A construction of weakly inverse semigroups |
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Authors: | Bing Jun Yu Yan Li |
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Affiliation: | (1) College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610066, P. R. China;(2) Department of Mathematics, Yang-En University, Quanzhou, 362014, P. R. China |
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Abstract: | Let S o be an inverse semigroup with semilattice biordered set E o of idempotents and E a weakly inverse biordered set with a subsemilattice E P = {e ∈ E | ∀ f ∈ E, S(f, e) ⊆ ω(e)} isomorphic to E o by θ: E P → E o. In this paper, it is proved that if ∀f, g ∈ E, f ⟷ g ⟹ f o θ D S° g o θ and there exists a mapping ϕ from E P into the symmetric weakly inverse semigroup P ℐ (E∪S o) satisfying six appropriate conditions, then a weakly inverse semigroup Σ can be constructed in P ℐ (S o), called the weakly inverse hull of a weakly inverse system (S o,E, θ, ϕ) with I(gS) ≅ S o, E(Σ) ≃ E. Conversely, every weakly inverse semigroup can be constructed in this way. Furthermore, a sufficient and necessary condition for two weakly inverse hulls to be isomorphic is also given. Supported by an NSF Grant of China #10471112 |
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Keywords: | weakly inverse semigroup VP(Vagner-Preston’ s) representation weakly inverse biordered set weakly inverse system weakly inverse hull |
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