Proper regular semigroups

Author:
F. E. Masat

Journal:
Proc. Amer. Math. Soc. **71** (1978), 189-192

MSC:
Primary 20M10

DOI:
https://doi.org/10.1090/S0002-9939-1978-0480796-1

MathSciNet review:
0480796

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In a recent paper, D. B. McAlister gave several characterizations of proper inverse semigroups. In this paper, the concept of proper is extended to the class of regular semigroups. This is done by requiring that the set of idempotents of the semigroup coincides with the kernel of the minimum group congruence on the semigroup. A theorem is presented which contains several characterizations of proper regular semigroups, and the related result of McAlister then follows as a corollary. The paper concludes with some open questions and examples.

**[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]**-,*The algebraic theory of semigroups*. II, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R. I., 1967; reprinted with corrections, 1971.**[3]**T. E. Hall,*On regular semigroups whose idempotents form a subsemigroup*, Bull. Austral. Math. Soc.**1**(1969), 195–208. MR**0249527**, https://doi.org/10.1017/S0004972700041447**[4]**J. M. Howie,*An introduction to semigroup theory*, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. L.M.S. Monographs, No. 7. MR**0466355****[5]**J. M. Howie and G. Lallement,*Certain fundamental congruences on a regular semigroup*, Proc. Glasgow Math. Assoc.**7**(1966), 145–159. MR**0197598****[6]**Francis E. Masat,*Right group and group congruences on a regular semigroup*, Duke Math. J.**40**(1973), 393–402. MR**0318378****[7]**D. B. McAlister,*Groups, semilattices and inverse semigroups. I, II*, Trans. Amer. Math. Soc.**192**(1974), 227–244; ibid. 196 (1974), 351–370. MR**0357660**, https://doi.org/10.1090/S0002-9947-1974-0357660-2**[8]**R. R. Stoll,*Homomorphisms of a semigroup onto a group*, Amer. J. Math.**73**(1951), 475–481. MR**0041128**, https://doi.org/10.2307/2372188**[9]**Tôru Saitô,*Proper ordered inverse semigroups*, Pacific J. Math.**15**(1965), 649–666. MR**0191977**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
20M10

Retrieve articles in all journals with MSC: 20M10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0480796-1

Keywords:
Proper,
group congruence,
reflexiveness,
inverse semigroups,
unitary semigroups,
orthodox semigroups

Article copyright:
© Copyright 1978
American Mathematical Society