Regular Semigroups Satisfying Certain Conditions on Idempotents and Ideals

The structure of regular semigroups is studied (1) whose poset of idempotents is required to be a tree or to satisfy a weaker condition concerning the behavior of idempotents in different <!-- MATH $\mathcal{D}$ --> -classes, or (2) all of whose ideals are categorical or satisfy a variation thereof. For this purpose the notions of -majorization of idempotents, where is a <!-- MATH $\mathcal{D}$ --> -class, <!-- MATH $\mathcal{D}$ --> -majorization, <!-- MATH $\mathcal{D}$ --> -categorical ideals, and completely semisimple semigroups without contractions are introduced and several connections among them are established. Some theorems due to G. Lallement concerning subdirect products and completely regular semigroups and certain results of the author concerning completely semisimple inverse semigroups are either improved or generalized.