Precoherent quantale completions of partially ordered semigroups

  • Bin ZhaoEmail author
  • Changchun Xia


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 \(a_1\cdots a_n\le b\).


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



  1. 1.
    Abramsky, S., Vickers, S.: Quantles, observational logic and process semantics. Math. Struct. Comput. Sci. 3, 161–227 (1993)CrossRefzbMATHGoogle Scholar
  2. 2.
    Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. The Joy of Cats. Wiley, New York (1990)zbMATHGoogle Scholar
  3. 3.
    Eklund, P., García, J.G., Höhle, U., Kortelainen, J.: Semigroups in Complete Lattices: Quantales, Modules and Related Topics. Developments in Mathematics, vol. 54. Springer, Berlin (2018)zbMATHGoogle Scholar
  4. 4.
    Frink, O.: Ideals in partially ordered sets. Am. Math. Mon. 61, 223–234 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hofmann, D., Waszkiewicz, P.: Approximation in quantale-enriched categories. Topol. Appl. 158, 963–977 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Han, S.W., Zhao, B.: \(Q\)-fuzzy subsets on ordered semigroups. Fuzzy Sets Syst. 210, 102–116 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Han, S.W., Zhao, B.: The quantale completion of ordered semigroup. Acta Math. Sin. Chin. Ser. 51, 1081–1088 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)zbMATHGoogle Scholar
  9. 9.
    Keimel, K.: A unified theory of minimal prime ideals. Acta Math. Acad. Sci. Hung. 23, 51–69 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lambek, J., Barr, M., Kennison, J.F., Raphael, R.: Injective hulls of partially ordered monoids. Theory Appl. Categ. 26, 338–348 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Larchey-Wendling, D., Galmiche, D.: Quantales as completions of ordered monoids: revised semantics for intuitionistic linear logic. Electron. Notes Theor. Comput. Sci. 35, 94–108 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Paseka, J.: Regular and normal quantales. Arch. Math. 22, 203–210 (1986)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Rosenthal, K.I.: Quantales and Their Applications. Longman Scientific and Technical, Harlow (1990)zbMATHGoogle Scholar
  14. 14.
    Resende, P.: Étale groupoids and their quantales. Adv. Math. 208, 147–209 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stubbe, I.: Categorical structures enriched in a quantaloid: categories, distributors and functors. Theory Appl. Categ. 14, 1–45 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Solovyov, S.A.: From quantale algebroids to topological spaces: fixed-and variable-basis approaches. Fuzzy Sets Syst. 161, 1270–1287 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sankappanavar, H.P., Burris, S.: A Course in Universal Algebra. Graduate Texts in Mathematics. Springer, Berlin (1981)zbMATHGoogle Scholar
  18. 18.
    Xia, C.C., Han, S.W., Zhao, B.: A note on injective hulls of posemigroups. Theory Appl. Categ. 32, 254–257 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Xia, C.C., Zhao, B., Han, S.W.: Further results of quantale completions of the partially ordered semigroup. Acta Math. Sin. Chin. Ser. 61, 811–822 (2018)Google Scholar
  20. 20.
    Yetter, D.N.: Quantales and (noncommutative) linear logic. J. Symb. Log. 55, 41–64 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhang, D.X.: An enriched category approach to many valued topology. Fuzzy Sets Syst. 158, 349–366 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, X., Laan, V.: Injective hulls for posemigroups. Proc. Estonian Acad. Sci 63, 372–378 (2014)CrossRefzbMATHGoogle Scholar

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© 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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