Semigroups with n + 1 Distinct Essentially n-Ary Term Operations

Abstract

In many papers idempotent algebras were characterized by their P n -sequences. This article as well as [19] yields the beginning of the investigation of nonidempotent algebras. Our aim is to give an equational description of all the semigroups without algebraic constants that have exactly n+1 different essentially n-ary term operations for every positive integer n. We prove that there are exactly two different varieties containing such the semigroups. The proof of this fact, as the reader can see, is not trivial and requires some work—mainly through lack of the idempotence of groupoids considered here.