McAlister Semigroups

This paper is the first in a sequence in which we investigate the algebraic structure of Kellendonkʹs one-dimensional tiling semigroups. However, as a prelude to studying the tiling semigroups themselves we study here a class of inverse semigroups which are generated by disjoint elements which we call “McAlister semigroups.” They were first introduced by Don McAlister as a by-product of his work on inverse semigroups separated over a subsemigroup.