The semigroup of nonempty finite subsets of rationals

Reuben Spake

International Journal of Mathematics and Mathematical Sciences, 1988, vol. 11, 1-6

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 A X to be the basis of 〈 X − min ( X ) 〉 and B X 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 A X = A Y and B X = B Y . 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 .

Date: 1988
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:364178

DOI: 10.1155/S0161171288000122

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