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Unit orthodox semigroups

Published online by Cambridge University Press:  18 May 2009

T. S. Blyth  and
R. McFadden
T. S. Blyth
Affiliation:
Mathematical Institute University of St. Andrews, Scotland
R. McFadden
Affiliation:
Department of Mathematical Science, Northern Illinois University De Kalb, Illinois 60115, U.S.A.
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Let S be a regular semigroup. Given x ∈ S, we shall say that a ∈ S is an associate of x if xax = x. The set of associates of x ∈ S will be denoted by A(x). Now suppose that S has an identity element 1. Let H1 denote the group of units of S. Then we say that u ∈ S is a unit associate of x whenever u ∈ A(x)∩Hl. In what follows we shall write U(x) = A(x)∩=H1, and we shall say that S is unit regular [1, 3] if (∀ x∈S)U(x)≠ ∅. Examples of unit regular semigroups include the full transformation semigroup on a finite set [1] and the semigroup of endomorphisms of a finite–dimensional vector space [3]. In this paper we shall be concerned with semigroups that are unit orthodox (i.e. unit regular and orthodox), and we shall describe completely the structure of those semigroups that are uniquely unit orthodox (i.e. orthodox and uniquely unit regular in the sense that, for every x∈S, the set U(x) is a singleton). It is worthy of mention that neither of the examples cited above is of this type.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 1 , January 1983 , pp. 39 - 42
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.D'Alarcao, H., Factorizable as a finiteness condition, Semigroup Forum 20 (1980), 281–282.CrossRefGoogle Scholar
2.Dubreil, P., Contribution à la théorie des demi-groupes, Mem. Acad. Sci. Inst. France (2) 63 (1941), 1–52.Google Scholar
3.Goodearl, K. R., von Neumann regular rings (Pitman, 1979).Google Scholar