Extreme Elements in Semigroups (Quasiorders, Green's).

Abstract

We study the elements of a semigroup which are minimal or maximal with respect to Green's quasiorders. Part 1 begins with a preliminary review. The sets of minimal elements are characterized in terms of minimal ideals. We discuss the relationship between the min set of a semigroup and the min set of a subsemigroup. The sets of maximal elements are characterized, and it is shown that these sets do not necessarily satisfy any inclusion relationships to each other. We discuss the max sets of subsemigroups and product semigroups. Conditions are given under which the max sets and min sets can intersect. We define the concept of a paved semigroup and present conditions under which homomorphisms preserve sets of maximal elements. The translational hull is discussed in Part 2. We compare the condition that a semigroup S is H paved with the condition that S = ESE, where E is the set of idempotents of S. We prove that if S is a subsemilattice of a finite semilattice T and if their max sets are equal, then the degree of S is at most the degree of T. Topological results appear in Part 3. Extreme sets of compact semigroups are discussed. An example is given in which the set of nonmaximal elements can be extended in more than one way. We compare the max set with various topological notions of boundary. Part 4 contains results on divisibility and on the Nambooripad partial order. Conditions are given under which the minimal sets inherit divisibility properties of the semigroup. We prove that divisibility of a semigroup very strongly implies divisibility of its max sets. Finally, we show that any element of a regular semigroup which is maximal with respect to the H, R, or L quasiorder is maximal with respect to the Nambooripad partial order.