[For the entire collection see Zbl 0549.00009.]
A functor F from a category of semigroups is called a (semigroup) expansion if there is a natural transformation \(\eta\) from the functor F to the identity functor such that each \(\eta_ S\) is surjective. Semigroup expansions were first used by J. Rhodes in complexity theory of semigroups and in recent years were extensively studied by a number of researchers. The paper is based on the author’s talk at the Marquette Conference on Semigroups and represents a survey of some results concerning the Birget-Rhodes expansion \(\tilde S^ R\) (and its dual) of a semigroup S and the expansion PR(S) of a completely simple semigroup S (there is not enough place here to reproduce definitions of these expansions). Necessary and sufficient conditions on S are given in order for \(\tilde S^ R\) to be a regular semigroup. For some special regular semigroups S the corresponding semigroups \(\tilde S^ R\) are described (e.g., if S is a group, then \(\tilde S^ R\) is an E-unitary inverse monoid; if S is a rectangular band, then \(\tilde S^ R\) is a normal band; if S is a left regular band, then \(\tilde S^ R\) is also a left regular band). One of the important properties of the expansions \(\tilde S^ R\) and PR(S) is that they lift certain free objects in various varieties of (regular) semigroups to free objects in larger varieties. Several such situations are described.