Ideals and finiteness conditions for subsemigroups

Significant information about a semigroup may be obtained by studying its ideal structure and various finiteness conditions related to it. Examples include the existence of minimal ideals, stability and the property of Green’s relations J and D coinciding. Such properties have been identified and investigated because of their usefulness in the study of finite semigroups. This has led to instances where theorems that were originally proved for finite semigroups have been extended to apply to wider classes. I will talk about infinite semigroups satisfying such finiteness properties relating to their ideal structure, and present some results about the preservation of such properties when passing to subsemigroups and extensions. The results I will present are joint work with V. Maltcev, J. D. Mitchell and N. Ruskuc.