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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A study of graph closed subsemigroups of a full transformation semigroup


Authors: R. G. Biggs, S. A. Rankin and C. M. Reis
Journal: Trans. Amer. Math. Soc. 219 (1976), 211-223
MSC: Primary 20M20; Secondary 05C20
MathSciNet review: 0404502
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Abstract: Let $ {T_X}$ be the full transformation semigroup on the set X and let S be a subsemigroup of $ {T_X}$. We may associate with S a digraph $ g(S)$ with X as set of vertices as follows: $ i \to j \in g(S)$ iff there exists $ \alpha \in S$ such that $ \alpha (i) = j$. Conversely, for a digraph G having certain properties we may assign a semigroup structure, $ S(G)$, to the underlying set of G. We are thus able to establish a ``Galois correspondence'' between the subsemigroups of $ {T_X}$ and a particular class of digraphs on X. In general, S is a proper subsemigroup of $ S \cdot g(S)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0404502-4
PII: S 0002-9947(1976)0404502-4
Keywords: Transformation semigroup, nil semigroup, idealizer, partially ordered set, digraph, algebraic graph, idealized graph
Article copyright: © Copyright 1976 American Mathematical Society