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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| Format: | Article |
| Language: | English |
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Wiley
1988-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171288000122 |
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| _version_ | 1832558013263642624 |
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| author | Reuben Spake |
| author_facet | Reuben Spake |
| author_sort | Reuben Spake |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-1070a4c0768e4605929ca8747e40fdd9 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1988-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-1070a4c0768e4605929ca8747e40fdd92025-02-03T01:33:22ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251988-01-01111818610.1155/S0161171288000122The semigroup of nonempty finite subsets of rationalsReuben Spake0Department of Mathematics, University of California, Davis 95616, California , USALet 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.http://dx.doi.org/10.1155/S0161171288000122greatest semilattice decompositionarchimedean components. |
| spellingShingle | Reuben Spake The semigroup of nonempty finite subsets of rationals International Journal of Mathematics and Mathematical Sciences greatest semilattice decomposition archimedean components. |
| title | The semigroup of nonempty finite subsets of rationals |
| title_full | The semigroup of nonempty finite subsets of rationals |
| title_fullStr | The semigroup of nonempty finite subsets of rationals |
| title_full_unstemmed | The semigroup of nonempty finite subsets of rationals |
| title_short | The semigroup of nonempty finite subsets of rationals |
| title_sort | semigroup of nonempty finite subsets of rationals |
| topic | greatest semilattice decomposition archimedean components. |
| url | http://dx.doi.org/10.1155/S0161171288000122 |
| work_keys_str_mv | AT reubenspake thesemigroupofnonemptyfinitesubsetsofrationals AT reubenspake semigroupofnonemptyfinitesubsetsofrationals |