On the number of finite algebraic structures

Abstract

Abstract. We prove that every clone of operations on a finite set A, if it contains a Malcev operation, is finitely related -i.e., identical with the clone of all operations respecting R for some finitary relation R over A. It follows that for a fixed finite set A, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term.

Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

“…Cube terms describe finite algebras having few subpowers (i.e., with a polynomial bound in n on the number of generators of subalgebras of n-th power). This result and many more interesting properties of algebras with cube terms can be found in [BIM + 10], [AMM14], and [KS12]. There are several Mal'cev conditions equivalent to having a cube term, e.g.…”

Higher commutator theory for congruence modular varieties

“…. , θ k−1 ) ∈ Con(A) k for A ∈ V. For a T -matrix h we will apply a composition of k − 1 many shift rotations, first at (0, 1), then at (1,2), ending at (k − 2, k − 1). For each stage there are n + 1 many choices of Day terms, each giving a different shift rotation.…”

“…Bulatov was interested in counting the number of distinct polynomial clones on a finite set that contain a Mal'cev operation. Although this problem was solved in [1] using other methods, higher commutators have found other important uses. In [3], supernilpotence is shown to be an obstacle to a Mal'cev algebra having a natural duality.…”

Finitely generated equational classes

“…The techniques that we use from order theory have been used in a similar way in [Aic10], and in an almost identical way as they are used in the present note in [AMM11]. To keep our presentation self-contained, we report these results with only a slight modification in notation.…”

“…The main technique for proving Theorem 1.1 will be an application of the arguments that were used in [Aic10,AMM11] to establish that every clone on a finite set that contains an edge operation is finitely related. In our setting, the role of clones is taken by a suitable encoding of the equational theory of a variety into a structure that we will call a clonoid.…”