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AAA80, BĘDLEWO, POLAND June 2-6, 2010

AAA80, BĘDLEWO, POLAND June 2-6, 2010 Janusz Czelakowski Ins t itute of Mathematics and Informatics Opole University, Poland e-mail: jczel@math.uni.opole.pl R ELATIVELY CONGRUENCE MODULAR QUASIVARIETIES AND THEIR RELATIVELY CONGRUENCE- DISTRIBUTIVE SUBQUASIVARIETIES.

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AAA80, BĘDLEWO, POLAND June 2-6, 2010

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  1. 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 1

  2. 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 alli I impliesh() = h() ). 2

  3. Keq- the equational logic associated with the classK. Keqis finitary if K finite set of finite algebras. Th (Keq) – the lattice of (closed) theoriesXof Keq

  4. QUASIVARIETIES – classes Q closed underS, P, andPu. S, P, andPu - operations of forming isomorphic copies of subalgebras, direct products, and ultraproducts, respectively. Qv(K)- the smallest quasivariety containingK; K generatesthe quasivarietyQv(K). (Mal’cev).Qv(K) = SPPu(K)for any class K. 4

  5. Q is finitely generated if Q = Qv(K) for a finite set K of finite algebras (i.e., Q = SP(K) forK as above). Then Keq = Qeq. 5

  6. Q quasivariety, A algebra, and  a congruence of A. • is a Q-congruence of A if A/Q. ConQ(A) := {Con(A) : A/Q}. 6

  7. 2. THE FINITE BASIS PROBLEMFOR 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. 7

  8. General answer: The complement ofSP(K) closed under the formation of ultraproducts. (Not a manageable condition) 8

  9. CONVENTION: The letter Q ranges over quasivarieties. Q is relatively congruence-distributive (RCD) ifthe lattice of Q-congruences on each A Q is distributive 9

  10. 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 mosta single quasi-equation. 

  11. 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], ……. 11

  12. 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 (inthe signature  ) finitely based? Conviction that the right track to a solution of the problem goesthrough commutator theory. 12

  13. 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]. 13

  14. 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.

  15. 3. THE EQUATIONAL COMMUTATOR Given : Q, two positive integersmandn, two m-tuples and two n-tuples of pairwise distinct individualvariables x = x1,...,xm,y= y1,...,ym and z = z1,..., zn,w=w1,…,wn, respectively. 15

  16. (x, y, z, w, u) (x, y, z, w, u) is called a commutator equation ofQ 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. 16

  17. In particular,if m = n = 1, then (x,y, z, w, u) (x,y, z, w, u) is called a quaternarycommutator equation ofQ (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 equationsofQ (in x, y and z, w). 17

  18. 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. 18

  19. Definition 3.1. (J.C. [2006]) Q quasivariety,A Q, • and are Q-congruences onA. Theequational commutatorof  and  onArelative to Q, [, ]A, is the leastQ-congruence onAthat 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(), andeA }.  19

  20. AAA 80 NOTE. The definition of the commutator also makes sense fortheories of Qeq.(Relevant in various syntactic investigations.)

  21. Theorem 3.2.Qquasivariety, AQ, and , ConQ(A).Then: (i) [, ]Ais a Q-congruence on A; • [, ]A; • [, ]A = [, ]A; (iv) The equational commutator is monotone in both arguments, i.e.,if 1, 2and are Q-congruences on A, then12implies [1, ]A [2, ]A.  21

  22. Two questions: (1) Does the equational commutator coincides with the“standard”commutator for RCM quasivarieties, defined byKearnes and McKenzie [1992]? (2) What new facts can be proved by means of the equationalcommutator? 22

  23. 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 onQ iff it is additive on all A Q. 

  24. We need one more property of the commutator: (C2) If h : ABis a surjective homomorphism between Q-algebras and, ConQ(A),then ker(h) +Q [, ] A= h1 ( [BQ(h), BQ(h)] B). Theorem 3.4. For any quasivariety Q, if the equational commutator forQ satisfies (C1), then it satisfies (C2). 

  25. 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 quaternarycommutatorequations forQ (possiblywith parameters u = u1, …, u k, k ) such that for every algebraAQand for every pair of setsX, 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. 25

  26. AAA80 Theorem 3.6. LetQ be RCM. Then the equational commutatorfor 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 ofTheorem 3.5, there exists a set (x, y, z, w, u)of quaternary commutatorequations which generates the commutator in Q.

  27. Corollary 3.7. For any RCM Q, the commutator possesses a generating set  (x, y, z, w, u) of quaternary commutator equations.  Crucial issue: Is  (x, y, z, w, u) finite? Theorem 3.8. LetQbe finitely generated (i.e., Q = SP(K)for some finite set K of finite algebras) with the additive equational commutator. The equational commutator in Q is generated by a finite set (x, y, z, w, u) of quaternary commutator equations. 

  28. Example. BA the variety of Boolean algebras, BA = SP(2). As BAis 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 Boolean operations of equivalence and join, respectively. 1 stands for the unit element.)

  29. 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. 

  30. This means that: For every Boolean algebraAand 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}).

  31. 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 = .

  32. A Q is prime (in Q) if 0Ais 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.

  33. Theorem 4.2.Let Q be quasivariety with the additive equational commutator anda 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). 

  34. AAA 80 Theorem 4.3.Let Q be a quasivariety with theadditiveequational commutator. (In particular, let Q be RCM.)The class SP(QPRIME) is the largestRCD quasivariety included in Q.

  35. MoreoverQPRIMEcoincides with the class of all relativelyfinitely subdirectly irreducible algebras in SP(QPRIME). SP(QPRIME) is axiomatized by any basis forQ augmented with a single quasi-identity.

  36. 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.

  37. 5. ITERATION PROCEDURES INVOLVING GENERATING SETS OF QUATERNARY COMMUTATOR EQUATIONS Q finitely generated. = (x, y, z, w, u) finite generating set of quaternarycommutator equations for Q. (x, y, z, w, u) –behaves like a generalizeddisjunction in AAL for sentential logics (but does not share all properties ofdisjunction like e.g. idempotency) binary operation symbol. Iterations: x1x2, (x1x2) x3, ((x1x2) x3) x4.

  38. We do the same with the set (x, y, z, w, u) (parameters may occur!) 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.

  39. Then a sequence of sets of equations 1, 2,… is inductively defined so that nis built in the variables x1,…, xn + 1, y1,…, yn + 1for n = 1,2,… . n = n(x1, y1,…, xn + 1, yn + 1).

  40. 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 replacing of 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,…, xn + 1, yn + 1), for all n.

  41. Theorem 5.2. Qfinitely generated with the additive equational commutator, (x, y, z, w, u) finite generating set of quaternary commutator equations. Suppose Q is generated by a finite class of algebras each ofwhich has at most m 1 elements, where m – 1 is a positiveinteger. 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, vn - 1). 

  42. 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, v5) is (*) 5(x1, x2, x1, x3, x1, x4, x2, x3, x2, x4, x3, x4, v5)

  43. 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, v5) . 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 terminatesin finitely many steps(further steps yield only secondary quasi-identities).

  44. Bibliography K. Baker [1974] Primitive satisfaction and equational problems for lattices and other algebras, Transactions of the American Mathematical Society 190, 125-150. [1977] Finite equational bases for finite algebras in a congruence distributive equational class, Advances in Mathematics 24, 207 – 243. W.J. Blok and D. Pigozzi [1986] A finite basis theorem for quasivarieties, Algebra Universalis 22, 1-13. J. Czelakowski [1998] Relative principal congruences in congruence-modular quasivarieties, Algebra Universalis 39, 81-101. [2006] General theory of the commutator for deductive systems. Part I. Basic facts, Studia Logica 83, 183-214. [2008] Additivity of the commutator and residuation, Reports on Mathematical Logic 43, 109-132.

  45. W. Dziobiak, M. Maróti, R. McKenzie, A. Nurakunov [2009] The weak extension property and finite axiomatizability of quasivarieties, Fundamenta Mathematicae 202, No.3, 199-222. R. Freese and R. McKenzie [1987] "Commutator Theory for Congruence Modular Varieties", London Mathematical Society Lecture Note Series 125, Cambridge University Press, Cambridge-New York. V. Gorbunov [1984] A characterization of residually small quasivarieties (in Russian), Doklady Akademii Nauk SSSR 275, 204-207. H.P. Gumm [1983] "Geometrical Methods in Congruence Modular Algebras", Memoirs of the American Mathematical Society 45, No. 286.

  46. H.P. Gumm and A. Ursini [1984] Ideals in universal algebras, Algebra Universalis 19, 45-54. J. Hagemann and C. Herrmann [1979] A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. der Math. (Basel) 32, 234-245. C. Herrmann [1979] Affine algebras in congruence modular varieties, Acta Sci. Math. (Seged) 41, 119-125. K. Kearnes [1990] Relatively congruence distributive subquasivarieties of a congruenece modular variety, Bulletin of the Australian Mathematical Society 41, 87-96. K. Kearnes and R. McKenzie [1992] Commutator theory for relatively modular quasivarieties,Transactions of the American Mathematical Society 331, No. 2, 465 – 502.

  47. K. Kearnes and Á. Szendrei [1998] The relationship between two commutators, Iternational Journal of Algebra and Computation 8, 497-531. R. Lyndon [1951] Identities in two-valued calculi, Transactions of the American Mathematical Society 71, 457 – 465. [1954] Identities in finite algebras, Proceedings of the American Mathematical Society 5, 9-9. M. Maróti and R. McKenzie [2004] Finite basis problems and results for quasivarieties, Studia Logica 78, 293-320. R. McKenzie [1987] Finite equational bases for congruence modular varieties, Algebra Universalis 24, 224-250.

  48. A. Nurakunov [2001] Quasi-identities of congruence-distributive quasivarieties of algebras, Sibirian Math. Journal 42, 108-118. D. Pigozzi [1988] Finite basis theorems for relatively congruence-distributive quasivarieties, Transactions of the American Mathematical Society 310, No. 2, 499-533. A. Ursini [1984] Prime ideals in universal algebra, Acta Universitatis Carolinae – Mathematica et Physica 25, No.1, 75-87. R. Villard [2000] A finite basis theorem for residually small, congruence meet-semidistributive varieties, Journal of Symbolic Logic 65, 187-200. [2001] Extending Baker’s theorem, Algebra Universalis 45, 335-344.

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