Ildikó Sain
István Németi
FORK ALGEBRAS IN USUAL AND IN
NON-WELL-FOUNDED SET THEORIES
PART II
Below we summarize some basic considerations about representation
theorems in algebra. For more detail on these ideas we refer to [19, item
2.7.46 (Part I, pp. 459–461)].
Remark 2.4.1 A class K1 of algebras is called abstract if it is closed
under taking isomorphic copies. A class K2 is called concrete if for any
A, B ∈ K2 , whenever A and B have the same universe, then A = B, that
is, the universe of A determines the operations of A. In mathematics,
representation theorems are of the following form. An abstract class K1
of algebras is considered together with a concrete class K2 ⊆ K1 . Now, a
representation theorem says that every algebra in the abstract class K1 is
isomorphic to a member of the concrete class K2 .
Examples of representation theorems are the representation of abstract Boolean algebras (as K1 ) in terms of concrete Boolean set algebras
(as K2 ), that of abstract groups as concrete permutation groups, pointdense relation algebras as set (or proper) relation algebras, locally finite
cylindric algebras as cylindric set algebras etc. In each of these cases, the
choice of K2 (e.g., Boolean set algebras, permutation groups etc.) was
a concrete class (e.g., the universe of a permutation group G completely
determines the operations of G).
In some cases, when real representation theorems are not available,
weak representation theorems are used as a substitute. In weak representation theorems, instead of requiring K2 to be concrete, the operations of
K2 are represented in terms of a third class K3 of structures, which in turn,
is defined abstractly again.
0 For
References see Part I BSL 24/3
of this remark were borrowed from [17, Remark 4].
1 Parts
182
In more detail, when K2 was concrete, then for each A ∈ K2 , the universe A of A completely determined the operations of A, that is, to each
operation, say g of A, there was an explicit set theoretic definition of g
using no other parameter than A. E.g., in Boolean set algebras “x ∧ y”
is the real set theoretic intersection of x and y. In case of weak representation theorems, the definition of K2 looks as if it were concrete, but in
reality it is not concrete. Here for each operation g of A ∈ K2 there is a
definition of g, but this definition is not absolute, that is, this definition
depends on something else and not only on A. Namely, a third class K3 is
introduced axiomatically (i.e., K3 is not concrete), and then the operations
(like g above) of K2 are defined in terms of the abstract class K3 . Since
K3 was chosen to be abstract, defining K2 in terms of K3 will not ensure
concreteness of K2 . The insight usually provided by strong representation
theorems comes from the fact that they interpret an abstract class by a
concrete class. As a contrast, weak representation theorems interpret an
abstract class (K1 ) in another abstract class (K2 ). Therefore, they do not
necessarily bridge the gap between “abstract” and “concrete”.
A typical example for K3 is the class
{hU, f i : U is a set and f : U × U >−→ U is an injective function}
of pairing structures of NFA’s. When discussing the difference between
strong and weak representation theorems (our terminology), [19] mentions
the Jónsson-Tarski Representation Theorem for Boolean algebras with operators as an example for a weak representation theorem: [19, Part I,
p. 460 line 12] writes that this weak representation theorem is “not what
could be regarded as a satisfactory representation theorem”. Nevertheless,
weak representation theorems can be useful.
Returning to our present algebras, SFA and SPA are concrete classes,
while NFA is not concrete, because the operation 5 f is defined only abstractly via the pairing structure hU, f i which was defined only axiomatically (and not concretely). Therefore, representing a class of algebras
in terms of SFA’s would be a strong representation theorem, while representing them in terms of NFA’s is only a weak representation theorem.
(Nevertheless, as we wrote before, it can be quite useful.) In particular,
then, Corollary 2.3 is a weak representation theorem.
A strong representation result directly implied by Corollary 2.3 says
that the RA-reducts of the finitely axiomatizable class I NFA are representable. This amounts to saying that those RA’s which can be expanded
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with a new operation 5 satisfying certain axioms are representable. Such
results are called pseudo-axiomatizability theorems, cf. e.g. [19, Part I, item
0.5.21, p.154]. No stronger (strong) representation theorem seems to be implied directly from Corollary 2.3. The original form of Corollary 2.3 in e.g.
[33] is a stronger (strong representation) result saying that a certain finitely
axiomatizable subclass QRA of RA’s is representable. (This is stronger in
two ways: (i) QRA is strictly bigger than the class of RA-reducts of I NFA,
and (ii) the QRA-result states finite axiomatizability, which is a stronger
property than pseudo-axiomatizability.)
The question of what a satisfactory representation theorem should
look like has been extensively investigated in algebraic logic. In the above
discussion we could outline only the “tip of the iceberg”. For more on
this subject we refer to e.g. [33, p. 62 lines 9–11, section 3.5 (p.56)], [25,
beginning with “Remark 2 (Finitization)”], [12], [30] [6, Bjarni Jónsson’s
problem on p. 441], and of course, [19, Part I, p.459].
2
Definition 2.5.
(i) An NFA is non-degenerate if the equation 1 ◦ (−Id ) ◦ 1 = 1 is true in
it. Those NFA’s that are not non-degenerate are called degenerate.
(ii) NFA+ denotes the class of isomorphic copies of non-degenerate NFA’s.
2
Intuitively, the equation 1 ◦ (−Id ) ◦ 1 = 1 excludes those algebras
(and only those) in which our “pairing universe” U has only one element,
implying f (a, a) = a for all a ∈ U .
Remark 2.6. Only the non-degenerate NFA’s are relevant to applications.
If an NFA A is degenerate then it has only 2 elements and it contains no
more information than the 2-element Boolean algebra.
2
Theorem 2.7. The equational theory Eq(NFA+ ) is hereditarily undecidable. That is, if an equational theory E contains Eq(NFA+ ), in symbols
E ⊇ Eq(NFA+ ), then E is undecidable, unless E is inconsistent in the
sense that E ` 1 = 0.
On the proof. A variant of this is already proved in [33] for a variety
called ORA strongly related to NFA+ . One possible proof for the present
theorem is based on the ideas in [33]. Another one is based on ideas from
[29] and a result in [2] or [3].
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Hint. Assume V is a subvariety of the variety S P NFA+ . It is enough
to concentrate on the class Sir(V) of subdirectly irreducible members of V.
Sir(V) ⊆ NFA+ . Let A ∈ Sir(V). Then to A there are U and
f : U × U >−→ U as in Definition 2.1 from Part I. Let
def
N1 = Rng(Id 5f − Id ).
def
Then N1 = {f (a, b) : a 6= b, a, b ∈ U }. Certainly, Id 1 = Id ∩ (N1 × N1 ) is
term-definable in A. Let
def
def
N2 = Rng(Id 1 5f − Id 1 ), Id 2 = Id ∩ (N2 × N2 ),
...
def
def
Nn+1 = Rng(Id n 5f − Id n ), Id n+1 = Id ∩ (Nn+1 × Nn+1 ).
...
One can check that {Id n : n ∈ ω} are infinitely many distinct elements
below Id , in A, hence we can apply [2, Thm. 1] or [3, Thm. 2.1].
The above theorem again points in the direction of the strong connection with [33].
Definition 2.8. ([4, section 7], based on [33])
def
PA0 = {hA, p, qi : A is a QRA with quasi-projection elements p, q}. PA is
the subvariety of PA0 defined by the axiom 1 ◦ −Id ◦ 1 = 1.
2
Let PA+ be the subvariety of PA defined by the single equation (p ◦
p−1 ) ∩ (q ◦ q −1 ) ≤ Id . PA+
0 is obtained from PA0 the same way.
Fact 2.9. ([4] and also [33]) PA is a finitely axiomatizable variety, and
therefore so is PA+ .
Let K1 and K2 be two classes of algebras. Then K1 ≡ K2 means that K1
and K2 are term-definitionally equivalent, cf. e.g. [22] or [19, polinomially
equivalent, Part I, p. 125].
Theorem 2.10. (implicit in [4] and in [33]) PA+ ≡ S P NFA+ .
Proof. Let A be a subdirectly irreducible PA+ . Then, by Lemma 8.1 of [4]
credited to [33] there, A is embeddable into some A0 = hP(U × U ), . . . , p, qi
whose RA-reduct is an SRA. But then, by the condition on p and q in the
def
definition of PA+ , p, q : U −→ U , moreover, letting f = {ha, b, ci ∈ U 3 :
p(c) = a & q(c) = b}, the relation f turns out to be an injective function
f : U × U >−→ U . Clearly, A0 is an NFA with pairing structure hU, f i.
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This proves that PA+ is term-definable in S P NFA+ . The other direction,
saying that S P NFA+ is term-definable in PA+ is immediate by the proof
of Theorem 1.14, Part I (x 5 y = (x ◦ p−1 ) ∩ (y ◦ q −1 ) etc.).
We note that a variant 2 of 5 was introduced and studied in [33, 8.4
(xii)] and [4, p. 427]. The connection is: x 5 y = (p∩q)−1 ◦(x2y) and x2y
= (p ◦ x) 5 (q ◦ y).
For the choice of the class FA of abstract (axiomatic) Fork Algebras
we see three reasonable choices. One would be FA1 = S P NFA , the other
would be FA2 = S P NFA+ , and the third would result from FA2 by adding
a few natural equations holding in SFA to the equations defining FA2 . Such
equations are
Id ∩ (Id 5 Id ) = 0,
1 ◦ −(1 ◦ (1 5 1)) ◦ 1 = 1.
(1)
(2)
Equation (2) above expresses that there are elements of the universe U
which are not ordered pairs. Let FA3 consist of those members of FA2
which satisfy equations (1) and (2) above.
In Definition 2.8 above we recalled the variety PA from [4] and [33].
PA+ was finitely axiomatized over PA (hence the results in the quoted works
carry over to PA+ ).
+
Theorem 2.11. FA1 ≡ PA+
0 and FA2 ≡ PA .
Proof. The proof of Theorem 1.14 works. Gyuris [17] documents this in
careful detail.
By this theorem, interesting properties of FA1 , FA2 can be found in
[33], [4] and related works.
We think that deciding which one of FA1 , FA2 , . . . should be FA,
might be premature. For any choice of FA ∈ {FA1 , FA2 , FA3 }, the situation
is analogous to that of abstract relation algebras (RA’s) and proper relation
algebras (SRA’s). The analogy is the following:
FA is analogous to RA,
SFA is analogous to SRA, and
NFA is analogous to the class K of those relativized SRA’s which satisfy the
axioms of RA. These relativized algebras are nonstandard “models”
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of RA theory, and their use stems from the fact that neither SRA nor
the variety generated by it is finitely axiomatizable.2
To summarize the analogy, RA is a variety which is a finitely axiomatizable approximation of the finitely not axiomatizable variety generated by
SRA. Similarly, we need a finitely axiomatizable approximation FA of the
non-axiomatizable variety generated by SFA. The intermediate class NFA is
also analogous to the complex-algebra representation of RA’s in that both
provide a weak representation of the non-representable algebras, and that
these weak representations are often very helpful for the intuition.3 This
distinction between useful weak representation and (real) representation
was discussed in Remark 2.4.
3. Candidates from the Literature for Playing the Rôles of SFA and SPA
We start with recalling from [21] the definition of True Pairing Algebras
(TPA’s).
Definition 3.1.
(i) An SPA A with greatest element U × U is called a True Pairing
Algebra (TPA) iff U is the smallest set such that W ⊆ U for some set
W , and U × U ⊆ U \ W .4
(ii) An SFA is called an SFA+ iff it is term-definitionally equivalent with
a TPA.
2
We note that SFA+ ⊆ SFA and TPA ⊆ SPA, and therefore Theorem
1.16 applies to SFA+ and TPA.
Indeed, in the early fork-algebra papers, pairing in SFA’s was represented via finite trees, which made them isomorphic with SFA+ ’s (cf. e.g.
[10], [34]), cf. Remark 1.4 and the footnote in it.
2 To be careful, we note that relativized SRA’s are somewhat less non-standard (i.e.,
more concrete, cf. Remark 2.4) than NFA’s.
3 A weak representation represents some of the operations of the algebra in real set
theoretic terms, while the remaining operations are “represented” only in terms of an
abstract structure like hU, f i with f : U × U −→ U in Definition 2.1.
4 In [21], “TPA” was reserved to denote the class of those TPA’s which are of the form
hP(U × U ) . . .i. However, this restriction does not change the variety or quasi-variety
generated by the class in question.
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In Definition 3.2 below, we recall from [10] the class GFA of free
grupoid-representable fork algebras.
Definition 3.2. By a free grupoid-representable fork algebra (GFA) we
understand an NFA A with pairing structure hU, f i such that hU, f i is a
free grupoid.
2
Proposition 3.3. I GFA = I SFA.
Theorem 3.4. (without Foundation) Eq(TPA), Eq(SFA+ ) and Eq(GFA)
are not recursively enumerable. Moreover, they all are Π11 -complete.
The Proof of the GFA-part follows from Proposition 3.3.
Theorem 3.5. (without Foundation)
(i) The varieties Var(TPA) and Var(SFA+ ) generated by TPA and SFA+
are not finitely axiomatizable.
(ii) The same holds for the generated quasi-varieties, too.
(iii) S P TPA and S P SFA+ are not axiomatizable by any set of first order
formulas.
Theorem 3.6.
ZF− , too.
Theorems 3.4, 3.5 remain true in ZFC− , moreover, in
By this last theorem, the negative results in e.g. [29], [23], [26] remain
true for the original algebras in [10], [34], even if we completely drop the
Axiom of Foundation from our set theory ZFC and do not “replace the
gap” by any kind of anti-foundation axiom.
In [10], GFA’s were suggested for the rôle of proper (or set) fork algebras. The negative results above seem to imply that this direction is not
likely to work in any set theory. The above negative results about freegrupoid representable fork algebras GFA’s and NFA’s suggest the question
that, perhaps, we could generalize the notion of a free grupoid so much that
the above negative results would go away. We will look into this below (but
cf. also Definition 3.9).
Definition 3.7.
(i) By a pairing structure we understand the pairing structure of some
NFA (cf. Definition 2.1 (ii)).
(ii) Let U = hU, f i be a pairing structure. For x ∈ U , we let f 0 (x) = x,
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and f n+1 (x) = f (f n (x), f n (x)) for each n ∈ ω. Now, U is called
loop-free iff U |= x 6= f n (x), for all n > 0.
(iii) By a loop-freely represented fork algebra (an LFA) we understand an
NFA whose pairing structure is loop-free.
2
We note that I LFA ⊇ GFA ∪ SFA+ is a very large class of fork algebras. The only restriction made in the definition of LFA is a very natural property of pairing (or of 5 ), namely, that a 6= ha, ai and similarly
a 6= hha, ai, ha, aii etc. Clearly, this condition or equivalently, x 6= f n (x) is
true in every (reasonable generalization of a) free grupoid.
Theorem 3.8. Let K ⊆ H S P LFA be such that hP(U × U ) . . .i ∈ K,
for some U with |U | > 1. Then Eq(K) as well as S K are not finitely
axiomatizable.
The above is a quite general result. It says that, as long as we keep
our proper (or set) FA’s to be loop-free, they will not support a nice representation theorem. Theorem 3.8 applies to SFA+ , TPA, GFA etc. as special
cases.
Besides grupoids, there is a further way of “representing” pairing (i.e.
fork), namely, by using trees. Finite trees will behave just as badly as SFA+
or TPA do, cf. Theorems 3.4–3.6.
Let us see if we can save our algebras by moving from finite trees to
arbitrary (possibly infinite) trees.
Definition 3.9.
(i) A pairing structure hU, f i is tree-like iff there is a transitive and irreflexive relation < ⊆ U × U on U such that (∀x, y ∈ U )(x < f (x, y)
& y < f (x, y)).
We note that a pairing structure is tree-like iff the pair-forming behavior of f is not “circular” (like x = f (x, x) or x = f (f (x, y), y)
etc.).
(ii) By a Tree Fork Algebra (TFA) we understand an NFA A such that its
pairing structure is tree-like.
Corollary 3.10. (ZFC− ) None of I TFA, S P TFA, H S P TFA is axiomatizable by any finite set of first order formulas. Eq(TFA) is not finitely
axiomatizable.
Proof. This result and the one below are corollaries of Theorem 3.8.
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Corollary 3.11. (ZFC− ) Let K ⊆ H S P TFA be such that there exists
A ∈ K with A = P(U × U ) for some U with |U | > 1. Then S K as well as
Eq(K) are not finitely axiomatizable.
4. On the concatenation-version of Proper
Fork Algebras
Definition 4.1. Let SFA’91 denote the variant of Proper Fork Algebras
defined in [34]. U consists of finite sequences and R 5 S = {hx, y ∩zi :
xRy & xSz} . Here y ∩z is concatenation of sequences.
2
Theorem 4.2. (ZF− )
(i) SFA’91 is not axiomatizable by any set of first order formulas. (Is
not closed under taking ultraproducts, hence it is not even pseudoaxiomatizable.)
(ii) The quasi-variety generated by SFA’91 is not finitely axiomatizable.
(iii) The same as (ii) but for the variety generated by SFA’91.
(iv) The first-order theory Th (SFA’91) is not finitely axiomatizable (not
even recursively enumerable).
Theorem 4.3. (ZF− )
(i) The equational theory Eq(SFA’91) is Π11 -complete.
(ii) Eq(SFA’91) is not recursively enumerable.
Proof. Follows from the proof method of [29]+Theorem 4.5 below, as
follows. The key observation in the proof is the following. The only use
of the Axiom of Foundation in [29] was to prove that all the binary trees
generated by any element b ∈ U of the base set of an SFA and the standard
projections p, q are finite. But this finiteness is built into the definition of
SFA’91, hence it need not be proved, thus the Axiom of Foundation is not
needed. The same train of thought applies to the free grupoid version and
the finite tree version of SFA, cf. e.g. [10].
In some of the papers of the fork algebra literature, proper fork algebras (SFA’s) are defined via free grupoids (see our section 3) or via finite
trees, cf. e.g. [10]. Theorems 4.2, 4.3 above remain true for both of these
variants.
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Theorem 4.4. Theorems 4.2, 4.3 remain true for both the free grupoid
and the finite tree version of proper fork algebras in place of Eq(SFA’91).
Theorem 4.5. (ZF− ) Eq(TPA) is interpretable in Eq(SFA’91).
def
Proof. Let suc = (Id 5 Id ). Then suc is a (derived) constant symbol
of SFA’91. Further, in any SFA’91 A, suc A = {hs, s∩si: the finite sequence
s is an element of the base set U of our algebra A}. Let p and q be
relation algebraic variable symbols. Let e1 (p) be the equation expressing
that (∀α, γ ∈ U )p(γ) = α =⇒ (∃δ ∈ U )γ = ααδ and let e2 (q) express that
(∀β, γ ∈ U )[q(γ) = β =⇒ ∃δ(γ = δββ)] . Such equations e1 , e2 exist (in the
language of SFA’91).
Since SFA’91 is a discriminator class, a conjunction of equations is
(equivalent with) a single equation. Let e3 be
{p−1 ◦ q = 1, (p−1 ◦ p) ∪ (q −1 ◦ q) ≤ Id , (p ◦ p−1 ) ∩ (q ◦ q −1 ) ≤ Id } .
Let e(x̄) be an arbitrary equation in the TPA language. Since the constant
symbols p, q of TPA are not explicitly available in SFA’91, e(x̄) may not be
in the SFA’91 language. Let e(p, q, x̄) be the same equation but with p, q
regarded as variable symbols. Then e(p, q, x̄) is in the SFA’91 language.
For this e(x̄), let its SFA’91 translation tr (e(x̄)) be defined as
[e1 (p) ∧ e2 (q) ∧ e3 (p, q)] → e(p, q, x̄) .
By being in a discriminator class, tr (e) has an equivalent equational form.
In tr (e) we should also add an e4 (p, q) to the effect that the empty sequence
∅ is not in Dom(p) ∪ Dom(q). There is a special constant symbol naming
{h∅, ∅i} in SFA’91 so this is no problem.
Claim. For any TPA-equation e(x̄), TPA |= e(x̄) iff SFA’91 |= tr (e)(p, q, x̄) .
For the proof of Claim we note that for γ ∈ U , p(γ), q(γ) are strictly
shorter than γ. So for any γ ∈ U , the tree
γ
p .
p . &
..
..
.
.
q
&
..
.
q
is a finite one. The rest is relatively easy. QED (Claim)
Claim proves Theorem 4.5.
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5. Positive results
Instead of only binary relations, consider all possible finitary relations over
some set U . Let us consider the possible algebras of these.
Notation 5.1. Ref (U ) is the set of all finitary relations over U , i.e.
def
Ref (U ) = {R : for some finite n, R is an n-ary relation over U }.
Theorem 5.2. ([26], [27]) It is possible to define set theoretic operations
f1 , . . . , f9 on finitary relations such that (i), (ii) below hold.
(i) The variety REL generated by {hRef (U ), f1 , . . . , f9 i : U is a set} is
finitely axiomatizable.
(ii) First order logic is expressible in the equational language of REL.
(iii) The operations of REL are logical in the sense of [33].
Remark 5.3. There is a version of the above theorem where the set
algebras already form a variety, so we do not have to generate a variety in
the new version of (i). This version as well as the theorem above are in [27]
and in [26].
2
Further positive results are in [28]. These results are simpler, and
involve simpler classes of algebras than REL.
ACKNOWLEDGEMENT. We would like to thank Ewa Orlowska for
calling our attention to the “Fork Algebra paper” [34] during the 1991 Warsaw Banach Semester. Thanks are due to Hajnal Andréka, Steve Givant,
Viktor Gyuris, Ágnes Kurucz and András Simon for careful reading and
helpful suggestions. We would like to thank those members of the Algebraic
Logic seminar and the Non-well-founded Sets seminar who double-checked
the proofs of these results (remaining mistakes are our fault, not theirs).
Mathematical Institute of the Hungarian Academy of Sciences
Budapest, Pf. 127
H–1364, Hungary
e-mail: [email protected], [email protected]
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