The semigroup of nonempty finite subsets of rationals
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 containi...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1988-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171288000122 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |