Extending congruences on semigroups
Author:
A. R. Stralka
Journal:
Trans. Amer. Math. Soc. 166 (1972), 147161
MSC:
Primary 22A15
MathSciNet review:
0294557
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Abstract: The two main results are: (1) Let S be a semigroup which satisfies the relation , let A be a subsemigroup of Reg S which is a band of groups and let be a congruence on A. Then can be extended to a congruence on S. (2) Let S be a compact topological semigroup which satisfies the relation , let A be a closed subsemigroup of Reg S and let be a closed congruence on A such that . Then can be extended to a closed congruence on S.
 [1]
L.
W. Anderson and R.
P. Hunter, Homomorphisms and dimension, Math. Ann.
147 (1962), 248–268. 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 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 0263964
(41 #8563), http://dx.doi.org/10.1090/S00029947197002639640
 [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]
George
Grätzer, Lattice theory. First concepts and distributive
lattices, W. H. Freeman and Co., San Francisco, Calif., 1971. MR 0321817
(48 #184)
 [7]
J.
M. Howie, Naturally ordered bands, Glasgow Math. J.
8 (1967), 55–58. MR 0205900
(34 #5726)
 [8]
Miyuki
Yamada and Naoki
Kimura, Note on idempotent semigroups. II, Proc. Japan Acad.
34 (1958), 110–112. MR 0098141
(20 #4603)
 [9]
J.
D. Lawson, Topological semilattices with small semilattices,
J. London Math. Soc. (2) 1 (1969), 719–724. MR 0253301
(40 #6516)
 [10]
M. Mislove, Four problems about compact semigroups, Dissertation, University of Tennessee, Knoxville, Tenn., 1969.
 [11]
Katsumi
Numakura, Theorems on compact totally
disconnected semigroups and lattices, Proc.
Amer. Math. Soc. 8
(1957), 623–626. MR 0087032
(19,290d), http://dx.doi.org/10.1090/S00029939195700870325
 [12]
Albert
R. Stralka, The Green equivalences and dimension in compact
semigroups, Math. Z. 109 (1969), 169–176. MR 0241563
(39 #2903)
 [13]
Albert
R. Stralka, The congruence extension property for compact
topological lattices, Pacific J. Math. 38 (1971),
795–802. MR 0304259
(46 #3394)
 [14]
Miyuki
Yamada, Regular semigroups whose idempotents satisfy permutation
identities, Pacific J. Math. 21 (1967),
371–392. MR 0227302
(37 #2887)
 [15]
Miyuki
Yamada, 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.
 [1]
 L. W. Anderson and R. P. Hunter, Homomorphisms and dimension, Math. Ann. 147 (1962), 248268. 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), 4758. 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), 327337. 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), 5558. MR 34 #5726. MR 0205900 (34:5726)
 [8]
 N. Kimura and M. Yamada, Note on idempotent semigroups. II, Proc. Japan Acad. 34 (1958), 110112. MR 20 #4603. MR 0098141 (20:4603)
 [9]
 J. D. Lawson, Topological semilattices with small semilattices, J. London Math. Soc. (2) 1 (1969), 719724. 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), 623626. MR 19, 290. MR 0087032 (19:290d)
 [12]
 A. R. Stralka, The Green equivalences and dimension in compact semigroups, Math. Z. 109 (1969), 169176. 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), 371392. MR 37 #2887. MR 0227302 (37:2887)
 [15]
 , On a regular semigroup in which the idempotents form a band, Pacific J. Math. 33 (1970), 261272. MR 0276391 (43:2138)
 [16]
 A. D. Wallace, Project MOB, Lecture Notes, University of Florida, Tallahassee, Fla., 1964, unpublished.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197202945579
PII:
S 00029947(1972)02945579
Keywords:
Topological semigroup,
semigroup,
congruence,
naturally ordered band,
Ninversive
Article copyright:
© Copyright 1972
American Mathematical Society
