THE SEMIGROUP OF NONEMPTY FINITE SUBSETS OF INTEGERS

ABSTRACT. Let Z be the additive group of integers and g the semigroup consisting of all nonempty finite subsets of Z with respect to the operation defined by A B {a+b a e A, b E B}, A, B e. For X, define AX to be the basis of <X-min(X)> and BX the basis of <max(X)-X>. In the greatest semilattice decomposition of g, let (X) denote the archimedean component containing X and define Go(X) [Y (X) min(Y) 0}. In this paper we examine the structure of and determine its greatest semilattice decomposition. In particular, we show that for X, Y (X) (Y) if and only if AX Ay and BX By. Furthermore, if X g is a non-singleton, then the idempotent-free (X) is isomorphic to the direct product of the (idempotent-free) power joined subsemigroup Go(X) and the group Z. KEY WORDS AND PHRASES. Greatest semilattice decomposition, archimedean component. 1980 AMS SUBJECT CLASSIFICATION CODE. 20M14, IOLI0, 05A99. I. INTRODUCTION. Let Z be the group of integers and g the semigroup conslsting of al