Abstract
In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup.
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REFERENCES
A. H. CLIFFORD and G. B. PRESTON, The Algebraic Theory of Semigroups, Vol. II, Math. Surveys No. 7, Amer. Math. Soc., Providence, 1967.
S. CrvenkoviĆ, On *-regular semigroups, in: Proc. 3rd Algebraic Conf. (Beograd, 1982), Institute of Mathematics, Novi Sad, 1983, 51-57.
J. CVETKOVI?, Structural Properties of a Class of Semigroups, M.Sc. Thesis, University of Novi Sad, 1987.
M. P. DRAZIN, Regular semigroups with involution, in: Proc. Symp. on Regular Semigroups (DeKalb, 1979), Northern Illinois University, DeKalb, 1979, 29-46
R. FEIGENBAUM, Regular semigroup congruences, Semigroup Forum 17 (1979), 373-377.
T. E. HALL, Congruences and Green's relations on regular semigroups, Glasgow Math. J. 13 (1972), 167-175.
J. M. HOWIE, Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1995.
T. IMAOKA, *-Congruences on regular *-semigroups, Semigroup Forum 23 (1981), 321-326.
J. C. MEAKIN, Congruences on regular semigroups, Semigroup Forum 1 (1970), 232-235.
J. C. MEAKIN, Congruences on orthodox semigroups, J. Austral. Math. Soc. (Ser. A) 12 (1971), 323-341.
K. S. S. NAMBOORIPAD and F. J. PASTIJN, Regular involution semigroups, in eds. Gy. Pollák, Št. Schwarz and O. Steinfeld, Semigroups (Szeged, 1981), Colloq. Math. Soc. János Bolyai Vol. 39, 199-249, North-Holland, Amsterdam, 1985.
T. E. NORDAHL and H. E. SCHEIBLICH, Regular *-semigroups, Semigroup Forum 16 (1978), 369-377.
F. J. PASTIJN and M. PETRICH, Congruences on regular semigroups, Trans. Amer. Math. Soc. 295 (1986), 607-633.
R. PENROSE, Generalized inverses for matrices, Proc. Cambridge Phil. Soc. 51 (1955), 406-413.
G. B. PRESTON, Inverse semigroups, J. London Math. Soc. 29 (1954), 396-403.
N. R. REILLY and H. E. SCHEIBLICH, Congruences on regular semigroups, Pacific J. Math. 23 (1967), 349-360.
H. E. SCHEIBLICH, Kernels of inverse semigroup homomorphisms, J. Austral. Math. Soc. (Ser. A) 18 (1974), 289-292.
M. StojakoviĆ, Sur une propriété des matrices quasiinverses, Mat. Vesnik 6 (1952), 155-158.
P. G. TROTTER, Normal partitions of idempotents of regular semigroups, J. Austral. Math. Soc. (Ser. A) 26 (1978), 110-114.
P. G. TROTTER, Congruences on regular and completely regular semigroups, J. Austral. Math. Soc. (Ser. A) 32 (1982), 388-398.
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Crvenković, S., Dolinka, I. Congruences on *-Regular Semigroups. Periodica Mathematica Hungarica 45, 1–13 (2002). https://doi.org/10.1023/A:1022355610931
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DOI: https://doi.org/10.1023/A:1022355610931