University of Sydney Algebra Seminar

James East (University of Western Sydney)

Friday 16 September, 12:05-12:55pm, Carslaw 175

Semigroups of singular mappings

John Howie showed in 1966 that any non-bijective function from a finite set to itself can be obtained by composing a number of idempotent functions. In semigroup-theoretic terms, this says that the singular part of a finite transformation semigroup is generated by its idempotents. I'll present a set of defining relations for this singular subsemigroup with respect to the generating set consisting of all corank 1 idempotents. Along the way, I'll also discuss some other extensions of Howie's results in the context of (some of) partial transformation semigroups, Brauer monoids, partition monoids, semigroups of integer matrices and semigroups of transformations of various kinds of spaces. If time permits, I'll say something about the infinite case.