Cube term blockers without finiteness

TL;DR

This paper characterizes the existence of cube terms in idempotent varieties through algebraic conditions, employing combinatorial and lattice-theoretic methods to establish sharp bounds and structural criteria.

Contribution

It provides a new characterization of cube terms in idempotent varieties using free algebra properties and Hall's Marriage Theorem, including sharp bounds on their dimensions.

Findings

A variety has a d-dimensional cube term iff its free algebra on two generators lacks a d-ary compatible cross.

Finite signature varieties have a d-dimensional cube term iff d=1+sum of (n_i-1), with this bound being sharp.

A pure cyclic term variety has a cube term iff it does not contain a 2-element semilattice.

Abstract

We show that an idempotent variety has a $d$-dimensional cube term if and only if its free algebra on two generators has no $d$-ary compatible cross. We employ Hall's Marriage Theorem to show that a variety of finite signature whose fundamental operations have arities $n_1, \ldots, n_k$ has a $d$-dimensional cube term if and only if it has one of dimension $d=1+\sum_{i=1}^k (n_i-1)$. This lower bound on dimension is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no $2$-element semilattice. We prove that the Maltsev condition "existence of a cube term" is join prime in the lattice of idempotent Maltsev conditions.