JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (6): 23-30.doi: 10.6040/j.issn.1671-9352.0.2021.566

Previous Articles    

Completions of S-semigroups

LIU Min, LI Yu-lin   

  1. School of Sciences, Changan University, Xian 710064, Shaanxi, China
  • Published:2022-06-10

PDF (PC)

343

Abstract: The concept of S-semigroup quantale completions for S-semigroups is introduced. It is proved that all S-semigroup quantale completions of an S-semigroup SA can be fully characterized as compatible quotients of the power-set S-semigroup quantale corresponding to SA. Three kinds of classical completion methods are given. Furthermore, S-algebra completions of double residuated S-semigroups are considered. The quotient with respect to the largest compatible nucleus on the down-set S-semigroup quantale is proved to be an S-algebra completion for an arbitrary double residuated S-semigroup.

Key words: partially ordered semigroup, Quantale, S-semigroup, S-semigroup quantale, S-algebra

CLC Number: 

  • O153.1
[1] WARD M. Structure residuation[J]. Annals of Mathematics, 1938, 39(3):558-569.
[2] WARD M, DILWORTH R P. Residuated lattices[J]. Transactions of the American Mathematical Society, 1939, 45(3):335-354.
[3] DILWORTH R P. Non-commutative residuated lattices[J]. Transactions of the American Mathematical Society, 1939, 46(3):426-444.
[4] BIRKHOFF G. Lattice theory[M]. 3rd ed. Providence, Rhode Island: American Mathematical Society, 1967.
[5] 韩胜伟,赵彬. Quantale理论基础[M]. 北京: 科学出版社, 2016. HAN Shengwei, ZHAO Bin. Foundation of Quantale theory[M]. Beijing: Science Press, 2016.
[6] ROSENTHAL K I. Quantales and their applications[M]. New York: Longman Scientific and Technical, 1990.
[7] GIRARD J Y. Linear logic[J]. Theoretical Computer Science, 1987, 50(1):1-101.
[8] EKLUND P, GARCÍA J G, HÖHLE U, et al. Semigroups in complete lattices: Quantales, modules and related topics[M]. Switzerland: Springer, 2018.
[9] 李永明,李志慧. Quantale与互模拟的进程语义[J]. 数学学报(中文版), 1999, 42(2):313-320. LI Yongming, LI Zhihui. Quantales and process semantics of bisimulation[J]. Acta Mathematica Sinica Chinese Series, 1999, 42(2):313-320.
[10] KILP M, KNAUER U, MIKHALEV A V. Monoids, acts and categories(with applications to wreath products and graphs)[M]. New York: De Gruyter, 2000.
[11] ZHANG X, PASEKA J. On injective constructions of S-semigroups[J]. Fuzzy Sets and Systems, 2019, 373:78-93.
[12] SOLOVYOV S A. A representation theorem for Quantale algebras[J]. Contributions to General Algebra, 2008, 18:189-198.
[13] SOLOVYOV S A. From Quantale algebroids to topological spaces: fixed-and variable-basis approaches[J]. Fuzzy Sets and Systems, 2010, 161(9):1270-1287.
[14] DAVEY B A, PRIESTLEY H A. Introduction to lattices and order[M]. 2nd ed. Cambridge: Cambridge University Press, 2002.
[15] ERNÉ M, REICHMAN J Z. Completions for partially ordered semigroups[J]. Semigroup Forum, 1986, 34(1):253-285.
[16] 韩胜伟, 赵彬. 序半群的Quantale完备化[J]. 数学学报(中文版), 2008, 51(6):1081-1088. HAN Shengwei, ZHAO Bin. The quantale completion of ordered semigroup[J]. Acta Mathematica Sinica Chinese Series, 2008, 51(6):1081-1088.
[17] 夏常春,赵彬,韩胜伟. 序半群Quantale完备化的进一步结果[J]. 数学学报(中文版), 2018, 61(5):811-822. XIA Changchun, ZHAO Bin, HAN Shengwei, Some further results on quantale completions of ordered semigroups[J]. Acta Mathematica Sinica Chinese Series, 2018, 61(5):811-822.
[18] RASOULI H. Completion of S-posets[J]. Semigroup Forum, 2012, 85:571-576.
[19] ZHANG X, PASEKA J. The MacNeille completions for residuated S-posets[J]. International Journal of Theoretical Physics, 2021, 60:667-676.
[20] XIA C C, ZHAO B. Sup-algebra completions and injective hulls of ordered algebras [J]. Algebra Universalis, 2018, 79(1):1-10.
[21] 谢祥云. 序半群引论[M]. 北京: 科学出版社, 2001. XIE Xiangyun. Introduction to partially ordered semigroups [M]. Beijing: Science Press, 2001.
[1] LUO Qing-jun. L-fuzzy ideals in Quantale [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(12): 63-67.
[2] LIANG Shao-hui. The category of E-quantales [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(12): 47-53.
[3] SHEN Chong, YAO Wei. A one-to-one correspondence between fuzzy G-ideals and fuzzy Galois connections on fuzzy complete lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(04): 36-41.
[4] LIU Min, ZHAO Bin*. Localic nuclei and conuclei on Quantales [J]. J4, 2013, 48(2): 81-87.
[5] LIANG Shao-hui. Researches on some properties of L-fuzzy quantale [J]. J4, 2012, 47(4): 104-109.
[6] LI Ling-qiang,QIU Jin . Many-valued filter and its application [J]. J4, 2007, 42(12): 29-32 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd