Congruences on regular semigroups

Authors:
Francis Pastijn and Mario Petrich

Journal:
Trans. Amer. Math. Soc. **295** (1986), 607-633

MSC:
Primary 20M10; Secondary 08A30, 20M17

MathSciNet review:
833699

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a regular semigroup and let be a congruence relation on . The kernel of , in notation , is the union of the idempotent -classes. The trace of , in notation , is the restriction of to the set of idempotents of . The pair is said to be the congruence pair associated with . Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple is said to be the congruence triple associated with . Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple.

The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of . For congruence relations and , put if and only if . Then and are complete congruences on the congruence lattice of and .

**[1]**Garrett Birkhoff,*Lattice Theory*, American Mathematical Society, New York, 1940. MR**0001959****[2]**A. H. Clifford, and G. B. Preston,*The algebraic theory of semigroups*, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R. I., Vol. I, 1961, Vol. II, 1967.**[3]**Ruth Feigenbaum,*Regular semigroup congruences*, Semigroup Forum**17**(1979), no. 4, 373–377. MR**532428**, 10.1007/BF02194336**[4]**T. E. Hall,*On the lattice of congruences on a regular semigroup*, Bull. Austral. Math. Soc.**1**(1969), 231–235. MR**0257254****[5]**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****[6]**Gérard Lallement,*Demi-groupes réguliers*, Ann. Mat. Pura Appl. (4)**77**(1967), 47–129 (French). MR**0225915****[7]**W. D. Munn,*Decompositions of the congruence lattice of a semigroup*, Proc. Edinburgh Math. Soc. (2)**23**(1980), no. 2, 193–198. MR**597123**, 10.1017/S0013091500003060**[8]**K. S. S. Nambooripad,*Structure of regular semigroups. I*, Mem. Amer. Math. Soc.**22**(1979), no. 224, vii+119. MR**546362**, 10.1090/memo/0224**[9]**F. J. Pastijn and P. G. Trotter,*Lattices of completely regular semigroup varieties*, Pacific J. Math.**119**(1985), no. 1, 191–214. MR**797024****[10]**Mario Petrich,*Inverse semigroups*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR**752899****[11]**N. R. Reilly and H. E. Scheiblich,*Congruences on regular semigroups*, Pacific J. Math.**23**(1967), 349–360. MR**0219646****[12]**H. E. Scheiblich,*Certain congruence and quotient lattices related to completely 0-simple and primitive regular semigroups*, Glasgow Math. J.**10**(1969), 21–24. MR**0244425****[13]**H. E. Scheiblich,*Kernels of inverse semigroup homomorphisms*, J. Austral. Math. Soc.**18**(1974), 289–292. MR**0360887****[14]**P. G. Trotter,*Normal partitions of idempotents of regular semigroups*, J. Austral. Math. Soc. Ser. A**26**(1978), no. 1, 110–114. MR**510594**

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0833699-3

Article copyright:
© Copyright 1986
American Mathematical Society