David Easdown (University of Sydney)

Friday 12th May, 12.05-12.55pm, Carslaw 159

Bosets and fundamental semigroups

No, that is not a misprint! The term boset was coined by Patrick Jordan as an abbreviation for biordered set, which refers to a set equipped with two intertwining reflexive and transitive relations which satisfy certain axioms. When these relations coincide the boset becomes a poset. Bosets were invented by Nambooripad (in the 1970s), who gave his own version of a general theory of regular fundamental semigroups, encompassing the classical theory of fundamental inverse semigroups via semilattices, due to Munn (in the 1960s).

A fundamental semigroup has the property that it can't be shrunk homomorphically without disrupting the skeleton of idempotents, which forms a boset. Fundamental semigroups and bosets are natural candidates for basic building blocks because every semigroup is a coextension of a fundamental semigroup in which the boset of idempotents remains intact.

Brad Roberts (2005) has produced a large class of (nonregular) fundamental semigroups built from bosets, substantially enlarging the class explained by Nambooripad's theory (in the regular case). Roberts' construction involves a novel representation by transformations and dual transformations of certain quotients associated with equivalence relations on the underlying boset.

As well as introducing Roberts' construction, the talk will illustrate bosets through examples and discuss the free semigroup on a boset, which is a key ingredient in the abstract characterisation (in the 1980s) of systems of idempotents of semigroups. Very little is known about this wild and intriguing object!