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 be the full transformation semigroup on the set *X* and let *S* be a subsemigroup of . We may associate with *S* a digraph with *X* as set of vertices as follows: iff there exists such that . Conversely, for a digraph *G* having certain properties we may assign a semigroup structure, , to the underlying set of *G*. We are thus able to establish a ``Galois correspondence'' between the subsemigroups of and a particular class of digraphs on *X*. In general, *S* is a proper subsemigroup of .

**[1]**A. H. Clifford and G. B. Preston,*The algebraic theory of semigroups. Vol. I*, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR**0132791****[2]**Mario Petrich,*The translational hull in semigroups and rings*, Semigroup Forum**1**(1970), no. 4, 283–360. MR**0267017**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1976-0404502-4

Keywords:
Transformation semigroup,
nil semigroup,
idealizer,
partially ordered set,
digraph,
algebraic graph,
idealized graph

Article copyright:
© Copyright 1976
American Mathematical Society