E-unitary congruences on inverse semigroups

Reilly, N. R. ; Munn, W. D.

Glasgow mathematical journal, Tome 17 (1976) no. 1, pp. 57-75

Résumé

By an E-unitary inverse semigroup we mean an inverse semigroup in which the semilattice is a unitary subset. Such semigroups, elsewhere called ‘proper’ or ‘reduced’ inverse semigroups, have been the object of much recent study. Free inverse semigroups form a subclass of particular interest.An important structure theorem for E-unitary inverse semigroups has been obtained by McAlister [4, 5]. From a triple (G, ) consisting of a group G, a partially ordered set and a subset , satisfying certain conditions, he constructs an E-unitary inverse semigroup P(G, ). A semigroup of this type is called a P-semigroup. The structure theorem states that every E-unitary inverse semigroup is, to within isomorphism, of this form. A second theorem asserts that every inverse semigroup is isomorphic to a quotient of a Psemigroup by an idempotent-separating congruence. We refer below to these results as McAlister's Theorems A and B respectively. A triple (C, ) of the type used to construct a P-semigroup is here termed a “McAlister triple”. It is shown further, in [5], that there is essentially only one such triple corresponding to a given E-unitary inverse semigroup.

Détail

DOI : 10.1017/S0017089500002731

Bibliographie

[1] 1.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Math. Surveys of the Amer. Math. Soc. 7 (Providence, R.I., 1961 (vol. 1) and 1967 (vol. 2)) Google Scholar

[2] 2.Green, D. G., Extensions of a semilattice by an inverse semigroup, Bull. Austral. Math. Soc. 9 (1973), 21–31 Google Scholar | DOI

[3] 3.Howie, J. M., The maximum idempotent-separating congruence on an inverse semigroup, Proc. Edinburgh Math. Soc. (2) 14 (1964), 71–79 Google Scholar | DOI

[4] 4.McAlister, D. B., Groups, semilattices and inverse semigroups, Trans. Amer. Math. Soc. 192 (1974), 227–244 Google Scholar

[5] 5.McAlister, D. B., Groups, semilattices and inverse semigroups II, Trans. Amer. Math. Soc. 196 (1974), 351–370 Google Scholar | DOI

[6] 6.McAlister, D. B. and McFadden, R., Zig-zag representations and inverse semigroups, J. Algebra 32 (1974), 178–206 Google Scholar | DOI

[7] 7.McFadden, R. and O'Carroll, L., F-inverse semigroups, Proc. London Math. Soc. (3) 22 (1971), 652–666 Google Scholar

[8] 8.Munn, W. D., A class of irreducible matrix representations of an arbitrary inverse semigroup, Proc. Glasgow Math. Assoc. 5 (1961), 41–48 Google Scholar

[9] 9.Munn, W. D., Fundamental inverse semigroups, Quarterly J. Math. Oxford (2) 21 (1970), 157–170 Google Scholar | DOI

[10] 10.Munn, W. D., Free inverse semigroups, Semigroup Forum 5 (1973), 262–269. (Research announcement.) Google Scholar | DOI

[11] 11.Munn, W. D., Free inverse semigroups, Proc. London Math. Soc. (3) 29 (1974), 385–404 Google Scholar | DOI

[12] 12.O'Carroll, L., Reduced inverse semigroups, Semigroup Forum 8 (1974), 270–276. (Research announcement.) Google Scholar | DOI

[13] 13.O'Carroll, L., Reduced inverse and partially ordered semigroups, J. London Math. Soc. (2) 9 (1974), 293–301 Google Scholar | DOI

[14] 14.Preston, G. B., Free inverse semigroups, J. Austral. Math. Soc. 16 (1973), 443–453 Google Scholar

[15] 15.Reilly, N. R., Free generators of free inverse semigroups, Bull. Austral. Math. Soc. 7 (1972), 407–424 Google Scholar | DOI

[16] 16.Reilly, N. R., E-unitary inverse semigroups, Math. Dept. Research Report, Simon Fraser University (1975) Google Scholar

[17] 17.Reilly, N. R. and Scheiblich, H. E., Congruences on regular semigroups, Pacific J. Math. 23 (1967), 349–360 Google Scholar

[18] 18.Scheiblich, H. E., Free inverse semigroups, Semigroup Forum 4 (1972), 351–359. (Research announcement.) Google Scholar | DOI

[19] 19.Scheiblich, H. E., Free inverse semigroups, Proc. Amer. Math. Soc. 38 (1973), 1–7 Google Scholar | DOI

[20] 20.Saitô, T., Proper ordered inverse semigroups, Pacific J. Math. 15 (1965), 649–666 Google Scholar