Algebra universalis

, 79:13 | Cite as

Sup-algebra completions and injective hulls of ordered algebras

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Abstract

In this paper, we study the \(\sup \)-algebra completions of ordered algebras. Firstly, we show that the \(\sup \)-algebra completions of the ordered algebras \(\mathcal {A}\) are completely determined, up to isomorphism, by the topological nuclei on \(\mathcal {P}(A)\). Moreover, we give the concrete forms of three types of special topological nuclei. Finally, we give the concrete form of injective hulls for arbitrary ordered algebras.

Keywords

Ordered algebras Sup-algebra completions Injective hulls 

Mathematics Subject Classification

08B30 08C05 06F99 

Notes

Acknowledgements

The authors would like to thank Professor Warren William McGovern and the anonymous reviewer for their valuable comments and suggestions.

References

  1. 1.
    Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories. The joy of cats. Wiley, New York (1990)MATHGoogle Scholar
  2. 2.
    Bloom, S.L.: Varieties of ordered algebras. J. Comput. Syst. Sci. 13, 200–212 (1976)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Han, S.W., Zhao, B.: The quantale completion of ordered semigroup. Acta Math. Sin. Chin. Ser. 51, 1081–1088 (2008)MathSciNetMATHGoogle Scholar
  4. 4.
    Lambek, J., Barr, M., Kennison, J.F., Raphael, R.: Injective hulls of partially ordered monoids. Theory Appl. Category 26, 338–348 (2012)MathSciNetMATHGoogle Scholar
  5. 5.
    Paseka, J.: Projective sup-algebras: a general view. Topol. Appl. 155, 308–317 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Resende, P.: Tropological systems and observational logic in concurrency and specification. PhD thesis, IST, Universidade Técnica de Lisboa (1998)Google Scholar
  7. 7.
    Xia, C.C., Zhao, B., Han, S.W.: Some further results on quantale completion of ordered semigroup. Acta Math. Sin. Chin. Ser. Accepted (2017)Google Scholar
  8. 8.
    Xia, C.C., Han, S.W., Zhao, B.: A note on injective hulls of posemigroups. Theory Appl. Category 26, 254–257 (2017)MathSciNetMATHGoogle Scholar
  9. 9.
    Zhang, X., Laan, V.: Injective hulls for posemigroups. Proc. Est. Acad. Sci. 63, 372–378 (2014)CrossRefMATHGoogle Scholar
  10. 10.
    Zhang, X., Laan, V.: Quotients and subalgebras of sup-algebras. Proc. Est. Acad. Sci. 64, 311–322 (2015)CrossRefMATHGoogle Scholar
  11. 11.
    Zhang, X., Laan, V.: On injective hulls of \(S\)-posets. Semigroup Forum. 91, 62–70 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Zhang, X., Laan, V.: Injective hulls for ordered algebras. Algebra Univ. 76, 339–349 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShaanxi Normal UniversityXi’anChina

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