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Precoherent quantale completions of partially ordered semigroups

  • Bin ZhaoEmail author
  • Changchun Xia
RESEARCH ARTICLE
  • 40 Downloads

Abstract

Just as complete lattices can be viewed as the completions of posets, quantales can also be treated as the completions of partially ordered semigroups. Motivated by the study on the well-known Frink completions of posets, it is natural to consider the “Frink” completions for the case of partially ordered semigroups. For this purpose, we firstly introduce the notion of precoherent quantale completions of partially ordered semigroups, and construct the concrete forms of three types of precoherent quantale completions of a partially ordered semigroup. Moreover, we obtain a sufficient and necessary condition of the Frink completion on a partially ordered semigroup being a precoherent quantale completion. Finally, we investigate the injectivity in the category \(\mathbf {APoSgr}_{\le }\) of algebraic partially ordered semigroups and their submultiplicative directed-supremum-preserving maps, and show that the \(\mathscr {E}_{\le }\)-injective objects of algebraic partially ordered semigroups are precisely the precoherent quantales, here \(\mathscr {E}_{\le }\) denote the class of morphisms \(h:A\longrightarrow B\) that preserve the compact elements and satisfy that \(h(a_1)\cdots h(a_n)\le h(b)\) always implies \(a_1\cdots a_n\le b\).

Keywords

Partially ordered semigroup Precoherent quantale completion Algebraic consistent quantic nucleus Injective object 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anPeople’s Republic of China

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