Homomorphisms and dimension

Anderson, Lee; Hunter, R. P.

doi:10.1007/bf01470743

Cited by 20 publications

(5 citation statements)

References 25 publications

(18 reference statements)

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“…An 777,-semigroup is a compact topological semigroup such that the Schiitzenberger group of each of its Jif-classes is a Lie group. From [1] it follows that this property is equivalent to requiring that all maximal subgroups be Lie groups. Further work on 777_.-semigroups was done by Mislove in [10].…”

Section: Then Since (R(/)<7(x)£cr(/-x) We Conclude That (Rx Ry) E [Omentioning

confidence: 99%

Extending Congruences on Semigroups

Stralka¹

1972

Transactions of the American Mathematical Society

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Abstract. The two main results are: (1) Let 5 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 [] can be extended to a closed congruence on S.For a semigroup S let us define Reg (S) to be the set of those elements x of S for which there is an element y e S such that xyx = x and yxy =y. In this paper we consider the following question : Let 5 be a (compact topological) semigroup, let A be a (closed) subsemigroup of Reg (S) and let [9] he a (closed) congruence on A. When can [9] he extended to a (closed) congruence on S? That is, when is there a (closed) congruence [] on S such that [i>] n A x A = [9] ? It is known (cf. [6]) that any congruence on any sublattice of distributive lattice L can be extended to L. In fact this property, known as the congruence extension property, serves to characterize distributive lattices. The topological analog of this result for compact topological lattices of finite breadth was proved in [13]. In the same paper an example was given of a compact distributive topological lattice of infinite breadth which does not share this property.In [16] Wallace, in a result which he attributes to Borsuk, gives conditions under which, given a closed congruence [9] on a closed subsemigroup A of a compact semigroup S, [9] uASxS is a closed congruence on S. Further results of this nature were established by Borrego in [3]. In [12] the author showed implicitly that if 5 is a compact topological semigroup and [9] is a closed congruence on E(S), the set of idempotents of S, such that dim 9(E(S)) = 0 then [9] can be extended to a closed congruence on S.Our main result for nontopological semigroups is that if S is a semigroup which satisfies the relation abcd=acbd then any congruence on any subsemigroup A of Reg S where A is a band of groups can be extended to a congruence on S. For topological semigroups we obtain the result that if S is a compact topological semigroup which satisfies the relation abcd=acbd, Ais a closed subsemigroup of Reg (S) and [9] a closed congruence on A such that dim 9(A)/Jf=0, then [9]

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Section: Then Since (R(/)<7(x)£cr(/-x) We Conclude That (Rx Ry) E [Omentioning

confidence: 99%

Extending Congruences on Semigroups

Stralka¹

1972

Transactions of the American Mathematical Society

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“…Thus it follows from the induced homomorphism theorem ( [4] or [5]) that there is a homomorphism f* such that the diagram -t[a) is commutative. Now let G x = £fJ a , G 2 = f*{S^^a) and q> = /*.…”

Section: • a -+ H F(a) • F(a) Is A Homomorphism And F O {^Y A )mentioning

confidence: 99%

The ℋ-equivalence in a compact semigroup II

Anderson

Hunter

1963

J. Aust. Math. Soc.

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In [1] we considered various aspects of the quotient semigroup H. · H2 where H is an ℋ-class of a semigroup S. In particular, the action of the Schützenberger group of H upon SH was studied to obtain various results on the existence of subcontinua. Crucial in [1] was the notion of the (right handed) core of an ℋ-class which may be considered as a generalization of the notion of the core of a homogroup, [2].

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“…In [4] Cohen and Krule determined the homomorphic images of a semigroup with zero on an interval. Anderson and Hunter in [1] proved several theorems in this direction.-In general, the problem seems to be rather difficult. However, the difficulty is lessened somewhat if all of the homomorphisms of the semigroups in question must be monotone.…”

Section: Introductionmentioning

confidence: 99%

Monotone Homomorphisms of Compact Semigroups

Clark

1969

J. Aust. Math. Soc.

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The problem of determining the class of homomorphic images of a given class of topological semigroups seems to have received little attention in the literature. In [4] Cohen and Krule determined the homomorphic images of a semigroup with zero on an interval. Anderson and Hunter in [1] proved several theorems in this direction. In general, the problem seems to be rather difficult. However, the difficulty is lessened somewhat if all of the homomorphisms of the semigroups in question must be monotone. Phillips, [7], showed that every homomorphism of a standard thread is monotone and hence every homomorphic image of a standard thread is either a standard thread or a point. In this paper a larger class of topological semigroups which admit only monotone homomorphisms is given. These results are used to determine the topological nature of the homomorphic images of certain classes of topological semigroups. These include products of standard threads with min threads, certain semilattices on a two-cell, and compact connected lattices in the plane.

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Homomorphisms and dimension

Cited by 20 publications

References 25 publications

Extending Congruences on Semigroups

Extending Congruences on Semigroups

The ℋ-equivalence in a compact semigroup II

Monotone Homomorphisms of Compact Semigroups

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