The semigroup of nonempty finite subsets of integers

Reuben Spake

International Journal of Mathematics and Mathematical Sciences, 1986, vol. 9, 1-12

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 by A + B = { a + b : a ∈ A , b ∈ B } , A , B ∈ g . For X ∈ g , 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 g , let α ( X ) denote the archimedean component containing X and define α 0 ( 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 A X = A Y and B X = B Y . 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 α 0 ( X ) and the group Z .

Date: 1986
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:461819

DOI: 10.1155/S0161171286000765

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