Two Papers Mathcal H Coextensions Of Monoids And The Structure Of A Band Of Groups

Two Papers: $\mathcal H$-Coextensions of Monoids; and The Structure of a Band of Groups

In the second paper, the structure of a band of groups is studied by means of the techniques of the first paper.

Two Papers: H-coextensions of Monoids and The Structure of a Band of Groups

Structure of Regular Semigroups. I

The structure of regular semigroups is studied in full generality. The principal tool used in this is the concept of a (regular) biordered set which abstractly characterizes the set of idempotents of a regular semigroup. The category of inductive groupoids is then defined as the category whose objects are pairs consisting of an ordered groupoid and an order-preserving functor of the chain groupoid of a biordered set whose vertex map is a bijection, and whose morphisms are certain commutative diagrams in the category of ordered groupoids. It is shown by an explicit construction that every regular semigroup can be constructed from an inductive groupoid and that the category of inductive groupoids is equivalent to the category of all regular semigroups. This construction is then applied to obtain the structure of all fundamental regular semigroups and all idempotent generated regular semigroups. The paper ends with a study of biordered sets of some important classes of regular semigroups.