Abstract
According to an old theorem of Yeager (Trans Am Math Soc 215:253–267, 1976), a homomorphism \(h:X\rightarrow Y\) between compact Hausdorff topological Clifford semigroups is continuous if and only if for every subgroup \(H\subset X\) and every subsemilattice \(E\subset X\) the restrictions \(h|H\) and \(h|E\) are continuous. In this paper we extend this Yeager result beyond the class of compact topological Clifford semigroups.
Highlights
In terms of E H -continuity, Theorem 1.1 says that each E H -continuous homomorphism between compact Hausdorff topological Clifford semigroups is continuous
In [1] it was shown that the class of ditopological unosemigroups contains all compact Hausdorff topological unosemigroups, is closed under taking subunosemigroups, Tychonoff, reduced, and semidirect products, and has many other nice properties
The class of ditopological inverse semigroups contains all compact Hausdorff topological inverse semigroups, all topological groups, all topological semilattices, and is closed under taking inverse subsemigroups and Tychonoff products, see [1]. This is a class nicely extending the class of compact topological inverse semigroups and many results known for compact topological inverse semigroups extend to ditopological inverse semigroups, see [2]
Summary
Theorem 1.1 [13] A homomorphism h : X → Y between compact topological Clifford semigroups is continuous if and only if for any subgroup H ⊂ X and any subsemilattice E ⊂ X the restrictions h|H and h|E are continuous. In this paper we shall extend this Yeager’s theorem beyond the class of compact topological Clifford semigroups. In terms of E H -continuity, Theorem 1.1 says that each E H -continuous homomorphism between compact Hausdorff topological Clifford semigroups is continuous. This Yeager’s theorem suggests the following problem addressed in this paper.
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