In an inverse semigroup it is well known that the set of idempotents carries a strong algebraic/order-theoretic structure itself, namely it forms a semilattice. To elucidate the relevant structure for an arbitrary semigroup is harder, and it turns out that the relevant notion is that of a biorder. This in turn gives rise to a free object, namely the free idempotent generated semigroup over a given biorder. Since this object is built on the framework of idempotents, general semigroup theory guarantees that it is replete with subgroups, and it is natural to expect they somehow control the behaviour of the semigroup. In this talk I will discuss some recent results on this topic, including the description of subgroups that can arise in free idempotent generated semigroups, and to what extent they control the word problem of the semigroup.
Free idempotent generated semigroups Sponsored by the Meyer Fund
Oct. 27, 2015 2pm (MATH 220)
Nik Ruskuc (University of St Andrews)
X
A partially ordered set (or more generally a quasi-ordered set) is said to be well quasi-ordered if it has no infinite descending chains and no infinite antichains. It has been investigated a fair bit in different areas of combinatorics, where it often represents a demarcation line between tractable and intractable down sets of combinatorial structures. The key technical result on well quasi-orders is the so-called Higman's Lemma. In its best known incarnation it establishes a direct and strong link between well quasi-order and regular languages. Recently, I (together with various collaborators) have been deploying this link to investigate well quasi-ordered classes of permutations, and possible enumerative, computational and decidability consequences.
Well quasi-order: words and permutations Sponsored by the Meyer Fund
Coxeter Groups