Derived varieties of semigroups and groupoids

Abstract.

A hypersubstitution of type $\tau$ is a map which assigns to every fundamental operation symbol f _i of type $\tau$ a term $\sigma$ (f _i ) of the same arity as f _i . 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.