The Existence of Irr(X)

Abstract

If X is a compact totally ordered space, we obtain the existence of an irreducible semigroup with idempotents X, Irr(X), with the property that any irreducible semigroup with idempotents X is the idempotent separating surmorphic image of Irr(X). Furthermore, it is shown that the Clifford-Miller endomorphisn? on Irr(X) is an injection when restricted to each J-class of Irr(X). A construction technique for noncompact semigroups is given, and some results about the structure of such semigroups are obtained. Introduction. A semigroup S is irreducible if S is a compact connected semigroup with identity 1 having no proper compact connected subsemigroup containing 1 and meeting M(S), the minimal ideal of S. If X is a compact totally ordered space, (X, min) will denote the semigroup X under the operation xy = minimum lx, y} for each x, y e X. A semigroup S has idempotents X if E(S) C (X, min), E(S) being the set of idempotents of S. The main result of this paper obtains the existence of an irreducible semigroup with idempotents X, denoted Irr(X), with the property that a compact semigroup S is irreducible with idempotents X if and only if S is the idempotent separating surmorphic image of Irr(X). Hofmann and Mostert attempted to construct Jrr(X) in Chapter B, ?5 of [31, but there were errors in their construction. The first section of this paper is devoted to pointing out those errors. We next describe a technique which generalizes the construction of generalized hormoi to noncompact semigroups, and then establish some properties of the semigroups so constructed. In the main section of this work we establish the existence of Irr(X), and in the last section we give a counterexample to another proposed structure for Irr(X). The notation and terminology will be that of [31, and the reader is advised to review the definitions of a chainable collection and of a hormos on pp. 139-143 of that volume. The duality theory used will be Pontryagin Duality The. ory for locally compact abelian groups, and standard references are [21 and L5[. This work forms part of the author's doctoral dissertation, and he wishes to express his deep gratitude to J. H. Carruth for his many helpful suggestions and his patient listening during its preparation. Thanks go also to A. D. Wallace for his Presented to the Society, November 22, 1969; received by the editors July 6, 1970. AMS (MOS) subject classifications (1970). Primary 22A15, 22A25.