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Notes on congruences on regular semigroups.I

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

R. J. Koch  and
B. L. Madison
R. J. Koch
Affiliation:
Louisiana State University Baton Rouge, Louisiana 70803, U.S.A.
B. L. Madison
Affiliation:
University of Arkansas Fayetteville, Arkansas 72701, U.S.A.
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Abstract

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Four properties of congruences on a regular semigroup S are studied and compared. Let R, L and D denote Green's relations and let V = {(a, b) ∈ S × S|a and b are mutually inverse}. A congruence ρ on S is (1) rectangular provided ρ ∩ D = (ρ ∩ L) ° (ρ ∩ R), (2) V-commuting provided ρ ° V = V ° ρ, (3) (L, R)-commuting provided L ° ρ = ρ ° L, and R ° ρ = ρ ° R, and (4) idempotent-regular provided each idempotent ρ-class is a regular subsemigroup of S.

A rectangular congruence is (L, R)-commuting and a V-commuting congruence is idempotent-regular. If ρ is idempotent-regular and (L, R)-commuting then ρ is V-commuting. Examples and conditions are given to show what other implications among the four properties hold. In addition to characterizations of the properties, these are studied in the presence of other conditions on S. For example, if S is a stable regular semigroup, then each congruence under D is rectangular.

MSC classification

Secondary: 20M15: Mappings of semigroups 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 39 , Issue 2 , October 1985 , pp. 146 - 158
Copyright
Copyright © Australian Mathematical Society 1985

References

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