# Four Unsolved Problems in Congruence Permutable Varieties

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1 Four Unsolved Problems in Congruence Permutable Varieties Ross Willard University of Waterloo, Canada Nashville, June 2007 Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

2 Congruence permutable varieties Definition A variety V is congruence permutable (or CP) if for each A V, Con A is a lattice of permuting equivalence relations. θ, ϕ permute if θ ϕ = θ ϕ = ϕ θ. Examples of CP varieties: Any variety of... groups expansions of groups (e.g., rings, modules, non-associative rings, near rings, boolean algebras, etc.) But not: quasi-groups in the language {, /, \} lattices, semilattices, semigroups, unary algebras. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

3 Basic facts about CP varieties Fact 1 CP congruence modularity. Fact 2 (Mal tsev, 1954) For a variety V, TFAE: V is CP. V has a term m(x, y, z) satisfying, in all A V, m(x, x, z) = z and m(x, z, z) = x ( ) Definitions Mal tsev term: a term m(x, y, z) satisfying ( ). Mal tsev algebra: an algebra having a Mal tsev term. Mal tsev variety: a variety having a common Mal tsev term. Fact 2 (restated) CP varieties = Mal tsev varieties. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

4 Aim of lecture Mal tsev algebras and varieties are... not far removed from groups, rings, near-rings, quasi-groups, etc... old-fashioned, solved. Aim of this lecture: to correct this perception, by stating some open problems that: are general are of current interest are open are ripe for study in Mal tsev algebras and varieties. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

5 1. Subpower membership problem Fix a finite algebra A. Subpower membership problem for A Input: X A n and f A n (n 1) Question: is f Sg A n(x )? How hard can it be? HARD: Naive algorithm is EXPTIME There is no better algorithm (Friedman 1982; Bergman et al Added in proof: Kozik, announced 2007). However, for groups and rings the problem is solvable in polynomial time. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

6 Subpower membership problem for groups (adapted from Sims 1971; Furst, Hopcroft, Luks 1980) Fix a finite group G. Suppose H G n. Consider where H = H (0) H (1) H (n) = {e} H (i) = {g H : g = (e,..., e,,..., )}. } {{ } i Let M i be a transversal for the cosets of H (i) in H (i 1), including ê. Concretely: 1 g M i g = (e,..., e, a,,..., ) H. } {{ } i 1 2 Every such form witnessed in H is represented in M i exactly once. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

7 Put M = n i=1 M i. Facts: 1 M is small ( M = O(n)) 2 M = H. In fact, H = M 1 M 2 M n every element h H is uniquely expressible in the form h = g 1 g 2 g n with each g i M i. ( Canonical form ) 3 Given h H, we can find g i M i recursively, efficiently (knowing M). 4 Same algorithm tests arbitrary f G n for membership in H. 5 Thus the subpower membership problem for G is solvable in polynomial time if, given X G n, we can find such an M for H = X. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

8 Finding M. Rough idea. Given X G n : Start with M i = {ê} for each i (so M = {ê}). For each g X, attempt to find the canonical form for g relative to M. (Will fail.) Each failure suggests an addition to some M i. The addition is always from X. Action: increment this M i by the suggested addition. Repeat until each g X passes; i.e., X M 1 M 2 M n. Next, for each g, h M, attempt to find the canonical form for gh. Make additions to appropriate M i upon each failure. Loop until g, h M gh passes. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

9 When to stop: Lemma M 1 M 2 M n = X as soon as g, h M gh M 1 M 2 M n. Corollary The subpower membership problem is solvable in polynomial time for any finite group G. Remark. Similar technique works for any expansion of a group by multilinear operations (e.g., rings, modules, nonassociative rings). Corollary The subpower membership problem is solvable in polynomial time for any finite ring or module. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

10 Partial generalization to Mal tsev algebras (Adapted from A. Bulatov and V. Dalmau, A simple algorithm for Mal tsev constraints, SIAM J. Comput. 36 (2006), ) Fix a finite algebra A with Mal tsev term m(x, y, z). Definition An index for A n is a triple (i, a, b) {1, 2,..., n} A A. Definition A pair (g, h) A n A n witnesses (i, a, b) if g = (x 1,..., x i 1, a,,..., ) h = (x 1,..., x i 1, b,,..., ) Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

11 Consider B A n. Definition A structured signature for B is an n-tuple (M 1,..., M n ) where 1 (i = 1): M 1 B Each form (a,,..., ) B is represented exactly once in M 1. 2 (2 i n): M i B 2 Each (g, h) M i witnesses some index (i, a, b). Each index (i, a, b) witnessed in B is represented exactly once in M i Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

12 Suppose (M 1,..., M n ) is a structured signature for B A n. Let M be the set of all g A n mentioned in (M 1,..., M n ). Facts: 1 (M 1,..., M n ) and M are small ( M = O(n)) 2 Sg A n(m) = B. 3 In fact, every element h B is expressible in the canonical form h = m(m( m(m(f 1, g 2, h 2 ), g 3, h 3 ), ), g n, h n ) with f 1 M 1 and (g i, h i ) M i for 2 i n. Note: can also require g 2 (2) = f 1 (2) g 3 (3) = m(f 1, g 2, h 2 )(3), etc. 4 f 1, g 2, h 2,..., g n, h n as above are unique for h and can be found recursively and efficiently. 5 Same algorithm tests arbitrary f A n for membership in B. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

13 This was enough for Bulatov and Dalmau to give a simple polynomial-time solution to the CSP problem with Mal tsev constraints. Question: What about the subpower membership problem? Suppose X A n and put B = Sg A n(x ). We can mimic the group algorithm by attempting to grow a structured signature for B. Sticking point: knowing when to stop. Problem 1 Using structured signatures or otherwise, is the Subpower Membership Problem for finite Mal tsev algebras solvable in polynomial time? Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

14 2. The Pixley Problem Recall: An algebra is subdirectly irreducible (or s.i.) if it cannot be embedding in a direct product of proper homomorphic images. (Equivalently, if its congruence lattice is monolithic.) Definition A variety V is a Pixley variety if: its language is finite every s.i. in V is finite (i.e., V is residually finite) V has arbitrarily large (finite) s.i. s. Question (Pixley, 1984): Is there a congruence distributive Pixley variety? Answer (Kearnes, W., 1999): No. Problem: Generalize. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

15 What is the situation for groups, rings, etc.? 1 Commutative rings with 1. No Pixley varieties here, since principal ideals are first-order definable. 2 Groups. Ol shanskii (1969) described all residually finite varieties of groups. None are Pixley varieties. 3 Rings (with or without 1). McKenzie (1982) analyzed all residually small varieties of rings. None are Pixley varieties. 4 Modules. Goodearl (priv. comm.): if R is an infinite, f.g. prime ring for which all nonzero ideals have finite index, then all nonzero injective left R-modules are infinite. Kearnes (unpubl.): hence no variety of modules is Pixley. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

16 Commutator Theory Mal tsev varieties (and congruence modular varieties) have a well-behaved theory of abelianness, solvability, centralizers and nilpotency. Fundamental notions: θ centralizes ϕ (θ, ϕ Con A), i.e., [θ, ϕ] = 0. ϕ c = largest θ which centralizes ϕ. Frequently important: if A is s.i.: 1 Con A = µ c µ 0 Fact: if V is a CM Pixley variety, then (by the Freese-McKenzie theorem) for every s.i. in V, µ c is abelian. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

17 An argument Suppose V is a congruence modular variety in a finite language and having arbitrarily large finite s.i. s. Case 1: There exist arbitrarily large finite s.i. s A V with A/µ c bounded. Use the module result to get an infinite s.i. A V with A/µ c bounded. Case 2: Else. Hence: Define C(x, y, z, w) Cg(x, y) centralizes Cg(z, w). Assume C(x, y, z, w) is first-order definable in V. Then use compactness to get an s.i. A V with A/µ c infinite. Theorem (Kearnes, W., unpubl.) If V is congruence modular and C(x, y, z, w) is definable in V, then V is not a Pixley variety. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

18 Notes: Previous theorem handles all varieties of groups, rings and modules. Doesn t handle varieties of non-associative rings. Problem 2 Does there exist a congruence permutable Pixley variety? What about varieties of non-associative rings? Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

19 3. McNulty s Problem Definition A variety V is strange if its language is finite. V is locally finite. V is not finitely based. There exists a finitely based variety W having exactly the same finite members as V. Definition A finite algebra is strange if the variety it generates is. Question (Eilenberg, Schützenberger, 1976): Does there exist a strange finite algebra? McNulty has asked the same question for varieties. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

20 Lemma (Cacioppo, 1993) If A is strange, then it is inherently nonfinitely based (INFB). Theorem (McNulty, Székely, W., 2007?) If A can be shown to be INFB by the shift automorphism method, then A is not strange. Examples of algebras known to be INFB but not by the shift automorphism method: 1 (Added in proof thank you, George): INFB Semigroups. Characterized by Sapir; George has checked that none are strange. 2 Isaev s non-associative ring (1989). That s it! Problem 3 1 Is Isaev s algebra strange? 2 Find more INFB algebras that are expansions of groups. Are any of them strange? Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

21 4. Dualizability Definition A finite algebra A M is dualizable if there exists an alter ego M partial operations... relations... discrete topology ISP and IS c P contravariant hom-functors dual adjunction (D, E, e, ε)... AARRRGGHH!!! STOP THE INSANITY!! Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

22 Dualizability All that you need to know about dualizability (but were afraid to ask): Dualizability is a property that a finite algebra may, or may not, have. In practice, dualizability coincides with an apparently stronger property, called finite dualizability. By a theorem of Zádori and myself, finite dualizability can be characterized in purely clone-theoretic terms. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

23 Classical clone theory Fix a finite algebra A. Recall that: 1 Inv(A) := {r A n : r A n, n 1}. 2 Inv(A) determines Clo(A), in the sense that f : A n A, f Clo(A) iff f preserves every r Inv(A). 3 Can speak of a subset R Inv(A) determining Clo(A) Clo(A) being finitely determined. Old Theorem The following are equivalent: R determines Clo(A) Every r Inv(A) can be defined from R by a &atomic formula. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

24 Partial operations with c.a.d. domains Fix A. A subset D A n is c.a.d. (conjunction-atomic-definable) if it is definable in A by a &atomic formula. Definition Clo cad (A) := {all restrictions of term operations of A to c.a.d. domains}. Then: 1 Inv(A) determines Clo cad (A), in the same sense: f : D A with c.a.d. domain, f Clo cad (A) iff f preserves every r Inv(A). 2 Can speak of a subset R Inv(A) determining Clo cad (A) Clo cad (A) being finitely determined. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

25 Lemma/Definition The following are equivalent: 1 A is finitely dualizable ( dualizable) 2 Clo cad (A) is finitely determined. 3 There is a finite set R Inv(A) such that every hom-transparent r Inv(A) is &atomic definable from R. Def. r Inv(A) is hom-transparent (or balanced) if Every homomorphism h : r A is a coordinate projection, and No two coordinate projections are the same. Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

26 Dualizability problem: which finite A are (finitely) dualizable? 1 CD case: (finitely) dualizable A has a near-unanimity term by Baker-Pixley, by (Davey, Heindorf, McKenzie, 1995) 2 Commutative rings with 1: (finitely) dualizable R generates a residually small variety. (Clark, Idziak, Sabourin, Szabó, W., 2001) 3 Groups: (finitely) dualizable G generates a residually small variety. by (Quackenbush, Szabó, 2002), by (Nickodemus, 2007?) 4 Rings (with or without 1): (finitely) dualizable? R generates a residually small variety. by (Szabó, 1999), by recent work of Kearnes, Szendrei? 5 But: if G = S 3, then G G is not dualizable, yet generates a residually small variety (Idziak, unpubl., 1994) expansion of (Z 4, +) that is (finitely) dualizable, yet generates a residually large variety (Davey, Pitkethly, W., 2007?) Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

27 Problem 4 1 Which finite Mal tsev algebras are (finitely) dualizable? Can we at least answer this for expansions of groups? 2 Is the answer to (1) decidable? Ross Willard (Waterloo) Four Unsolved Problems Nashville, June / 27

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