International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 1, Pages 81-86
doi:10.1155/S0161171288000122

The semigroup of nonempty finite subsets of rationals

Reuben Spake

Department of Mathematics, University of California, Davis 95616, California , USA

Abstract

Let Q be the additive group of rational numbers and let be the additive semigroup of all nonempty finite subsets of Q. For X, define AX to be the basis of Xmin(X) and BX the basis of max(X)X. In the greatest semilattice decomposition of , let 𝒜(X) denote the archimedean component containing X. In this paper we examine the structure of and determine its greatest semilattice decomposition. In particular, we show that for X,Y, 𝒜(X)=𝒜(Y) if and only if AX=AY and BX=BY. Furthermore, if X is a non-singleton, then the idempotent-free 𝒜(X) is isomorphic to the direct product of a power joined subsemigroup and the group Q.