Bulletin of the Section of Logic
Volume 39:1/2 (2010), pp. 93–102
Ewa Graczyńska
M -HYPERQUASI-IDENTITIES OF FINITE ALGEBRAS ∗
Abstract
In [4] (see [2]) the notion of a hypersubstitution of a given type and the notion
of a derived variety D(V ) of a given variety V of a fixed type τ were invented.
Suitable hyperequational calculus appeared as a modification of G. Birkhoff calculus with additional rule called hypersubstitution rule (6). In Graczyńska E. and
Schweigert D. [6] we considered further generalization to M -hyperquasivarieties
of a given type. Basing on the result of A. Wroński [11] we modify A. Selman
calculus [10] and the Gentzen-style calculus of [11] presenting a finite axiomatization of M -hyperquasi-identities of a finite algebra (of a finite type τ ). The
results were presented on the conference AAA79 and CYA25 Olomouc (Czech
Republic), February 12–14, 2010.
1. Notation
Our nomenclature and notation is basically those of G. Birkhoff [1], E.
Graczyńska and D. Schweigert [4], A. Selaman [10] and A. Wroński [11].
We recall some fundamental concepts.
Definition 1.1. A type τ of an algebra A = (A, {fi : i ∈ I}) = (A, F )
is a function τ : I → IN from the indexing set I into the set IN of natural
numbers, where τ (i) = ni if fi is an ni -ary operation. A type τ is called
finite if the set I is finite.
∗ AMS Mathematical Subject Classification 2010:
Primary: 08B05, 08C15, 08B99, 08C99
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We deal only with universal algebras of a given finite type τ : I → IN ,
where I is a nonempty set and IN denotes the set of all integers.
Hypersubstitutions of a given type τ were invented by D. Schweigert
and the author in [4] (cf. [2]). Shortly speaking, they are mappings sending terms to terms by substituting variables by (the same) variables and
fundamental terms of the form fi (x0 , ..., xτ (i)−1 ) by terms of the same arities, i.e. σ(x) = x for any variable x, and for a given operation symbol fi ,
for i ∈ I assume that σ(fi (x0 , ..., xτ (i)−1 )) is a given term of the same arity
as fi , then σ acts on all terms of a given type τ in an inductive way:
σ(fi (p0 , ..., pτ (i)−1) )) = σ(fi (x0 , ..., xτ (i)−1 ))(σ(p0 ), ..., σ(pτ (i)−1 )), for i ∈
I, (where fi (x0 , ..., xτ (i)−1 ) denotes a fundamental term).
Consider an algebra A and the equivalence relation ≡A on the set H(τ )
defined by: σ1 ≡A σ2 in H(τ ) iff the identity σ1 (fi (x0 , ..., xτ (i)−1 ))
≈ σ2 (fi (x0 , ..., xτ (i)−1 )) is satisfied in A for every i ∈ I.
The following can be easily proved by induction on the complexity of
term p:
Theorem 1.2. If σ1 ≡A σ2 in H(τ ) then for every term p of type τ the
identity σ1 (p) ≈ σ2 (p) is satisfied in A.
It is clear that in a finite algebra A of a finite type τ there are only
finitely many non-equivalent hypersubstitutions of type τ .
H(τ ) = (H(τ ), ◦, σid ) denotes the monoid of all hypersubstitutions σ
of a given type τ with the operation ◦ of composition and the identity
hypersubstitution σid . M is a submonoid of H(τ ).
2. Derived algebras
Let V be a class of algebras of type τ . Derived algebras of a given type τ
were defined in [4].
Definition 2.1. Let A = (A, F ) be an algebra in V and σ a hypersubstitution in H(τ ). Then the algebra B = (A, (F )σ ) is a derived algebra of
A, with the same universe A and the set (F )σ of all derived operations of
F by σ. B is then denoted as Aσ .
D(V ) denotes the class of all derived algebras of type τ of all algebras
of V .
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Definition 2.2. A variety V is called solid if and only if D(V ) ⊆ V .
Let us note, that for a finite algebra A of a finite type τ , the class
D(A) is finite.
2.1. Quasi-identities and quasivarieties of algebras
We recall the notion invented by A. I. Mal’cev from [8]:
Definition 2.3.
A quasi-identity e is an implication of the form:
(1.2.1) (t0 ≈ s0 ) ∧ ... ∧ (tn−1 ≈ sn−1 ) → (tn ≈ sn ).
where ti ≈ si are k-ary identities of a given type, for i = 0, ..., n.
A quasi-identity above is satisfied in an algebra A of a given type if and
only if the following implication is satisfied in A: given a sequence a1 , ..., ak
of elements of A. If these elements satisfy the equations ti (a1 , ..., ak ) =
si (a1 , ..., ak ) in A, for i = 0, 1, ..., n − 1, then the equality tn (a1 , ..., ak ) =
sn (a1 , ..., ak ) is satisfied in A. In that case we write:
A |= (t0 ≈ s0 ) ∧ ... ∧ (tn−1 ≈ sn−1 ) → (tn ≈ sn ).
A quasi-identity e is satisfied in a class V of algebras of a given type,
if and only if it is satisfied in all algebras A belonging to V .
A hyperquasi-identity e (of a type τ ) is the same as quasi-identity (of
type τ ). The difference between quasi-identities and hyperquasi-identities
is in satisfaction. Similarly as in [5] we accept the following modification
of the satisfaction of a quasi-identity to the notion of a hypersatisfaction in
the following way:
Definition 2.4. A hyperquasi-identity e is satisfied (is hyper-satisfied,
holds) in an algebra A if and only if the following implication holds:
if σ is a hypersubstitution of type τ and the elements a1 , ..., ak ∈ A satisfy
the equalities σ(ti )(a1 , ..., ak ) = σ(si )(a1 , ..., ak ) in A, for i ∈ {0, 1, ..., k −
1}, then the equality σ(tn )(a1 , ..., ak ) = σ(sn )(a1 , ..., ak ) holds in A. In
symbols A |=H e.
If e is satisfied (as a hyperquasi-identity) in a class V of algebras of
type τ , then we write V |=H e.
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3. Solid quasivarieties
A reformulation of the notion of quasivariety invented by A. I. Mal’cev
in [8], p. 210 to the notion of hyperquasivariety of a given type τ was
considered by D. Schweigert and the author in [5] in a natural way:
Definition 3.1. A class K of algebras of type τ is called a hyperquasivariety if there is a set Σ of quasi-identities of type τ such that K consists
exactly of those algebras of type τ that hypersatisfy all the quasi-identities
of Σ.
Definition 3.2. A quasivariety V is called solid if and only if D(V ) ⊆ V .
It was proved in [5] that the notion of hyperquasivariety and solid
quasivariety coincides. We presented there some examples and theorems of
Mal’cev type for solid quasivarieties.
Definition 3.3. A hyperquasi-identity e is M -hyper-satisfied (holds) in
an algebra A if and only if the following implication is satisfied:
If σ is a hypersubstitution of M and the elements a1 , ..., an ∈ A satisfy
the equalities σ(ti )(a1 , ..., ak ) = σ(si )(a1 , ..., ak ) in A, for i = 0, 1, ..., n − 1,
then the equality σ(tn )(a1 , ..., ak ) = σ(sn )(a1 , ..., ak ) holds in A. We say
then, that e is an M -hyperquasi-identity of A and write:
A |=M
H (t0 ≈ s0 ) ∧ ... ∧ (tn−1 ≈ sn−1 ) → (tn ≈ sn ).
M HQId(A) denotes the set of all hyperquasi-identities satisfied in A (as
M -hyperquasi-identities).
Following [11] in the sequel we use the notation:
∆ → α, for a set ∆ = {pi ≈ qi : 0 ≤ i ≤ n − 1} and α = pn ≈ qn
instead of the quasi-identity: p0 ≈ q0 ∧ ... ∧ pn−1 ≈ qn−1 → pn ≈ qn .
We adopt the convention, that an identity p ≈ q may be regarded as a
quasi-identity e of the form ∅ → p ≈ q, where ∅ denotes the empty set. A
hyperquasi-identity e is M -hyper-satisfied (holds) in a class V if and only
if it is M -hypersatisfied in any algebra of V . We write then: V |=M
H e.
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97
We modify Selman calculus [10] by adding a new rule:
(4) an M -hypersubstitution rule
(t0 ≈s0 )∧...∧(tn−1 ≈sn−1 )→(tn ≈sn )
σ(t0 )≈σ(s0 )∧...∧σ(tn−1 )≈σ(sn−1 )→σ(tn )≈σ(sn ) ,
or in an equivalent notation:
(4) an M -hypersubstitution rule
{γ0 ,...,γn−1 }→β
{σ(γ0 ),...,σ(γn−1 )}→σ(β) ,
where σ ∈ {σj : j = 1, ..., m} for any hypersubstitutions σ1 , ..., σm ∈ M of
H(τ ) such that M/ ≡A = {[σ1 ] ≡A , ..., [σm ] ≡A } ⊆ H(τ )/ ≡A . We assume
that σj are pairwise nonequivalent.
In the sequel we will use the symbol Γ for a set {γ0 , ..., γn−1 } and the
set {σ(γ0 ), ..., σ(γn−1 )} will be denoted by σ(Γ), for σ ∈ M .
We modify the rule (4) to the following rule:
(4)A
Γ→δ
σj (Γ)→σj (δ) ,
for every j = 1, 2, ..., m.
The Selman calculus [10] with added inference rule (4)A will be called
M -hyperquasi-equational calculus.
Note, that for a trivial monoid M , the rules (4) and (4)A reduces to
the identity rule.
Remark. Let us note that in case M is a trivial (i.e. 1-element) monoid
of hypersubstitutions of a given type τ , then the satisfaction |=M
H gives rise
to the satisfaction |= and the operator DM to the identity operator.
In case M = H(τ ) we get the notions of the operator D and of the
satisfaction |=H considered in [4] and [5].
4. M -solid quasivarieties
In [6] we reformulated the notion of hyperquasivariety of a given type of [5]
to the case of M -hyperquasivariety of a given type in a natural way:
Definition 4.1. A class K of algebras of type τ is called an M -hyperquasivariety if there is a set Σ of hyperquasi-identities of type τ such that K
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Ewa Graczyńska
consists exactly of those algebras of type τ that M -hypersatisfy all the
hyperquasi-identities of Σ.
We accept the following definition:
Definition 4.2. Let QV be a quasivariety, then QV is M -solid if and
only if every M -derived algebra Aσ belongs to QV , for every algebra A in
QV and σ in M , i.e.
DM (QV ) ⊆ QV
The following was proved in [6]:
Theorem 4.3. A (quasi)variety K of algebras of a given type is an M hyper(quasi)variety if and only if it is M -solid.
In [6] we presented some Mal’cev type theorems for M -hyperquasivarieties.
Let us recall from [6] that:
Remark. M HQId(A) = QId(DM (A))
5. Gentzen-style calculus
We use the notation of [11], p. 74. For every n = 1, 2, ... we accept the rule
Gn invented by A. Wroński:
(Gn )
x0 ≈x1 ,Γ→δ;...;x0 ≈xn ,Γ→δ;xn−1 ≈xn ,Γ→δ
.
Γ→δ
For every n = 1, 2, ... let Gn be Gentzen-style calculus invented in [11]
by adding the rule Gn to the calculus of A. Selman [10].
We modify Gentzen-style axiomatization Gn (A) of quasi-identities
A. Wroński to the system MHGn (A) resulting from Selman calculus by
adding the rules Gn and (4)A and all the axioms (1), (2), (3) to obtain a
finite axiomatization of all M -hyperquasi-identities of the algebra A. In
fact we could agree to call this axiomatization as hyper Gentzen-style, as
we use an additional inference rule of hypersubstitution.
Let n be a natural number and F (x1 , ..., xn ) be absolutely free algebra
of terms of type τ in variables x1 , ...xn . Let A be an algebra of type
τ whose number of elements does not exceed n. The quotient algebra
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F (x1 , ..., xn )/M HId(A) ∼
= F (x1 , ..., xn )/Id(DM (A)) is an n-generated free
algebra in the variety HSP DM (A). For every term t ∈ F (x1 , ..., xn ) we
pick a term r(t) ∈ [t]M HId(A) and call it the representative of t. Thus
whenever A |=M
H t1 ≈ t2 and V ar(t1 , t2 ) ⊆ {x1 , ..., xn } then t1 and t2 must
have the same representative i.e. r(t1 ) = r(t2 ).
In order to axiomatize the M -hyperquasi-identities of A we adopt similar axioms (1), (2) and (3) as in [11], p. 74, 75:
(1) xi ≈ r(xi ), for every i = 1, ..., n;
(2) O(r(t1 ), ..., r(tk )) ≈ r(O(t1 , ..., tk )), for every k-ary operation O of
type τ and terms t1 , ..., tk in variables x1 , ..., xn , for every k = 1, ....
The following consequences of the above definitions are similar ones to
that of (i) and (ii) of [11], p. 75:
(i) For every term t of type τ in variables x1 , ..., xn , the identity t ≈ r(t)
can be derived from axioms (1), (2) by Birkhoff calculus.
(ii) For all terms t1 ,t2 of type τ in variables x1 , ..., xn , if
M
A |=M
H t1 ≈ t2 then A |=H σ(t1 ) ≈ σ(t2 ) and all the identities σ(t1 ) ≈
σ(t2 ) can be derived from axioms (1), (2) by Birkhoff calculus, for every
σ ∈ M ⊆ H(τ ).
Similarly as in A. Wroński [11], in addition to (1) and (2) we adopt as
axioms all quasi-identities of the form:
(3) Γ → δ such that A |=M
H Γ → δ and each member of Γ ∪ {δ} is
an identity of the form r(t1 ) ≈ r(t2 ) where t1 , t2 are terms of type τ in
variables x1 , ..., xn .
It is clear from [11], p. 75 that the cardinality of axioms (1)–(3) is finite.
Moreover, for a finite algebra A of a finite type there are only finitely many
nonequivalent hypersubstitutions of type τ . For a given monoid M ⊆ H(τ )
we choose for one representative from M in every equivalence class by the
relation ≡A |M in M . Assume that σ1 , ..., σm are all representatives, where
σ1 = σid . Therefore the cardinality of (4)A is also finite.
The following observation is similar to (iii) of [11], p. 75:
(iii) For every quasi-identity Γ → δ of type τ in variables x1 , ..., xn , if
A |=M
H Γ → δ then the quasi-identity σ(Γ) → σ(δ) can be derived from
axioms (1), (2) and (3) by Selman calculus, for every σ ∈ M .
Proof. For every σ ∈ M the quasi-identities of the form σ(Γ) → σ(δ) we
get that A |=M
H σ(Γ) → σ(δ) by the definition of M -hyperquasi-satisfaction.
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From the other hand they are consequences of quasi-identities of the form
Γ → δ by the inference rules (1), (2) and (3) by Selman calculus, in the
following way: assume σ ≡A σj in M , for some j: 1 ≤ j ≤ m. Then
A |=M
H σj (Γ) → σj (δ), for 0 ≤ j ≤ m. By Selman calculus to each of such
quasi-identity one may adopt the proof of (iii) of [11], p. 75 in order to
obtain that it can be derived from axioms (1), (2) and (3) as follows:
First, using (1), (2) and Selman calculus we derive σj (Γ) → r(σj (Γ))
and r(σj (δ)) → σj (δ). We also have r(σj ((Γ)) → r(σj (δ)) by (3). Thus
the result σj (Γ) → σj (δ) can be achieved by cut and the desired result by
Selman calculus. We have the following slight modification of Theorem 2 of A. Wroński
[11], p.77:
Theorem 5.1. A |=M
H Γ → δ iff MHGn (A) proves Γ → δ
iff MHGn (A) proves σj (Γ) → σj (δ), for j = 1, ..., m
iff MHGn (A) proves σ(Γ) → σ(δ), for every σ ∈ M .
Proof. The part “if” follows from the fact that if MHGn (A) proves Γ →
σ then it proves σj (Γ) → σj (δ) for all 1 ≤ j ≤ m by the inference rule (4)A .
Moreover, each quasi-identity σ(Γ) → σ(δ), for σ ∈ M is a consequence of
quasi-identities of the form σj (Γ) → σj (δ) for some 1 ≤ j ≤ m by Selman
calculus. We conclude that every implication σ(Γ) → σ(δ) is provable from
MHGn (A), for σ ∈ M . Thus A |=M
H Γ → σ, as all the rules are admissible
in A.
To prove the “only if” part suppose that Γ → δ is a quasi-identity in
variables x1 , ..., xk , for k ∈ IN , such that A |=M
H Γ → δ, thus A |= Γ → δ by
therefore
by
a similar argument as in the
the definition of satisfaction |=M
H
proof of Theorem 2 of [11], p. 77 we conclude that Γ → δ is provable by the
calculus MHGn (A) (without an application of the rule (4)A ). Therefore
every implication of the form σj (Γ) → σj (δ) is provable by the calculus
MHGn (A), for all 0 ≤ j ≤ m. In consequence every implication of the
form σ(Γ) → σ(δ) is provable by the calculus MHGn (A), for all σ ∈ M ,
by the similar argument as in the part “if”. Remark. Let us note that the calculus MHGn (A) reduces to the calculus
Gn (A) of A. Wroński [11] for the trivial monoid M .
M -Hyperquasi-Identities of Finite Algebras
101
The next Theorem follows from Theorem above by a finite application
of hypersubstitution rule (4)A to each axiom obtained in Theorem 2. Then
the rule (4)A may be omitted as the resulting set is already closed under
that rule. This fact follows from an observation that the rule (4)A commutes with all the rules of Selman calculus and the rule (Gn ). Therefore
the closure of a set of axioms closed under (4)A under Selman and (Gn )
inference rules is already closed under that rule.
The following is a slight modification of Theorem 1 of A. Wroński [11],
p. 74:
Theorem 5.2. For every n = 1, 2, ... the calculus Gn allows for a finite
axiomatization of all M -hyperquasi-identities of every algebra of a finite
type whose number of elements does not exceed n.
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Ewa Graczyńska
[10] A. Selman, Completeness of calculi for axiomatically defined classes
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Opole University of Technology, Institute of Mathematics
ul. Luboszycka 3, 45-036 Opole, Poland
e-mail: [email protected]
http://www.egracz.po.opole.pl/