International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 1, Pages 81-86

The semigroup of nonempty finite subsets of rationals

Reuben Spake

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

Received 8 December 1986

Copyright © 1988 Reuben Spake. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.