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| Título | Semigroups of transformations with restricted range |
| Publication Type | Unpublished |
| Year of Publication | 2010 |
| Authors | Fernandes VH, Sanwong J |
| Series Title | Preprint |
| Palavras-chave | Green’s relations, rank, regular elements, restricted range, Stirling numbers, transformations |
| Abstract | Let PT(X) be the semigroup of all partial transformations on X (under composition) and let T(X) and I(X) be the subsemigroups of PT(X) of all full transformations on X and of all injective partial transformations on X, respectively. Given a nonempty subset Y of X, let PT(X,Y)={α ın PT(X)| Xα \subset Y}, T(X,Y)=PT(X,Y) ∩ T(X) and I(X,Y)=PT(X,Y) ∩ I(X). In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined the Green's relations of T(X,Y). In this paper, we present analogous results for both PT(X,Y) and I(X,Y). For a finite set X such that |X| ≥ 3, the ranks of PT(X)= PT(X,X), T(X)=T(X,X) and I(X)=I(X,X) are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of PT(X,Y), T(X,Y) and I(X,Y), for any proper nonempty subset Y of X. Stirling numbers of the second kind play an essential role in these computations. |
| URL | http://www.dm.fct.unl.pt/sites/www.dm.fct.unl.pt/files/preprints/2010/24_10.pdf |