Relatively inherently nonfinitely Q-based semigroups

Abstract

We prove that every semigroup S whose quasivariety contains a 3-nilpotent semigroup or a semigroup of index more than 2 has no finite basis for its quasi-identities provided that one of the following properties holds: • S is finite; • S has a faithful representation by injective partial maps on a set; • S has a faithful representation by order preserving maps on a chain. As a corollary it is shown that, in an asymptotic sense, almost all finite semigroups and finite monoids admit no finite basis for their quasi-identities. © 2008 American Mathematical Society.