International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 3, Pages 605-616
doi:10.1155/S0161171286000765
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.
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 byA+B={a+b:a∈A, b∈B}, A,B∈g.For X∈g, 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 α(X) denote the archimedean component containing X and define α0(X)={Y∈α(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,Y∈g, α(X)=α(Y) if and only if AX=AY and BX=BY. Furthermore, if X∈g 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.