Thirty-Third Annual Victorian Algebra Conference, Sydney, Australia, 2015

Register for: VAC33

Details of talk

TitleCompletely meet irreducible pseudovarieties.
PresenterMarcel Jackson (La Trobe University)
Author(s)Dr Marcel Jackson

A semigroup pseudovariety is a class of finite semigroups closed under
subsemigroups, homomorphic images and finitary direct products.  

In the book ``Q-Theory of Finite Semigroups'', Benjamin Steinberg and John
Rhodes ask if the lattice of order increasing continuous self maps of the
semigroup pseudovariety lattice has atoms.  This turns out to be equivalent to
finding compact completely meet irreducible elements in the semigroup
pseudovariety lattice.  In turn, this is equivalent to finding a pair of finite
semigroups $S$ and $T$ such that a pseudovariety properly contains the
pseudovariety of $S$ if and only if it contains $T$.  We show that there are
infinitely many such pairs $(S,T)$, thus providing infinitely many of the
desired atoms.

This is joint work with Egri-Nagy, Steinberg and Rhodes.