Representations of certain compact semigroups by semigroups
Authors:
J. H. Carruth and C. E. Clark
Journal:
Trans. Amer. Math. Soc. 149 (1970), 327337
MSC:
Primary 22.05
MathSciNet review:
0263964
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Abstract: An HLsemigroup is defined to be a topological semigroup with the property that the Schützenberger group of each class is a Lie group. The following problem is considered: Does a compact semigroup admit enough homomorphisms into HLsemigroups to separate points of S; or equivalently, is S isomorphic to a strict projective limit of HLsemigroups? An affirmative answer is given in the case that S is an irreducible semigroup. If S is irreducible and separable, it is shown that S admits enough homomorphisms into finite dimensional HLsemigroups to separate points of S.
 [1]
L.
W. Anderson and R.
P. Hunter, The \𝑐𝑎𝑙𝐻equivalence in
compact semigroups, Bull. Soc. Math. Belg. 14 (1962),
274–296. MR 0151939
(27 #1920)
 [2]
L.
W. Anderson and R.
P. Hunter, The \𝑐𝑎𝑙𝐻equivalence in
a compact semigroup. II, J. Austral. Math. Soc. 3
(1963), 288–293. MR 0162232
(28 #5431)
 [3]
L.
W. Anderson and R.
P. Hunter, Homomorphisms and dimension, Math. Ann.
147 (1962), 248–268. MR 0146804
(26 #4324)
 [4]
C.
E. Clark, Certain types of congruences on compact commutative
semigroups, Duke Math. J. 37 (1970), 95–101. MR 0255717
(41 #377)
 [5]
A.
H. Clifford and G.
B. Preston, The algebraic theory of semigroups. Vol. I,
Mathematical Surveys, No. 7, American Mathematical Society, Providence,
R.I., 1961. MR
0132791 (24 #A2627)
 [6]
Haskell
Cohen, A cohomological definition of dimension for locally compact
Hausdorff spaces, Duke Math. J. 21 (1954),
209–224. MR 0066637
(16,609b)
 [7]
Karl
Heinrich Hofmann and Paul
S. Mostert, Elements of compact semigroups, Charles E. Merr ll
Books, Inc., Columbus, Ohio, 1966. MR 0209387
(35 #285)
 [8]
R.
P. Hunter, On the structure of homogroups with applications to the
theory of compact connected semigroups, Fund. Math.
52 (1963), 69–102. MR 0144318
(26 #1864)
 [9]
R.
P. Hunter and N.
J. Rothman, Characters and cross sections for certain
semigroups, Duke Math. J. 29 (1962), 347–366.
MR
0142110 (25 #5503)
 [10]
Witold
Hurewicz and Henry
Wallman, Dimension Theory, Princeton Mathematical Series, v.
4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
(3,312b)
 [11]
L.
S. Pontryagin, Topological groups, Translated from the second
Russian edition by Arlen Brown, Gordon and Breach Science Publishers, Inc.,
New York, 1966. MR 0201557
(34 #1439)
 [1]
 L. W. Anderson and R. P. Hunter, The equivalence in compact semigroups, Bull. Soc. Math. Belg. 14 (1962), 274296. MR 27 #1920. MR 0151939 (27:1920)
 [2]
 , The equivalence in compact semigroups. II, J. Austral. Math. Soc. 3 (1963), 288293. MR 28 #5431. MR 0162232 (28:5431)
 [3]
 , Homomorphisms and dimension, Math. Ann. 147 (1962), 248268. MR 26 #4324. MR 0146804 (26:4324)
 [4]
 C. E. Clark, Certain types of congruences on compact commutative semigroups, Duke Math. J. (to appear). MR 0255717 (41:377)
 [5]
 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. 1, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R. I., 1961. MR 24 #A2627. MR 0132791 (24:A2627)
 [6]
 H. Cohen, A cohomological definition of dimension for locally compact Hausdorff spaces, Duke Math. J. 21 (1954), 209224. MR 16, 609. MR 0066637 (16:609b)
 [7]
 K. H. Hofmann and P. S. Mostert, Elements of compact semigroups, Charles E. Merrill, Columbus, Ohio, 1966. MR 35 #285. MR 0209387 (35:285)
 [8]
 R. P. Hunter, On the structure of homogroups with applications to the theory of compact connected semigroups, Fund. Math. 52 (1963), 69102. MR 26 #1864. MR 0144318 (26:1864)
 [9]
 R. P. Hunter and N. Rothman, Characters and cross sections for certain semigroups, Duke Math. J. 29 (1962), 347366. MR 25 #5503. MR 0142110 (25:5503)
 [10]
 W. Hurewicz and H. Wallman, Dimension theory, Princeton Math. Series, vol. 4, Princeton Univ. Press, Princeton, N. J., 1941. MR 3, 312. MR 0006493 (3:312b)
 [11]
 L. S. Pontrjagin, Topological groups, 2nd ed., GITTL, Moscow, 1954; English transl., Gordon & Breach, New York, 1966. MR 17, 171 ; MR 34 #1439. MR 0201557 (34:1439)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197002639640
PII:
S 00029947(1970)02639640
Keywords:
Compact semigroup,
Lie group,
Schützenberger group,
Hclass,
representation,
irreducible semigroup,
projective limit
Article copyright:
© Copyright 1970 American Mathematical Society
