Abstract
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.
ACKNOWLEDGMENT
Marcel Jackson was supported by ARC Discovery Project Grant DP0342459.
Notes
1Strictly, z 0 xz 1 should be either z 0(xz 1) or (z 0 x)z 1, but we can work with semigroup words instead of terms.
2Note also, that the definability of I a and J a are obviously equivalent to the definability of the corresponding congruences. For example, if θ a denotes the Rees congruence corresponding to I a and Φ(x, y) defines principal ideals, then the sentence Φ′(x, y, z): = (x ≈ y)∨Φ(z, x) & Φ(z, y) defines θ in the sense that (c, d) ∈ θ a if and only if A⊨Φ′(c, d, a).
3In this article we do not consider the one element algebra to be subdirectly irreducible.
4The statement of Corollary 2.8 of Demlová and Koubek [Citation14] requires the phrase “where P is in normal form” to be added for it to read correctly.
Communicated by V. Gould.