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

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.