The semigroup of nonempty finite subsets of rationals

[摘要] LetQbe the additive group of rational numbers and letℛbe the additive semigroup of all nonempty finite subsets ofQ. ForX∈ℛ, defineAXto be the basis of〈X−min(X)〉andBXthe basis of〈max(X)−X〉. In the greatest semilattice decomposition ofℛ, let?(X)denote the archimedean component containingX. In this paper we examine the structure ofℛand determine its greatest semilattice decomposition. In particular, we show that forX,Y∈ℛ,?(X)=?(Y)if and only ifAX=AYandBX=BY. Furthermore, ifXis a non-singleton, then the idempotent-free?(X)is isomorphic to the direct product of a power joined subsemigroup and the groupQ.

[学科分类] 数学（综合）