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On Semigroups of Transformations Acting Transitively on a Set

Published online by Cambridge University Press:  20 November 2018

E.J. Tully Jr.
E.J. Tully Jr.*
Affiliation:
University of California, Davis
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We call a semigroup S transitive if S is isomorphic to a semigroup T of transformations of some set M into itself, where T acts on M transitively, that is in such a manner that for all x, y ∊ M we have Xπ = y for some transformation π∊T. In [4] the author showed that S is transitive if and only if there exists a right congruence σ (i.e., an equivalence relation for which a σ b always implies ac σ bc for all c ∊ S) on S, satisfying:

  1. (1)There exists a left identity modulo σ, that is an element e such that ea σ a for all a ∊ S .

  2. (2)Each σ-class meets each right ideal, or, equivalently, for all a, b ∊ S we have ac σ b for some c ∊ S .

  3. (3)The relation σ contains ( i. e. , is less fine than) no left congruence except the identity relation (in which each class consists of a single element).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 4 , October 1966 , pp. 417 - 420
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Dubreil, P., Contribution à la théorie des demi-groupes. Mem. Acad. Sci. Inst. France (2) 63, no. 3, 52 pages, (1941).Google Scholar
2. Teissier, M., Sur les équivalences régulières dans les demi-groupes. C.R. Acad. Sci. Paris 232, (1951), pages 1987-1989.Google Scholar
3. Tully, E.J. Jr., Representation of a semigroup by transformations of a set. Dissertation, Tulane University, (1960),Google Scholar
4. Tully, E.J. Jr., Representation of a semigroup by transformations acting transitively on a set. Amer. J. Math. 83, (1961), pages 533-541. Errata, Amer. J. Math, 84, (1962), page 386.Google Scholar
5. Tully, E.J. Jr., Congruence relations on a set with semigroup of transformations. Unpublished.Google Scholar