Relatively inherently nonfinitely q-based semigroups

Jackson, Marcel; Volkov, Mikhail

doi:10.1090/S0002-9947-08-04798-3

Relatively inherently nonfinitely q-based semigroups
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by Marcel Jackson and Mikhail Volkov PDF

Trans. Amer. Math. Soc. 361 (2009), 2181-2206 Request permission

Abstract:

We prove that every semigroup $\mathbf {S}$ 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:

$\mathbf {S}$ is finite;

$\mathbf {S}$ has a faithful representation by injective partial maps on a set;

$\mathbf {S}$ 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.

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