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- JOEL BERMAN, PAWE L IDZIAK, PETAR MARKOVIĆ, RALPH MCKENZIE, MATTHEW VALERIOTE
- 2006

The Constraint Satisfaction Problem Dichotomy Conjecture of Feder and Vardi (1999) has in the last 10 years been profitably reformulated as a conjecture about the set SP fin (A) of subalgebras of finite Cartesian powers of a finite universal algebra A. One particular strategy, advanced by Dalmau in his doctoral thesis (2000), has confirmed the conjecture… (More)

In this paper we study first-order definability in the lattice of equational theories of commutative semigroups. In a series of papers, J. Ježek, solving problems posed by A. Tarski and R. McKenzie, has proved, in particular , that each equational theory is first-order definable in the lattice of equational theories of a given type, up to automorphism, and… (More)

Let D, ≤ be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We characterize the order ideals in D, ≤ that are well-quasi-ordered by embeddability, and thus characterize the members of D that belong to at least one infinite anti-chain in D.

We investigate definability in the set of isomorphism types of finite semilattices ordered by embeddability; we prove, among other things, that every finite semilattice is a definable element in this ordered set. Then we apply these results to investigate definability in the closely related lattice of universal classes of semilattices; we prove that the… (More)

Maltsev families of varieties which are closed under join or Maltsev product are investigated. New Maltsev conditions for congruence semi-distributivity are given.

Let D be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite distributive lattice D, the set {d, d opp } is definable, where d and d opp are the isomorphism types of D and its opposite (D turned… (More)

- ROSS WILLARD, R. McKenzie
- 2009

Let % be a class of (universal) algebras of fixed type. 5t denotes the class obtained by augmenting each member of .X by the ternary discrim-the closure of 5t under the formation of subalgebras, homomorphic images, and arbitrary Cartesian products. For example, the class of Boolean algebras is definitionally equivalent to V(Xt) where % consists of a… (More)

- Ralph McKenzie
- 2011

This is a very modest paper. My aim is to have a look at some problems that arise in the regions where lattice theory and equa-tional logic share common ground. The list of problems is selected from my own mathematical experience, and is not intended to be in any way comprehensive or definitive. Throughout the paper, L(A) denotes the lattice of equational… (More)

- M Droste, D Kuske, R Mckenzie, R Pp Oschel
- 2007

In ZFC, it is shown that every relational clone on a set A closed under complementation is a Krasner clone if and only if A is at most countable. This is achieved by solving an equivalent problem on locally invertible monoids: A partially ordered set is constructed whose endomorphism monoid is not contained in the local closure of its au-tomorphism group.

The sub algebra functor Sub A is faithful for Boolean algebras (Sub A •-Sub B implies A • B, see D. Sachs [7]), but it is not faithful for bounded distributive lattices or unbounded distributive lattices. The automorphism functor Aut A is highly unfaithful even for Boolean algebras. The endomorphism functor End A is the most faithful of all three. B. M.… (More)