The semigroup of nonempty finite subsets of integers
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:a∈A, b∈B}, A,B∈g.For X∈g, define AX to be the basis of 〈X−min(X)〉 and BX the basis of 〈max(X)−X〉. In the greatest semilattice decomposition...
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| Language: | English |
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Wiley
1986-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171286000765 |
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| _version_ | 1832559154849382400 |
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| author | Reuben Spake |
| author_facet | Reuben Spake |
| author_sort | Reuben Spake |
| collection | DOAJ |
| description | 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:a∈A, b∈B}, A,B∈g.For X∈g, define AX to be the basis of 〈X−min(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)={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 AX=AY and BX=BY. 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 α0(X) and the group Z. |
| format | Article |
| id | doaj-art-8fd38e09915a43a69e7b842956c8e591 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1986-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-8fd38e09915a43a69e7b842956c8e5912025-02-03T01:30:41ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251986-01-019360561610.1155/S0161171286000765The semigroup of nonempty finite subsets of integersReuben Spake0Department of Mathematics, University of California, Davis 95616, California, USALet 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:a∈A, b∈B}, A,B∈g.For X∈g, define AX to be the basis of 〈X−min(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)={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 AX=AY and BX=BY. 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 α0(X) and the group Z.http://dx.doi.org/10.1155/S0161171286000765greatest semilattice decompositionarchimedean component. |
| spellingShingle | Reuben Spake The semigroup of nonempty finite subsets of integers International Journal of Mathematics and Mathematical Sciences greatest semilattice decomposition archimedean component. |
| title | The semigroup of nonempty finite subsets of integers |
| title_full | The semigroup of nonempty finite subsets of integers |
| title_fullStr | The semigroup of nonempty finite subsets of integers |
| title_full_unstemmed | The semigroup of nonempty finite subsets of integers |
| title_short | The semigroup of nonempty finite subsets of integers |
| title_sort | semigroup of nonempty finite subsets of integers |
| topic | greatest semilattice decomposition archimedean component. |
| url | http://dx.doi.org/10.1155/S0161171286000765 |
| work_keys_str_mv | AT reubenspake thesemigroupofnonemptyfinitesubsetsofintegers AT reubenspake semigroupofnonemptyfinitesubsetsofintegers |