The semigroup of nonempty finite subsets of rationals

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.