Groups, semilattices and inverse semigroups. II

McAlister, D. B.

doi:10.1090/S0002-9947-74-99950-4

Groups, semilattices and inverse semigroups. II
HTML articles powered by AMS MathViewer

by D. B. McAlister PDF

Trans. Amer. Math. Soc. 196 (1974), 351-370 Request permission

Abstract:

An inverse semigroup is called proper if the equations $ae = e = {e^2}$ together imply ${a^2} = a$. In a previous paper, with the same title, the author proved that every inverse semigroup is an idempotent separating homomorphic image of a proper inverse semigroup. In this paper a structure theorem is given for all proper inverse semigroups in terms of partially ordered sets and groups acting on them by order automorphisms. As a consequence of these two theorems, and Preston’s construction for idempotent separating congruences on inverse semigroups, one can give a structure theorem for all inverse semigroups in terms of groups and partially ordered sets.

References

T. S. Blyth, Residuated inverse semigroups, J. London Math. Soc. (2) 1 (1969), 243–248. MR 244122, DOI 10.1112/jlms/s2-1.1.243
D. G. Green, Extensions of a semilattice by an inverse semigroup, Bull. Austral. Math. Soc. 9 (1973), 21–31. MR 323928, DOI 10.1017/S0004972700042829
J. M. Howie, The maximum idempotent-separating congruence on an inverse semigroup, Proc. Edinburgh Math. Soc. (2) 14 (1964/65), 71–79. MR 163976, DOI 10.1017/S0013091500011251
B. P. Kočin, The structure of inverse ideal-simple $omega$-semigroups, Vestnik Leningrad. Univ. 23 (1968), no. 7, 41–50 (Russian, with English summary). MR 0227296
D. B. McAlister, $0$-bisimple inverse semigroups, Proc. London Math. Soc. (3) 28 (1974), 193–221. MR 364507, DOI 10.1112/plms/s3-28.2.193

Groups, semilattices and inverse semigroups

D. B. McAlister, Groups, semilattices and inverse semigroups, Proceedings of a Symposium on Inverse Semigroups and their Generalisations (Northern Illinois Univ., DeKalb, Ill., 1973) Northern Illinois Univ., DeKalb, Ill., 1973, pp. 64–76. MR 0384969
D. B. McAlister and R. McFadden, Zig-zag representations and inverse semigroups, J. Algebra 32 (1974), 178–206. MR 369590, DOI 10.1016/0021-8693(74)90180-X
R. McFadden and L. O’Carroll, $F$-inverse semigroups, Proc. London Math. Soc. (3) 22 (1971), 652–666. MR 292978, DOI 10.1112/plms/s3-22.4.652
W. D. Munn, A class of irreducible matrix representations of an arbitrary inverse semigroup, Proc. Glasgow Math. Assoc. 5 (1961), 41–48. MR 153762, DOI 10.1017/S2040618500034286
W. D. Munn, Regular $\omega$-semigroups, Glasgow Math. J. 9 (1968), 46–66. MR 229742, DOI 10.1017/S0017089500000288
W. D. Munn, Fundamental inverse semigroups, Quart. J. Math. Oxford Ser. (2) 21 (1970), 157–170. MR 262402, DOI 10.1093/qmath/21.2.157
G. B. Preston, Inverse semi-groups, J. London Math. Soc. 29 (1954), 396–403. MR 64036, DOI 10.1112/jlms/s1-29.4.396
N. R. Reilly, Bisimple $\omega$-semigroups, Proc. Glasgow Math. Assoc. 7 (1966), 160–167 (1966). MR 190252, DOI 10.1017/S2040618500035346
N. R. Reilly, Congruences on a bisimple inverse semigroup in terms of $RP$-systems, Proc. London Math. Soc. (3) 23 (1971), 99–127. MR 306375, DOI 10.1112/plms/s3-23.1.99
N. R. Reilly and H. E. Scheiblich, Congruences on regular semigroups, Pacific J. Math. 23 (1967), 349–360. MR 219646, DOI 10.2140/pjm.1967.23.349
H. E. Scheiblich, Free inverse semigroups, Proc. Amer. Math. Soc. 38 (1973), 1–7. MR 310093, DOI 10.1090/S0002-9939-1973-0310093-1
Tôru Saitô, Proper ordered inverse semigroups, Pacific J. Math. 15 (1965), 649–666. MR 191977, DOI 10.2140/pjm.1965.15.649
R. J. Warne, $I$-regular semigroups, Math. Japon. 15 (1970), 91–100. MR 288200