Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Groups, semilattices and inverse semigroups. II


Author: D. B. McAlister
Journal: Trans. Amer. Math. Soc. 196 (1974), 351-370
MSC: Primary 20M10
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] T. S. Blyth, Residuated inverse semigroups, J. London Math. Soc. (2) 1 (1969), 243-248. MR 39 # 5439. MR 0244122 (39:5439)
  • [2] D. G. Green, Extensions of a semilattice by an inverse semigroup, Bull. Austral. Math. Soc. 9 (1973), 21-32. MR 0323928 (48:2281)
  • [3] J. M. Howie, The maximum idempotent-separating congruence on an inverse semigroup, Proc. Edinburgh Math. Soc. (2) 14 (1964/65), 71-79. MR 29 # 1275. MR 0163976 (29:1275)
  • [4] B. P. Kočin, The structure of inverse ideal-simple $ \omega $-semigroups, Vestnik Leningrad. Univ. 23 (1968), no. 7, 41-50. (Russian) MR 37 # 2881. MR 0227296 (37:2881)
  • [5] D. B. McAlister, 0-bisimple inverse semigroups, Proc. London Math. Soc. (to appear). MR 0364507 (51:761)
  • [6] -, Groups, semilattices and inverse semigroups, Trans. Amer. Math. Soc. (to appear).
  • [7] -, Groups, semilattices and inverse semigroups, Proc. Sympos. on Inverse Semigroups, Northern Illinois University, DeKalb, Ill., 1973. MR 0384969 (52:5839)
  • [8] D. B. McAlister and R. McFadden, Zig-zag representations and inverse semigroups, J. Algebra (to appear). MR 0369590 (51:5823)
  • [9] R. McFadden and L. O'Carroll, F-inverse semigroups, Proc. London Math. Soc. (3) 22 (1971), 652-666. MR 45 # 2059. MR 0292978 (45:2059)
  • [10] W. D. Munn, A class of irreducible matrix representations of an arbitrary inverse semigroup, Proc. Glasgow Math. Assoc. 5 (1961), 41-48. MR 27 # 3723. MR 0153762 (27:3723)
  • [11] W. D. Munn, Regular $ \omega $-semigroups, Glasgow Math. J. 9 (1968), 46-66. MR 37 #5316. MR 0229742 (37:5316)
  • [12] -, Fundamental inverse semigroups, Quart. J. Math. Oxford Ser. (2) 21 (1970), 157-170. MR 41 # 7010. MR 0262402 (41:7010)
  • [13] G. B. Preston, Inverse semigroups, J. London Math. Soc. 29 (1954), 396-403. MR 16, 215. MR 0064036 (16:215c)
  • [14] N. R. Reilly, Bisimple $ \omega $-semigroups, Proc. Glasgow Math. Assoc. 7 (1966), 160-167. MR 32 #7665. MR 0190252 (32:7665)
  • [15] -, Congruences on a bisimple inverse semigroup in terms of RP-systems, Proc. London Math. Soc. (3) 23 (1971), 99-127. MR 46 # 5501. MR 0306375 (46:5501)
  • [16] N. R. Reilly and H. E. Scheiblich, Congruences on regular semigroups, Pacific J. Math. 23 (1967), 349-360. MR 36 # 2725. MR 0219646 (36:2725)
  • [17] H. E. Scheiblich, Free inverse semigroups, Proc. Amer. Math. Soc. 38 (1973), 1-7. MR 0310093 (46:9196)
  • [18] T. Saitô, Proper ordered inverse semigroups, Pacific J. Math. 15 (1965), 649-666. MR 33 # 204. MR 0191977 (33:204)
  • [19] R. J. Wame, I-regular semigroups, Math. Japon. 15 (1970), 91-100. MR 44 # 5398. MR 0288200 (44:5398)

Similar Articles

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

Retrieve articles in all journals with MSC: 20M10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-74-99950-4
PII: S 0002-9947(74)99950-4
Keywords: Inverse semigroup, proper inverse semigroup, semilattice, idempotent separating congruence, fundamental inverse semigroup, group action, order automorphisms
Article copyright: © Copyright 1974 American Mathematical Society