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AAA80, BĘDLEWO, POLAND

June 2-6, 2010

Janusz Czelakowski Institute of Mathematics and Informatics

Opole University, Poland

e-mail: jczel@math.uni.opole.pl

RELATIVELY CONGRUENCE MODULAR QUASIVARIETIES

AND THEIR RELATIVELY CONGRUENCE- DISTRIBUTIVE SUBQUASIVARIETIES

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EQUATIONAL LOGICS

K - class of algebras.

Keq – the consequence operation on the set of equations Eq() determined by the class K:

Keq ({ i i : i I}) if and only if

(A K)(h Hom(Te, A)) (h( i ) = h( i) for all i I

implies h( ) = h( ) ).

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Keq - the equational logic associated with

the class K.

Keq is finitary if K finite set of finite algebras.

Th (Keq ) – the lattice of (closed) theories X of Keq

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QUASIVARIETIES – classes Q closed under S, P, and Pu.

S, P, and Pu - operations of forming isomorphic copies of

subalgebras, direct products, and ultraproducts, respectively.

Qv(K) - the smallest quasivariety containing K;

K generates the quasivariety Qv(K).

(Mal’cev). Qv(K) = SPPu(K) for any class K.

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Q is finitely generated if Q = Qv(K) for a finite

set K of finite algebras

(i.e., Q = SP(K) for K as above).

Then Keq = Qeq.

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Q quasivariety, A algebra, and a congruence

of A.

is a Q-congruence of A if A/ Q.

ConQ(A) := { Con(A) : A/ Q}.

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2. THE FINITE BASIS PROBLEM FOR QUASIVARIETIES

Many positive results for varieties:

Lyndon [1951, [1954], Baker [1974], [1977], Jónsson [1978],

McKenzie [1987],….

Problem: K finite set of finite algebras.

Find plausible conditions implying that the quasivariety

SP(K) is characterized by a finite set of quasi- identities.

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General answer:

The complement of SP(K) closed under the formation of ultraproducts.

(Not a manageable condition)

 

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CONVENTION: The letter Q ranges over quasivarieties.

Q is relatively congruence-distributive (RCD) if the lattice of Q-congruences on each A Q is distributive

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Main result:

Theorem 2.1 (Pigozzi [1988]). finite signature.

Every finitely generated RCD quasivariety Q is finitely based.

Improvement by Czelakowski and Dziobiak [1992]:

Q has a base consisting of a finite set of equations and at most a single quasi-equation.

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AAA80, Będlewo, June 2-6, 2010

Generalizations of Theorem 2.1:

R. Villard [2000],

M. Maróti and McKenzie[2004],

A. Nurakunov [2001]

W. Dziobiak, M. Maróti, R. McKenzie, A. Nurakunov [2009], …….

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Q is relatively congruence-modular (RCM) if the lattice

of Q-congruences on each A Q is modular.

The crucial unsolved problem: Let be a finite algebraic

signature. Is every finitely generated RCM quasivariety

(in the signature ) finitely based?

Conviction that the right track to a solution of the problem

goes through commutator theory.

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Commutator theory – well developed for varieties

(J.D.P. Smith, J. Hagemann, C. Herrmann, H.P. Gumm,

W. Taylor, R. Freese, R. McKenzie, K. Kearnes,

Á. Szendrei ….).

Commutator for quasivarieties – pioneering work of

Kearnes and McKenzie [1992].

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J.C. [2006] – new definition of the commutator within

the framework of AAL (fits for an arbitrary deductive system

in any dimension).

Below – how it works for quasivarieties and how it is

related to the commutator in the sense adopted in universal

algebra.

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3. THE EQUATIONAL COMMUTATOR

Given : Q,

two positive integers m and n,

two m-tuples and two n-tuples of pairwise distinct individual variables

x = x1,..., xm, y = y1,..., ym

and

z = z1,..., zn, w = w1,…, wn,

respectively.

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(x, y, z, w, u) (x, y, z, w, u)

is called a commutator equation of Q in the variables x, y and

z, w if

(x, x, z, w, u) (x, x, z, w, u)

and

(x, y, z, z, u) (x, y, z, z, u)

are valid in Q.

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In particular, if m = n = 1, then

(x, y, z, w, u) (x, y, z, w, u)

is called a quaternary commutator equation of Q (in the

variables x, y and z, w) if

(x, x, z, w, u) (x, x, z, w, u)

and

(x, y, z, z, u) (x, y, z, z, u)

are valid in Q.

Qeq (x y) Qeq (z w)

is the set of quaternary commutator equations of Q

(in x, y and z, w).17

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If is a congruence of A and

a = a1,..., am , b = b1,..., bm Am,

we write a b ()

to indicate ai bi () for i = 1,..., m.

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Definition 3.1. (J.C. [2006]) Q quasivariety, A Q, and are Q-congruences on A.

The equational commutator of and on A relative to Q,

[, ]A,

is the least Q-congruence on A that includes the pairs:

{ (a, b, c, d, e), (a, b, c, d, e) : (x, y, z, w, u) (x, y, z, w, u)

is a commutator equation for Q, a b (), c d (),

and e A }.

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AAA 80

NOTE. The definition of the commutator also makes sense for theories of Qeq. (Relevant in various syntactic investigations.)

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Theorem 3.2. Q quasivariety, A Q, and

, ConQ (A). Then:

(i) [, ]A is a Q-congruence on A;

(ii) [, ]A ;

(ii) [, ]A = [, ]A;

(iv) The equational commutator is monotone in both arguments, i.e., if 1, 2 and are Q-congruences on A, then 1 2 implies [1, ]A [2, ]A .

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Two questions:

(1) Does the equational commutator coincides

with the “standard”commutator for RCM quasivarieties, defined

by Kearnes and McKenzie [1992]?

(2) What new facts can be proved by means of the

equational commutator?

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Definition 3.3. Q quasivariety, A Q. The equational

commutator is additive on A if

(C1) [supQ {i : i I }, ] A = supQ{ [i, ] A : i I}

in the lattice ConQ (A).

The equational commutator is additive on Q iff it is additive

on all A Q.

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We need one more property of the commutator:

(C2) If h : A B is a surjective homomorphism between

Q-algebras and , ConQ (A), then

ker(h) +Q [, ] A = h 1 ( [BQ (h), B

Q (h)] B ).

Theorem 3.4. For any quasivariety Q, if the equational

commutator for Q satisfies (C1), then it satisfies (C2).

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Theorem 3.5. The following are equivalent for any Q:

(1) The equational commutator is additive on Q;

(2) There exists a set (x, y, z, w, u) of quaternary commutator equations for Q (possibly with parameters u = u1, …, u k,

k ) such that for every algebra A Q and for every pair of sets X, Y A2,

[Q (X), Q (Y)]A = Q({ (a, b, c, d, e), (a, b, c, d, e):

, a, b X, c, d Y, e Ak}). If (2) holds, (x, y, z, w, u) is said to generate the equational commutator in the algebras of Q.

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AAA80Theorem 3.6. Let Q be RCM. Then the equationalcommutator for Q and the commutator for Q in the sense of Kearnes-McKenzie coincide.

But the crucial fact concerning the Kearnes-McKenzie commutator is that any RCM quasivariety it satisfies (C1).

Thus, in view of Theorem 3.5, there exists a set (x, y, z, w, u) of quaternary commutator equations which generates the commutator in Q.

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Corollary 3.7. For any RCM Q, the commutator possesses a generating set (x, y, z, w, u) of quaternary commutatorequations.

Crucial issue: Is (x, y, z, w, u) finite?

Theorem 3.8. Let Q be finitely generated (i.e., Q = SP(K) forsome finite set K of finite algebras) with the additive equationalcommutator.

The equational commutator in Q is generated by a finite set(x, y, z, w, u) of quaternary commutator equations.

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Example. BA the variety of Boolean algebras, BA = SP(2). As BA is congruence-distributive, the commutator of any two

congruences on a Boolean algebra coincides with the meet of the two congruences.

Let be the equation

(x y) (z w) 1.

(Here , and 1 stand for the operation symbols of Booleanoperations of equivalence and join, respectively. 1 stands for

the unit element.) 

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is a quaternary commutator equation for BA.

The singleton set

(x, y, z, w) := { }

generates the (equational) commutator for BA.

( x, y, z, w) does not contain parametric variables.

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This means that:

For every Boolean algebra A and for every pair of sets

X, Y A2,

[(X), (Y)]A =

({ (a, b, c, d), (a, b, c, d): a, b X, c, d Y}).

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4. PRIME ALGEBRAS

Definition 4.1. Q – quasivariety, A Q, - Q-congruence on A.

is prime (in the lattice ConQ(A)) if, for any

1, 2 ConQ(A),

[1, 2]A = implies 1 = or 2 = .

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A Q is prime (in Q) if 0A is prime in ConQ(A),

i.e., [1, 2]A = 0A holds for no pair of nonzero congruences 1, 2 ConQ(A).

QPRIME - the class of prime algebras in Q.

Generally QPRIME QRFSI.

If Q is RCD, then QPRIME = QRFSI.

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Theorem 4.2. Let Q be quasivariety with the additive

equational commutator and a generating set (x, y, z, w, u).

Suppose A Q. The following conditions are equivalent:

(i) A is prime.

(ii) A (⊨ xyzw)( (u) (x, y, z, w, u) x y z w).

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Theorem 4.3. Let Q be a quasivariety with the additive equational commutator. (In particular, letQ be RCM.) The class

SP(QPRIME)

is the largest RCD quasivariety included in Q.

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Moreover QPRIME coincides with the class of all

relatively finitely subdirectly irreducible algebras in

SP(QPRIME).

SP(QPRIME) is axiomatized by any basis for Q

augmented with a single quasi-identity.

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R - the variety of rings. (The existence of unit is not assumed.)

R is congruence permutable and hence congruence modular.

Dziobiak [1990] - various characterizations of RCD

quasivarieties contained in R.

Kearnes [1990] - other characterizations of RCD

subquasivarieties of congruence modular varieties.

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5. ITERATION PROCEDURES INVOLVING GENERATING SETS OF QUATERNARY COMMUTATOR EQUATIONS

Q finitely generated. = (x, y, z, w, u) finite generating set of quaternary commutator equations for Q.

(x, y, z, w, u) – behaves like a generalized disjunction in AAL for sentential logics (but does not share all properties of disjunction like e.g. idempotency)

binary operation symbol. Iterations:

x1 x2,

(x1 x2) x3,

((x1 x2) x3) x4.

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We do the same with the set (x, y, z, w, u) (parameters mayoccur!)To simplify matters parameters are discarded. So

= (x, y, z, w)

Two infinite sequences of variables (different from x, y, z and w).

x1, x2, x3, …, xn, xn + 1, …

y1, y2, y3, …, yn, yn + 1, ….

are selected.

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Then a sequence of sets of equations

1, 2,…

is inductively defined so that n is built in the variables

x1,…, xn + 1 , y1,…, yn + 1 for n = 1,2,… .

n = n (x1, y1,…, x n + 1, y n + 1).

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Defintion 5.1.

(a) 1(x1, y1, x2, y2) := (x/ x1, y/y1, z/ x2, w/y2),

i.e., 1(x1, y1, x2, y2) is the result of the uniform replacingof x by x1, y by y1, z by x2, w by y2. (b) n + 1: = { (x/, y/, z/x n + 1, w/y n + 1) : n }, for all n 1.

Since the set (x, y, z, w) is finite, so are the sets n (x1, y1,…, x n + 1, y n + 1), for all n.

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Theorem 5.2. Q finitely generated with the additive equationalcommutator, (x, y, z, w, u) finite generating set of quaternarycommutator equations. Suppose Q is generated by a finite classof algebras each of which has at most m 1 elements, where m – 1 is a positive integer. Let

x 1,… , x m

be a sequence of individual variables of length m and let

y i, z i (1 i n, n = m(m 1)/ 2)

be an enumeration of the pairs xi, x j, where 1 i j m. Then Q obeys the equations

n - 1 (y1, z1, y2, z2,…, yn, zn, v n - 1).

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For instance for m = 4, and the variables

x1, x2, x3, x4,

we have that n = 6 and we get 6 pairs

x1, x2, x1, x3, x1, x4, x2, x3, x2, x4,

x3, x4.

The resulting set of equations obtained from

5 (x1, y1, x2, y2, x3, y3, x4, y4, x5, y5, x6, y6, v 5)is(*) 5 (x1, x2, x1, x3, x1, x4, x2, x3, x2, x4, x3, x4, v 5)

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Thus if Q generated by a finite set of at most three-element

algebras, Q satisfies equations

(*) 5 (x1, x2, x1, x3, x1, x4, x2, x3, x2, x4, x3, x4, v 5) .

Theorem 5.1 is a key element of a procedure of producing

certain iterations of quasi-identities of Q .

Theorem 5.1 implies that this procedure terminates in finitely

many steps (further steps yield only secondary quasi-identities).

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