International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1986 / Article

Open Access

Volume 9 |Article ID 461819 | https://doi.org/10.1155/S0161171286000765

Reuben Spake, "The semigroup of nonempty finite subsets of integers", International Journal of Mathematics and Mathematical Sciences, vol. 9, Article ID 461819, 12 pages, 1986. https://doi.org/10.1155/S0161171286000765

The semigroup of nonempty finite subsets of integers

Received16 Dec 1985
Revised13 Feb 1986

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 byA+B={a+b:aA,bB},A,Bg.For Xg, define AX to be the basis of Xmin(X) and BX the basis of max(X)X. In the greatest semilattice decomposition of g, let α(X) denote the archimedean component containing X and define α0(X)={Yα(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,Yg, α(X)=α(Y) if and only if AX=AY and BX=BY. Furthermore, if Xg is a non-singleton, then the idempotent-free α(X) is isomorphic to the direct product of the (idempotent-free) power joined subsemigroup α0(X) and the group Z.

Copyright © 1986 Hindawi Publishing Corporation. 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.


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