International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 3, Pages 605-616

The semigroup of nonempty finite subsets of integers

Reuben Spake

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 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 byA+B={a+b:aA,bB},A,Bg.For Xg, define AX to be the basis of Xmin(X) and BX the basis of max(X)X. In the greatest semilattice decomposition of g, let α(X) denote the archimedean component containing X and define α0(X)={Yα(X):min(Y)=0}. In this paper we examine the structure of g and determine its greatest semilattice decomposition. In particular, we show that for X,Yg, α(X)=α(Y) if and only if AX=AY and BX=BY. Furthermore, if Xg is a non-singleton, then the idempotent-free α(X) is isomorphic to the direct product of the (idempotent-free) power joined subsemigroup α0(X) and the group Z.