Samuel Herman

Date of Award

2021

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

Henckell, Karsten

Area of Concentration

Mathematics

Abstract

We develop a general framework for functorial detection of pseudovariety membership by way of dualities between lattices of endofunctors on the category of finite semigroups and the lattice of pseudovarieties. These dualities arise when the action of these endofunctors may be described as pointwise selections of elements from functorially assigned finite lattices. To illustrate this phenomenon, we consider two examples. The first example involves the family of universal image functors, which send a finite semigroup to its maximal homomorphic image belonging to a given pseudovariety. The second example—which is the focus of much of the paper—involves the family of pointlike functors, which send a finite semigroup to its semigroup of pointlike subsets with respect to a given pseudovariety. In each case, these families of functors are exhibited as the fixed points of an antitone Galois connection between a larger lattice of functors and the lattice of pseudovarieties. We also provide new proofs of the decidability of pointlike sets for the pseudovarieties of R- and L-trivial semigroups, as well as for pseudovarieies of semigroups whose subgroups all belong to some specified pseudovariety of groups.

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