Derived varieties of semigroups and groupoids

Abstract

A hypersubstitution of type \( \tau \) is a map which assigns to every fundamental operation symbol fi of type \( \tau \) a term \( \sigma \) (fi) of the same arity as fi. For any algebra \( \langle A; f_i \rangle _{i \in I} \) and any hypersubstitution \( \sigma \) both of type \( \tau \) we can form the derived algebra \( \langle A ; \sigma (f_i) \rangle _{i \in I} \). In this paper we consider derived algebras and derived varieties, along with the related concepts of semisolidity and mutual solidity of varieties and derivation diagrams. In particular we present a number of examples, based on varieties of semigroups and groupoids.