arXiv subjects

Clifford Bergman

Publications and source records attributed to Clifford Bergman.

4 recordsLinked to original sources

arXiv2026

Semilattice sums of algebras and Mal'tsev products of varieties

The Mal'tsev product of two varieties of similar algebras is always a quasivariety. We consider the question of when this quasivariety is a variety. The main result asserts that if $\mathcal{V}$ is a strongly irregular variety with no nullary operations and at least one non-unary operation, and $\mathcal{S}$ is the variety, of the same type as $\mathcal{V}$, equivalent to the variety of semilattices, then the Mal'tsev product $\mathcal{V} \circ \mathcal{S}$ is a variety. It consists precisely of semilattice sums of algebras in $\mathcal{V}$. We derive an equational base for the product from an equational base for $\mathcal{V}$. However, if $\mathcal{V}$ is a regular variety, then the Mal'tsev product may not be a variety. We discuss various applications of the main result, and examine some detailed representations of algebras in $\mathcal{V} \circ \mathcal{S}$.

math.RA↗

arXiv2022

Universal Algebraic Methods for Constraint Satisfaction Problems

After substantial progress over the last 15 years, the "algebraic CSP-dichotomy conjecture" reduces to the following: every local constraint satisfaction problem (CSP) associated with a finite idempotent algebra is tractable if and only if the algebra has a Taylor term operation. Despite the tremendous achievements in this area (including recently announce proofs of the general conjecture), there remain examples of small algebras with just a single binary operation whose CSP resists direct classification as either tractable or NP-complete using known methods. In this paper we present some new methods for approaching such problems, with particular focus on those techniques that help us attack the class of finite algebras known as "commutative idempotent binars" (CIBs). We demonstrate the utility of these methods by using them to prove that every CIB of cardinality at most 4 yields a tractable CSP.

cs.LO↗

arXiv2019

Random Models of Idempotent Linear Maltsev Conditions. I. Idemprimality

We extend a well-known theorem of Murski\vı to the probability space of finite models of a system $\mathcal{M}$ of identities of a strong idempotent linear Maltsev condition. We characterize the models of $\mathcal{M}$ in a way that can be easily turned into an algorithm for producing random finite models of $\mathcal{M}$, and we prove that under mild restrictions on $\mathcal{M}$, a random finite model of $\mathcal{M}$ is almost surely idemprimal. This implies that even if such an $\mathcal{M}$ is distinguishable from another idempotent linear Maltsev condition by a finite model $\mathbf{A}$ of $\mathcal{M}$, a random search for a finite model $\mathbf{A}$ of $\mathcal{M}$ with this property will almost surely fail.

math.LO↗

arXiv2015

Commutative, idempotent groupoids and the constraint satisfaction problem

A restatement of the Algebraic Dichotomy Conjecture, due to Maroti and McKenzie, postulates that if a finite algebra A possesses a weak near-unanimity term, then the corresponding constraint satisfaction problem is tractable. A binary operation is weak near-unanimity if and only if it is both commutative and idempotent. Thus if the dichotomy conjecture is true, any finite commutative, idempotent groupoid (CI groupoid) will be tractable. It is known that every semilattice (i.e., an associative CI groupoid) is tractable. A groupoid identity is of Bol-Moufang type if the same three variables appear on either side, one of the variables is repeated, the remaining two variables appear once, and the variables appear in the same order on either side (for example, $x(x(yz))\approx(x(xy))z$). These identities can be thought of as generalizations of associativity. We show that there are exactly 8 varieties of CI groupoids defined by a single additional identity of Bol-Moufang type, derive some of their important structural properties, and use that structure theory to show that 7 of the varieties are tractable. We also characterize the finite members of the variety of CI groupoids satisfying the self-distributive law $x(yz)\approx(xy)(xz)$, and show that they are tractable.

math.GR↗