On the ranks of semigroups of transformations on a finite set with restricted range
Let PT (X) be the semigroup of all partial transformations on X, 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 non-empty subset Y of X, let PT (X; Y ) = {α ∈ PT (X) : Xα ⊆ Y}, T (X; Y ) = PT (X;...
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| Main Authors: | , |
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| Format: | Journal |
| Published: |
2018
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| Online Access: | https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=84903307086&origin=inward http://cmuir.cmu.ac.th/jspui/handle/6653943832/45628 |
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| Institution: | Chiang Mai University |
| Summary: | Let PT (X) be the semigroup of all partial transformations on X, 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 non-empty subset Y of X, let PT (X; Y ) = {α ∈ PT (X) : Xα ⊆ 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 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 with jXj - 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 non-empty subset Y of X. © 2014 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University. |
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