stability, in Hee. Since T is aperiodic, we
conclude eewe
ee = ee. Hence, by (5.3)
showing ze ∈ T 0 and so z ∈ T 0 ϕ−1 .
ee = eeze,
Corollary 5.5. Let M be a small monoid with group of units H and 0-simple
0-minimal ideal I. Let L ⊆ I \ 0 be an L-class such that LH ∪ 0 ∈
/ EA.
0
0
Let G be a non-trivial group in hE(LH ∪ 0)i. Then G is a G ∗ A-pointlike
m (G ∗ A) and so M does not have
subset of M . In particular, M ∈
/ A
complexity 1.
Proof. Let e ∈ G0 be the identity and let L0 be the L-class of e in LH ∪H ∪0.
Then G0 ⊆ L0 . By Lemma 5.4, (e, L0 H ∪ H) is an A-stabilizer pair. Let
g ∈ G0 . Then g = e1 · · · en with ei ∈ E(LH). Let ei = xi hi with xi ∈ L and
hi ∈ H. Then xi R ei in LH ∪H ∪0 and so xi is regular in LH ∪H ∪0. Thus
xi L e in LH ∪H ∪0, whence xi ∈ L0 . Thus ei ∈ L0 H so G0 ⊆ hE(hL0 H ∪Hi)i.
Hence G0 is a G-pointlike subset of hL0 H ∪ Hi [30, 11] and so eG0 = G0 is
G ∗ A-pointlike by Lemma 5.3.
The final statement follows from the Aperiodicity Lemma.
28
BENJAMIN STEINBERG
5.2. Proof of Sufficiency. To prove sufficiency, we shall find it expedient
to use Rees coordinates [6, 16]. In particular, we shall use some fundamental
results of Graham [9]. We recall the necessary definitions and theorems.
If G is a group, A, B are sets and C : B × A → G0 is a matrix (where
0
G = G∪0), then M0 (G, A, B, C) denotes the Rees matrix semigroup [6, 16]
with structure group G and structure matrix C. We write bCa instead of
(b, a)C to reflect the symmetry of the situation. One calls C regular if each
row and column contains a non-zero entry. In this case M0 (G, A, B, C) is
(completely) 0-simple. See [6, 16] for more on 0-simple semigroups.
A renormalization of a 0-simple semigroup is a change of Rees coordinates,
i.e. of Rees matrix representations. We recall [16] that all renormalizations
of a regular Rees matrix semigroup that do not permute rows or columns
are of the form
(a, g, b) 7−→ (a, ga ggb , b)
where a 7→ ga , b 7→ gb are functions A → G, B → G, respectively. If
C : B × A → G0 is the original matrix, then the matrix C 0 : B × A → G0 in
the new coordinate system is given by
bC 0 a = gb−1 (bCa)ga−1 .
Suppose first that C is regular. One says that b1 , b2 ∈ B are attached if
there is an element a ∈ A such that b1 Ca 6= 0 6= b2 Ca. Being attached is
a reflexive and symmetric relation. We use TCA to denote the transitive
closure of being attached. By a TCA block, we mean a block of the TCA
partition. Similarly attached and TCA can be defined for elements of A.
The TCA blocks of A and B are connected in the following way. For each
TCA block Bi ⊆ B, there is a unique TCA block Ai such that bCa 6= 0 for
some b ∈ Bi and a ∈ Ai ⊆ A. We shall say that Bi and Ai are in this case
attached. Sometimes the Ai × Bi will be referred to as the TCA blocks of
C. All multi-defined terminology should be clear from the context.
The easiest way to think about this is to construct a bipartite graph [9]
with vertices A ∪ B and with an edge from a to b if bCa 6= 0. The connected
components of this graph correspond exactly to the TCA blocks.
If we arrange the TCA blocks in order A1 × B1 , . . . , An × Bn and permute
the rows and columns of C accordingly, we obtain a new coordinatization
of J 0 for which the structure matrix is block diagonal and each block is a
regular Rees matrix. Graham proved further the following result [9].
Theorem 5.6. A 0-simple semigroup S is isomorphic to one of the form
M0 (G, A, B, C), such that:
(1) C is the direct sum of regular Bi × Ai matrices Ci (where the Ai × Bi
are the attached
TCA-blocks);
S
(2) hE(S)i = M0 (Gi , Ai , Bi , Ci ) where Gi is the subgroup of G generated by the entries of Ci .
In particular, S ∈ EA if and only if S has a matrix representation in which
all entries are either 0 or 1.
ON APERIODIC RELATIONAL MORPHISMS
29
Such a matrix representation is called a Graham normalization.
Graham’s results can be made to hold for non-regular Rees matrix semigroups in the following manner. First of all, if C is a general matrix,
then by reordering the rows and columns so that all zero rows appear at
the bottom and all zero columns appear at the right hand side, we can
write C = Creg ⊕ Cnull where Creg is a regular Rees matrix and Cnull is
a matrix of zeroes. One then puts Creg in Graham normal form to obtain a Graham normalization for C. The idempotent generated subsemigroup of M0 (G, A, B, C) is then the idempotent generated subsemigroup of
M0 (G, Areg , Breg , Creg ) where Creg : Breg × Areg → G0 is the restriction.
Let M = H ∪I be a non-aperiodic small monoid. Set I = M0 (G, A, B, C)
and J = I \0. Recalling that B can be identifieed with the set of L-classes of
J, the condition on L-classes of Theorem 5.1 can be restated in the following
equivalent forms:
(1) For all b ∈ B the restriction CbH : bH × A → G0 of C can be
normalized to 00 s and 10 s;
(2) For all b ∈ B, M0 (G, A, bH, CbH ) ∈ EA.
We prove that if (1) occurs, then M has complexity 1. By the Fundam (G ∗ A).
mental Lemma of Complexity [25, 8], it suffices to prove M ∈ A By Theorem 4.3, it suffices to show each regular R-class has a cross-section
over G ∗ A.
First observe that ρ : M −→
◦ H given by
(
m m ∈ H,
mρ =
H otherwise.
is a cross-section for H. Thus we need only find a cross-section for a regular
R-class R of J = I \ 0. By the Presentation Lemma–Semigroup Version, it
suffices to find a presentation (Φ, P) for R over A.
Recall that we have a complete transformation group (B, H) via RLMJ |H .
Let B/H be the set of orbits under the H-action. Recall [8], that if Q is a
set and q ∈ Q, then qe : Q → Q denotes the constant map with Qe
q = q and
one sets
e = {e
Q
q | q ∈ Q} ∪ {1Q }.
We proceed to define a parameterized relational morphism
]
Φ : (R, M ) → (B/H, B/H).
] is a
Set xϕ1 = Lx H for x ∈ R. To see that ϕ1 : (R, M ) −→
◦ (B/H, B/H)
−1
relational morphism, suppose x ∈ bHϕ1 (with x ∈ R) and m ∈ M . We
may take m 6= 0, since any element covers 0. We then have two cases. Set
(
1B/H m ∈ H,
(5.4)
m̂ =
L]
m H otherwise.
−1
In the first case, xm ∈ bH and so xm ∈ bH1B/H ϕ−1
1 = bH m̂p1 , as desired.
For the second case, if xm is not defined, we are done. If xm is defined,
30
BENJAMIN STEINBERG
then Lxm = Lm by stability. But then
−1
] −1
xm ∈ Lm Hϕ−1
1 = bH Lm Hϕ1 = bH m̂ϕ1 .
So m̂ covers m in all cases.
e by
Define ϕ2 : M −→
◦ Q
(5.5)


1B/H
mϕ2 = L]
mH

e
Q
m ∈ H,
m ∈ J,
m = 0.
Using the stability of M , it is straightforward to verify that ϕ2 is a relational
morphism. The argument of the previous paragraph shows that Φ = (ϕ1 , ϕ2 )
is a parameterized relational morphism.
Let us decongest notation for DΦ . First observe that since ϕ1 is a function,
we can canonically identify #ϕ1 with R via x ↔ (x, Lx H), that is, we
] is
can take DΦ = (R, DΦ ). Also ϕ2 |M \0 is a function and (B/H, B/H)
complete. Thus, to ease notation, for m ∈ M \ 0, we write the arrow
(LH, (m, mϕ2 ), LHmϕ2 ) as simply (LH, m).
We now define a partition P on R by x P y if and only if
(5.6)
Lx H = Ly H,
(5.7)
y = xm where m ∈ hE(Lx H ∪ 0)i.
Notice that (5.6) can be restated as xϕ1 = yϕ1 and so x P y implies
xϕ1 = yϕ1 . The remaining verifications that P is an admissible partition
are shown via Rees coordinates. That is we take I = M0 (G, A, B, C).
Suppose R = a0 × G × B. We remind the reader that if Lx = b0 , then
Lx H ∪ 0 = M0 (G, A, b0 H, Cb0 H ).
By hypothesis, there is a Graham normalization M0 (G, A, b0 H, P (b0 H))
given by
(5.8)
a 7→ ga , b 7→ gb , a ∈ A, b ∈ b0 H
such that bP (b0 H)a ∈ {0, 1} for all a ∈ A, b ∈ b0 H. By defining gb = 1 for
b ∈ B \ b0 H, we can extend P (b0 H) to B × A to obtain a renormalization
M0 (G, A, B, P (b0 H)) of I such that P (b0 H)b0 H is in Graham normal form.
However, we caution the reader that P (b0 H) will not in general be a Graham
normalization of I.
Lemma 5.7. P is a partition. More specifically, suppose x, y ∈ R with
Lx , Ly ∈ b0 H. Then using the Rees coordinates M0 (G, A, B, P (b0 H)),
x = (a0 , g, b) P (a0 , g 0 , b0 ) = y ⇐⇒ g = g 0 and b TCA b0 in P (b0 H)b0 H .
Proof. We prove the second statement, as the first is an immediate consequence. Suppose y = xm with
m = (a, g, b) ∈ hE(M0 (G, A, b0 H, P (b0 H)b0 H ))i.
ON APERIODIC RELATIONAL MORPHISMS
31
Then bP (b0 H)a 6= 0 and b = b0 . By Theorem 5.6, a is in the same TCA block
as b (in P (b0 H)b0 H ) and g = 1, so we conclude b TCA b0 (in P (b0 H)b0 H )
and
g 0 = gbP (b0 H)ag = g
(using that bP (b0 H)a = 1).
For the converse, suppose x = (a0 , g, b), y = (a0 , g, b0 ) with b, b0 ∈ b0 H and
b TCA b0 in P (b0 H)b0 H . Then there exists a such that bP (b0 H)a 6= 0 and
hence bP (b0 H)b0 H a 6= 0. Thus a and b0 are in corresponding TCA-blocks of
P (b0 H)b0 H and so, by Theorem 5.6,
But
m = (a, 1, b0 ) ∈ hE(M0 (G, A, b0 H, P (b0 H)b0 H ))i.
(a0 , g, b)(a, 1, b0 ) = (a0 , gbP (b0 H)a1, b0 ) = (a0 , g, b0 )
since bP (b0 H)a = 1. So y = xm, as desired.
One could prove that P is a partition without coordinates along the lines
of [47]. However, we shall make use of the precise description of P given
above to show that (Φ, P) is a presentation. First we point out that P is
injective on H-classes of R, that is, (3) of Definition 4.9 is satisfied.
Lemma 5.8. Suppose x, y ∈ R with x H y and x P y. Then x = y.
Proof. Since x P y, by Lemma 5.7, x = (a0 , g, b), y = (a0 , g, b0 ) in coordinates M0 (G, A, B, P (b0 H)) for I, where b0 H = Lx H = Ly H. Since x H y,
we conclude b = b0 and so x = y.
We now show that P is an automaton congruence.
Lemma 5.9. P is an automaton congruence.
Proof. Suppose x, y ∈ R with x P y. Let t ∈ DΦ with xt, yt defined; we
need to show xt P yt. Then, for some b0 ∈ B, Lx H = b0 H = Ly H. There
are two cases.
First suppose t = (b1 H, h) with h ∈ H. The assumption that xt, yt are
defined means that yϕ1 = xϕ1 = b0 H = b1 H. In this case xt = xh, yt = yh.
Note that xh, yh ∈ b0 H = Lx H = Ly H.
Since x P y, y = xm with m ∈ hE(Lx H ∪ 0)i. Then
yt = yh = xmh = xh(h−1 mh) = xt(h−1 mh)
So to show that yt P xt, it suffices to show that h−1 mh ∈ hE(Lx H ∪ 0)i.
Let e ∈ E(Lx H); so e = zh0 with z ∈ Lx and h0 ∈ H. Then h−1 z ∈ Lx ,
whence h−1 eh = (h−1 z)h0 h ∈ Lx H and so, in fact, h−1 eh ∈ E(Lx H). A
straightforward induction argument implies that h−1 mh ∈ hE(Lx H ∪ 0)i, as
desired.
Suppose now that t = (b1 H, s) with s ∈ J. Then xt = xs ∈ R and
yt = ys ∈ R. Thus, using Rees coordinates M0 (G, A, B, P (b0 H)), let (using
Lemma 5.7)
x = (a0 , g, b), y = (a0 , g, b0 ), s = (a, g, b)
32
BENJAMIN STEINBERG
with
bP (b0 H)a 6= 0 6= b0 P (b0 H)a.
Since b, b0 ∈ b0 H, we have, in fact,
bP (b0 H)a = 1 = b0 P (b0 H)a.
Thus
xs = (a0 , gg, b) = ys.
We conclude that P is an automaton congruence.
We now complete the proof that (Φ, P) is a presentation by establishing
that P is an injective congruence.
Lemma 5.10. P is an injective congruence.
Proof. Suppose x, y ∈ R and t ∈ DΦ with xt P yt. Again we have two cases.
If t = (b1 H, h), then define t0 = (b1 H, h−1 ). Then x, y ∈ b1 H, (xt)t0 =
xhh−1 = x and (yt)t0 = yhh−1 = y. So x P y by the previous lemma.
Thus we are left with the case t = (b1 H, s) with s ∈ J. Since xt and yt
are defined, xs, ys ∈ R. By stability, xs L s L ys. So xs H ys, whence, by
Lemma 5.8, xs = ys. Since xt, yt are defined, Lx H = b1 H = Ly H. Choose
the Rees coordinate system I = M0 (G, A, B, P (b1 H)).
Suppose x = (a0 , g, b), y = (a0 , g 0 , b0 ) and s = (a, g, b). Then xs = ys 6= 0
implies that
bP (b1 H)a = 1 = b0 P (b1 H)a
(recall b, b0 ∈ b1 H) and
(a0 , gg, b) = xs = ys = (a0 , g 0 g, b).
Thus g 0 = g and b, b0 are attached. It follows x P y by Lemma 5.7.
We have thus shown (Φ, P) is a presentation, completing the proof of
Theorem 5.1.
Just to show that there are indeed complexity 2 small monoids, let M be
given as follows. Let H = Z2 = hzi and let I = M0 (Z2 , 2, 2, C) where
1 1
C=
1 −1
and where we define
(a, g, 1)z = (a, g, 2)
(a, g, 2)z = (a, −g, 1)
z(1, g, b) = (2, g, b)
z(2, g, b) = (1, −g, b).
There is just one H orbit on B. It is straight forward to verify I ∈
/ EA.
Thus M has complexity 2 by Theorem 5.1; M is in fact known to be a smallest order semigroup of complexity 2. Similarly M has reverse complexity
ON APERIODIC RELATIONAL MORPHISMS
33
m (G ∗ ∗A) where ∗∗ denotes the
2. In fact, one can prove that M ∈
/ A
two-sided semidirect product. In fact the entire proof of Theorem 5.1 can
be adapted to show the following.
Theorem 5.11. Let M be a small monoid with group of units H and 0m (G ∗ ∗A) if and only if:
minimal ideal M0 (G, A, B, C). Then M ∈ A For each a ∈ A, b ∈ B, M0 (G, Ha, bH, C|Ha×bH ) ∈ EA.
We remark that Rhodes has proven that the two-sided complexity of
any 2J -semigroup is at most 1 (private communication). The difference
between the above theorem and Rhodes’s result is that Rhodes places all 2J m (G∗∗(G∗∗A)), which is strictly bigger than A m (G∗∗A).
semigroups in A The key to adapting the proof is to replace V-stabilizers with a two-sided
version, to use the kernel category instead of the derived category and to
replace all uses of derived transformation semigroups with a new two-sided
construction that is an automaton rather than a transformation semigroup
(notice that we never use that DΦ is a semigroup).
More specifically, to construct a cross-section from M to G ∗ ∗A, we
consider T = (H/A × B/H) ∪ 1 where H/A × B/H is given the rectangular
band multiplication. There is a natural relational morphism from M to T
relating H to 1, (a, g, b) to Ha × bH and 0 to everything.
The associated automaton K has states J × J. Edges are of the form
(bH, m, Ha) : (m1 , mm2 ) → (m1 m, m2 ) with m ∈ M , mm2 , m1 m ∈ J, m1 ∈
bH, m2 ∈ Ha. A partition is defined on J × J by setting (s1 , s2 ) P (s01 , s02 )
if and only if Ls1 H = Ls01 H = b0 H, HRs2 = HRs02 = Ha0 and there exists
m ∈ hE(M0 (Ha0 , G, b0 H, C|Ha0 ×b0 H )i with s01 m = s1 , ms02 = s2 . A similar
argument to the above shows that P is an injective automaton congruence.
If we embed the quotient in a permutation automata with transition group
G, arguments similar to the above show that there is a relational morphism
from M to GT which is a cross-section for J.
6. Complexity is Not Local
In this section, we present Rhodes’s tall fork semigroup F [27, 5]. This is
m G whose local submonoids have
a complexity two semigroup in (A ∗ G) strictly smaller complexity.
Recall that if S is a semigroup, then the local submonoids of S are the
submonoids of the form eSe with e ∈ E(S). If V is a pseudovariety, then
LV is the pseudovariety of all semigroups whose local submonoids belong
to V. A pseudovariety is local in the sense of Eilenberg [8] if LV = V. Let
D be the pseudovariety of semigroups whose idempotents form a right zero
semigroup; D ⊆ A. A consequence of Tilson’s Delay Theorem is that a
pseudovariety V generated by monoids is local in the sense of Tilson if and
only if LV = V ∗ D [48]. Since Cn ∗ D = Cn , a complexity pseudovariety is
local in the sense of Eilenberg if and only if it is local in the sense of Tilson.
Our goal is to prove, using F , the following theorem of Rhodes [27, 5].
34
BENJAMIN STEINBERG
Theorem 6.1 (Rhodes). C1 is neither local in the sense of Eilenberg nor
local in the sense of Tilson.
Our Presentation Lemma shall allow for a simpler, less coodinate-heavy
proof. We retain the notation of [5] for the talk fork. The tall fork has the
following J -class structure, giving rise to its name:
J1
J2
J0
J
{0}
Figure 2. J -class structure of the tall fork
We first describe J 0 as a Rees matrix semigroup M0 (G, A, B, P ). Set
A = {a1 , a2 , a3 , a4 , a5 , a6 , a7 }, B = {1, 2, 3, 4, 10 , 30 }, G = Z2 = {1, −1} and
1
2
P = 3
4
10
30
a1 a2 a3 a4 a5 a6 a7
1 0 0 1 0 0 0
1 1 0 0 0 0 0
0 1 1 0 0 0 0
0 0 1 1 0 0 0
0 0 0 0 0 1 1
0 0 0 0 1 0 1
We present the tall fork F as a submonoid of the translational hull T (J 0 )
of J 0 [6]. The translational hull [6, 16] of J 0 consists of all linked pairs (C, R)
where C is an A × A column monomial matrix over G0 , R is a B × B row
monomial matrix over G0 and where being linked means RP = P C. Since
P has no proportional rows or columns, P embeds into its translational
hull [16]. Set h = (Ch , Rh ), z = (Cz , Rz ), and t = (Ct , Rt ) where
ON APERIODIC RELATIONAL MORPHISMS
35
a1
a2
a
Ch = 3
a4
a5
a6
a7
a1
0
1
0
0
0
0
0
a2
0
0
1
0
0
0
0
a3
0
0
0
1
0
0
0
a4
1
0
0
0
0
0
0
a5
0
0
0
0
0
0
0
a6
0
0
0
0
0
0
0
a7
0
1
0
2
0
Rh = 3
0
4
0
10
0
30
0
1
0
0
0
1
0
0
2
1
0
0
0
0
0
3
0
1
0
0
0
0
4 10
0 0
0 0
1 0
0 0
0 0
0 0
30
0
0
0
0
0
0
a1
a2
a
Cz = 3
a4
a5
a6
a7
a1
0
0
0
0
0
0
0
a2
0
0
0
0
0
0
0
a3
0
0
0
0
0
0
0
a4
0
0
0
0
0
0
0
a5
0
0
0
0
0
1
0
a6
0
0
0
0
1
0
0
a7
0
1
0
2
0
Rz = 3
0
4
0
10
0
30
1
1
0
0
0
0
0
0
2
0
0
0
0
0
0
3
0
0
0
0
0
0
4 10
0 0
0 0
0 0
0 0
0 0
0 1
30
0
0
0
0
1
0
1
0
0
0
0
−1
0
2
0
0
0
0
0
0
3
0
0
0
0
0
1
4 10
0 0
0 0
0 0
0 0
0 0
0 0
30
0
0
0
0
0
0
a1
a2
a
Ct = 3
a4
a5
a6
a7
a1
0
0
0
0
0
−1
0
a2
0
0
0
0
1
0
0
a3
0
0
0
0
1
0
0
a4
0
0
0
0
0
−1
0
a5
0
0
0
0
0
0
0
a6
0
0
0
0
0
0
0
a7
0
1
0
2
0
Rt = 3
0
4
0
10
0
30
0
Then the tall fork is F = hJ 0 , h, z, ti ≤ T (J 0 ). The J -structure of F is
as in Figure 2, where
J1 = {h, h2 , h3 , h4 = e} = Z4 , J2 = {z, z 2 = f } = Z2
J 0 = {t, th, th2 , th3 , zt, zth, zth2 , zth3 }
and
(6.1)
J1 J2 = J2 J1 = J 0 J 0 = J 0 J2 = J1 J 0 = 0.
We remark that hE(F )i = {e, f } ∪ hE(J 0 )i ∈ A and so F ∈ EA.
mG
Proposition 6.2 (Rhodes [27]). F ∈ (A ∗ G) Proof. Consider the relational morphism ϕ : F −→
◦ Z2 given by J1 7→ 1,
J2 7→ 1, J 0 7→ Z2 and J 0 7→ −1. Then the inverse image of 1 is S = F \ J 0 .
It is a straightforward computation to verify that the self-conjugate core of
S is hE(S)i = hE(F )i ∈ A.
Once we have shown that F has complexity two, this result will imply
A ∗ G is not local in the Tilson sense [48], since if it were, then
m G = (A ∗ G) ∗ G = A ∗ G
(A ∗ G) 36
BENJAMIN STEINBERG
would hold [11]; but then F could not have complexity two. The author
showed [39] that, in fact, A ∗ G has infinite vertex rank; the construction
was inspired by F .
We now show that every local submonoid of F has complexity at most
one. Of course 0F 0 has complexity zero. If e0 ∈ J is an idempotent, then
e0 F e0 is a group with 0 and so has complexity one. On the other hand
it is straightforward to see from (6.1) that eF e and f F f are 2J -monoids.
Since F ∈ EA, it immediately follows from Tilson’s 2J -class Theorem that
eF e, f F f have complexity one.
We now aim to show that there can be no presentation for R = a7 ×G×B
over A. Our argument, in fact, can be used to show, via Theorem 4.10, that
a7 × G × {1, 2, 3, 4} is G ∗ A-pointlike.
First we prove a standard lemma on aperiodic pointlikes.
Lemma 6.3. Let Φ : (X, S) −→
◦ (Q, T ) be a parameterized relational morphism of gts with T aperiodic. Let x ∈ X and suppose H ≤ S is a subgroup.
Then there is an idempotent e0 ∈ E(T ) and q ∈ Q such that H ≤ e0 ϕ−1
2 ,
−1
0
qe = q and xH ⊆ qϕ1 .
Proof. As we saw in the proof of the Aperiodicity Lemma, there must be an
0
idempotent e0 of T with H ≤ e0 ϕ−1
2 . Let q ∈ xϕ. Then
0 0 −1
xH ⊆ q 0 ϕ−1
1 H ⊆ (q e )ϕ1 .
Setting q = q 0 e0 completes the proof.
The following lemma is the version of Rhodes’s “Tie Your Shoes” Lemma
[5] corresponding to our notion of a presentation.
Lemma 6.4 (Tie Your Shoes). Suppose R is a regular R-class of a semigroup S belonging to a J -class J. Suppose J 0 = M0 (G, A, B, C) is a Rees
coordinatization, Φ : (R, S) −→
◦ (Q, T ) a parameterized relational morphism
and P an admissible partition on DΦ . Let R be the R-class corresponding
to a ∈ A. Suppose b1 Ca0 6= 0 6= b2 Ca0 and
x = (a, g(b1 Ca0 )−1 , b1 ), y = (a, g(b2 Ca0 )−1 , b2 ) ∈ qϕ−1
1 .
Then (x, q) P (y, q).
Proof. Set z = (a0 , 1, b1 ). Then xz = (a, g, b1 ) = yz ∈ R, so if t ∈ zϕ2 , then
qt is defined and
(x, q)(q, (z, t), qt) = (xz, qt) = (yz, qt) = (y, q)(q, (z, t), qt).
Since P is an injective automaton congruence on DΦ , (x, q) P (y, q).
We can now obtain the following result of Rhodes [27, 5]; our proof is
simpler since we avoid what Rhodes calls projectivized cross-sections.
Theorem 6.5 (Rhodes [27]). The tall fork F has complexity two.
ON APERIODIC RELATIONAL MORPHISMS
37
Proof. By the Depth Decomposition Theorem [45], F has complexity at
most two. Let R be the R-class associated to a7 ; we show that there is
no presentation over A for R; it will then follow that F has complexity
two by our Presentation Lemma. Suppose (Φ, P) is a presentation for R,
Φ : (R, S) −→
◦ (Q, T ) with T ∈ A.
Set x = (a7 , 1, 10 ), y = (a7 , 1, 30 ). Notice that xJ2 = yJ2 = {x, y}. So
Lemma 6.3 shows that there exists q ∈ Q such that x, y ∈ qϕ−1
1 . Since
10 P a7 = 1 = 30 P a7 , it follows from “Tie Your Shoes” that (x, q) P (y, q).
Choose t0 ∈ tϕ2 . By Lemma 6.3, there exists e0 ∈ E(T ) with J1 ≤ e0 ϕ−1
2 .
Set X = {x, y}tJ1 . Since
{x, y}t = {(a7 , −1, 1), (a7 , 1, 3)},
(6.2)
we see that
X = a7 × G × {1, 2, 3, 4}.
Also, since X ⊆ R, we have that q 0 = qt0 e0 is defined and X ≤ q 0 ϕ−1
1 .
Consider (x, q)(q, (te, t0 e0 ), q 0 ) and (y, q)(q, (te, t0 e0 , q 0 )). Since P is an automaton congruence, (x, q) P (y, q) and te = t, it follows from (6.2) that
((a7 , −1, 1), q 0 ) P ((a7 , 1, 3), q 0 ). Repeated application of “Tie Your Shoes”
yields:
((a7 , 1, 3), q 0 ) P ((a7 , 1, 4), q 0 ) P ((a7 , 1, 1), q 0 ) P ((a7 , 1, 2), q 0 ) and
((a7 , −1, 1), q 0 ) P ((a7 , −1, 2), q 0 ) P ((a7 , −1, 3), q 0 ) P ((a7 , −1, 4), q 0 ).
We conclude that X × q 0 belongs to a single partition block of P. This
contradicts the fact that (Φ, P) is a presentation, since X contains the Hclasses a7 × G × i, for i = 1, 2, 3, 4. This contradiction proves that F has
complexity two.
So F ∈ LC1 \ C1 , finishing the proof of Theorem 6.1.
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School of Mathematics and Statistics, Carleton University, 1125 Colonel
By Drive, Ottawa, Ontario K1S 5B6, Canada
E-mail address: [email protected]