The Division Relation: Congruence Conditions and Axiomatisability
Marcel Jackson, Belinda Trotta · Communications in Algebra · 2010
We examine a universal algebraic abstraction of the semigroup theoretic concept of “divides:” a divides b in an algebra A if for some n ∈ ω, there is a term t(x, y 1,…, y n ) involving all of the listed variables, and elements c 1,…, c n such that t A (a, c 1,…, c n ) = b. The first order definability of this relation is shown to be a very broad generalisation of some familiar congruence properties, such as definability of principal congruences. The algorithmic problem of deciding when a finitely generated variety has this relation definable is shown to be equivalent to an open problem concerning flat algebras. We also use the relation as a framework for establishing some results concerning the finite axiomatisability of finitely generated varieties.