2016/09/08 by Keith A. Kearnes, Kearnes, Keith A., Ágnes Szendrei +1
Computer Science · Mathematics · #08B20 #Advanced Algebra and Logic #FOS: Mathematics #Primary: 08B05 #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #Secondary: 08A30 #semigroups and automata theory
paper · doi:10.48550/arxiv.1609.02605
openalex publication_date 2016/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 n1, …, nk has a d-dimensional cube term if and only if it has one of dimension d=1+∑i=1k (ni-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.