In this series of talks, I'll present the basics of combinatorial semigroup theory, starting with elementary results and ending in recent research using high-powered tools. I'll begin by giving an overview of the elements of semigroup theory, including the analogue of Cayley's theorem, eggbox diagrams, Green's relations, inverse semigroups, and a famous result due to Green & Penrose. In the subsequent talk, I'll present the elements of presentations of semigroups, free (inverse) semigroups, Munn trees, and rewriting systems, leading into the fundamental problem central to combinatorial semigroup theory: the word problem. In the next talk, I'll dive into a particular class of semigroups called "special" monoids, and give proofs via rewriting systems due to Zhang (1990s) of famous results due to Adian (1960s), giving a solution to the word problem in all monoids given by a single defining relation of the form w=1. In the final talk (if there is time) I will dip our toes into how rewriting systems can compute the (co)homology of a monoid, and give new proofs via the spectral sequence of certain rewriting systems (forthcoming) of homological results due to Gray & Steinberg (2023).
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