International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 3, Pages 605-616
The semigroup of nonempty finite subsets of integers
Department of Mathematics, University of California, Davis 95616, California, USA
Received 16 December 1985; Revised 13 February 1986
Copyright © 1986 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 be the additive group of integers and the semigroup consisting of all nonempty finite subsets of with respect to the operation defined byFor , define to be the basis of and the basis of . In the greatest semilattice decomposition of , let denote the archimedean component containing and define . In this paper we examine the structure of and determine its greatest semilattice decomposition. In particular, we show that for , if and only if and . Furthermore, if is a non-singleton, then the idempotent-free is isomorphic to the direct product of the (idempotent-free) power joined subsemigroup and the group .