We prove that when a digraph G has a Maltsev polymorphism, then G also has a ma jority polymorphism. We consider the consequences of this result for the structure of Maltsev digraphs and the complexity of the Constraint Satisfaction Problem.

This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but that in general these problems are EXPTIME complete. In addition, we provide sharp bounds in terms of the siz...

A kernelization algorithm for a computational problem is a procedure which compresses an instance into an equivalent instance whose size is bounded with respect to a complexity parameter. For the constraint satisfaction problem (CSP), there exist many results concerning upper and lower bounds for kernelizability of specific problems, but it is safe to say that we lack general methods to determi...

The theory of protomodular categories provides a simple and general context in which the basic theorems needed in homological algebra of groups, rings, Lie algebras and other non-abelian structures can be proved [2] [3] [4] [5] [6] [7] [9] [20]. An interesting aspect of the theory comes from the fact that there is a natural intrinsic notion of normal monomorphism [4]. Since any internal reflexi...

We develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvability theory for the nonmodular TC commutator. These theories are used to show that each of the following sets of statements are equivalent for a variety V of algebras. (I) (a) V satisfies a nontrivial congruence identity. (b) V satisfies an idempotent Maltsev cond...

We improve on and generalize a 1960 result of Maltsev. For field $F$, we denote by $H(F)$ the Heisenberg group with entries in $F$. Maltsev showed that there is copy $F$ defined $H(F)$, using existential formulas an arbitrary non-commuting pair $(u,v)$ as parameters. show interpreted computable $\Sigma_1$ no give two proofs. The first existence proof, relying Harrison-Trainor, Melnikov, R. Mill...

The Birkhoff-Maltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasiva-riety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the Birkhoff-Maltsev problem are also considered, including ones that stem from the ...

Let B be a finite, core relational structure and let A be the algebra associated to B, i.e. whose terms are the operations on the universe of B that preserve the relations of B. We show that if A generates a so-called arithmetical variety then CSP(B), the constraint satisfaction problem associated to B, is solvable in Logspace; in fact ¬CSP(B) is expressible in symmetric Datalog. In particular,...

Let A be an idempotent algebra, α ∈ ConA such that A/α has few subpowers, and m be a fixed natural number. There is a polynomial time algorithm that can transform any constraint satisfaction problem over A with relations of arity at most m into an equivalent problem which is m consistent and in which each domain is inside an α block. Consequently if the induced algebras on the blocks of α gener...