Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Extending congruences on semigroups


Author: A. R. Stralka
Journal: Trans. Amer. Math. Soc. 166 (1972), 147-161
MSC: Primary 22A15
DOI: https://doi.org/10.1090/S0002-9947-1972-0294557-9
MathSciNet review: 0294557
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The two main results are: (1) Let S be a semigroup which satisfies the relation $ abcd = acbd$, let A be a subsemigroup of Reg S which is a band of groups and let $ [\varphi ]$ be a congruence on A. Then $ [\varphi ]$ can be extended to a congruence on S. (2) Let S be a compact topological semigroup which satisfies the relation $ abcd = acbd$, let A be a closed subsemigroup of Reg S and let $ [\varphi ]$ be a closed congruence on A such that $ \dim \,\varphi (A)\vert\mathcal{H} = 0$. Then $ [\varphi ]$ can be extended to a closed congruence on S.


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

  • [1] L. W. Anderson and R. P. Hunter, Homomorphisms and dimension, Math. Ann. 147 (1962), 248-268. MR 26 #4324. MR 0146804 (26:4324)
  • [2] -, On the infinite subsemigroups of certain compact semigroups (to appear).
  • [3] J. T. Borrego, Adjunction semigroups, Bull. Austral. Math. Soc. 1 (1969), 47-58. MR 39 #7021. MR 0245715 (39:7021)
  • [4] J. H. Carruth and C. E. Clark, Representations of certain compact semigroups by HLsemigroups, Trans. Amer. Math. Soc. 149 (1970), 327-337. MR 41 #8563. MR 0263964 (41:8563)
  • [5] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vols. 1, 2, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R. I., 1961, 1967. MR 24 #A2627; MR 36 #1558.
  • [6] G. Gratzer, Lectures on lattice theory, Freeman, San Francisco, Calif., 1971. MR 0321817 (48:184)
  • [7] J. M. Howie, Naturally ordered bands, Glasgow Math. J. 8 (1967), 55-58. MR 34 #5726. MR 0205900 (34:5726)
  • [8] N. Kimura and M. Yamada, Note on idempotent semigroups. II, Proc. Japan Acad. 34 (1958), 110-112. MR 20 #4603. MR 0098141 (20:4603)
  • [9] J. D. Lawson, Topological semilattices with small semilattices, J. London Math. Soc. (2) 1 (1969), 719-724. MR 40 #6576. MR 0253301 (40:6516)
  • [10] M. Mislove, Four problems about compact semigroups, Dissertation, University of Tennessee, Knoxville, Tenn., 1969.
  • [11] K. Numakura, Theorems on compact totally disconnected semigroups and lattices, Proc. Amer. Math. Soc. 8 (1957), 623-626. MR 19, 290. MR 0087032 (19:290d)
  • [12] A. R. Stralka, The Green equivalences and dimension in compact semigroups, Math. Z. 109 (1969), 169-176. MR 39 #2903. MR 0241563 (39:2903)
  • [13] -, The congruence extension property for compact topological lattices, Pacific J. Math. (to appear). MR 0304259 (46:3394)
  • [14] M. Yamada, Regular semigroups whose idempotents satisfy permutation identities, Pacific J. Math. 21 (1967), 371-392. MR 37 #2887. MR 0227302 (37:2887)
  • [15] -, On a regular semigroup in which the idempotents form a band, Pacific J. Math. 33 (1970), 261-272. MR 0276391 (43:2138)
  • [16] A. D. Wallace, Project MOB, Lecture Notes, University of Florida, Tallahassee, Fla., 1964, unpublished.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22A15

Retrieve articles in all journals with MSC: 22A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0294557-9
Keywords: Topological semigroup, semigroup, congruence, naturally ordered band, N-inversive
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society