International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 1, Pages 81-86
doi:10.1155/S0161171288000122
The semigroup of nonempty finite subsets of rationals
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.
Abstract
Let Q be the additive group of rational numbers and let ℛ be the additive semigroup of all nonempty finite subsets of Q. For X∈ℛ, define AX to be the basis of 〈X−min(X)〉 and BX the basis of 〈max(X)−X〉. In the greatest semilattice decomposition of ℛ, let 𝒜(X) denote the archimedean component containing X. In this paper we examine the structure of ℛ and determine its greatest semilattice decomposition. In particular, we show that for X,Y∈ℛ, 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.