Mal’tsev condition satisfaction problems for conditions which imply edge terms

Abstract

This paper considers the complexity of the problem of determining whether a finite idempotent algebra generates a variety which satisfies various strong Mal'tsev conditions. In particular we show that if the strong Mal'tsev condition under consideration implies the existence of an edge term, then deciding whether a finite idempotent algebra generates a variety which satisfies the Mal'tsev condition is in the complexity class NP. This extends an earlier result of Kazda, Opršal, Valeriote and Zhuk concerning minority terms to a much broader class of conditions, notably including all nontrivial linear strong Mal'tsev conditions in which every equation relates a term to a variable. The result of Kazda et al. being an application of a result of Mayr's concerning subpower membership problems, we note that our new result may similarly be seen as an application of the results on subpower membership problems obtained by Bulatov, Mayr and Szendrei. We also show that for fixed k ≥ 2 there is a polynomial-time procedure which takes an idempotent algebra A and produces the operation table of a k-edge term operation of A if such a term exists or else correctly returns the answer that no such term exists, again extending an earlier result of Kazda et al.