Abstract
The notion of a (semi)topological residuated lattice is introduced, and its properties are investigated. Some separation axioms on topological residuated lattices are studied. The notion of completion of a residuated lattice is introduced and characterized by means of the inverse limit of an inverse system. A residuated lattice with a given system of filters is illustrated with the linear topology, and it is shown that a compact and Hausdorff residuated lattice with the linear topology is complete.
Similar content being viewed by others
References
Blount K, Tsinakis C (2003) The structure of residuated lattices. Int J Algebra Comput 13(04):437–461
Borzooei R, Rezaei G, Kouhestani N (2011) On (semi) topological BL-algebras. Iran J Math Sci Inf 6(1):59–77
Choban M (1993) Some topics in topological algebra. Topol Appl 54(1–3):183–202
Choe TH (1969) On compact topological lattices of finite dimension. Trans Am Math Soc 140:223–237
Choe TH, Greechie RJ (1993) Profinite orthomodular lattices. Proce Am Math Soc0 118(4):1053–1060
Ciungu LC (2006) Classes of residuated lattices. Ann Univ Craiova-Math Comput Sci Ser 33:189–207
Di Nola A, Georgescu G, Iorgulescu A (2002) Pseudo-BL algebras: part I. Multiple Valued Logic 8(5/6):673–716
Engelking R (1989) General topology. Heldermann, Berlin
Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics, vol 151. Elsevier, Amsterdam
Grätzer G (2008) Universal algebra. Springer, Berlin
Hoo C (1997) Topological mv-algebras. Topol Appl 81(2):103–121
Husain T (1966) Introduction to topological groups. W. B. Saunders Co., Philadelphia
Jipsen P, Tsinakis C (2002) A survey of residuated lattices. In: Ordered algebraic structures. Springer, pp 19–56
Kouhestani N, Borzooei RA (2014) On (semi) topological residuated lattices. Ann Univ Craiova-Math Comput Sci Ser 41(1):15–29
Nganou JB, Tebu SF (2015) Topological FL\(_{ew}\)-algebras. J Appl Logic 13(3):259–269
Rasouli S, Zarin Z (2019) On residuated lattices with left and right internal state. Fuzzy Sets Syst 373:37–61. https://doi.org/10.1016/j.fss.2019.03.001
Strauss DP (1968) Topological lattices. Proc Lond Math Soc 3(2):217–230
Warner S (1989) Topological fields, vol 157. Elsevier, Amsterdam
Warner S (1993) Topological rings, vol 178. Elsevier, Amsterdam
Willard S (2004) General topology. Courier Corporation, North Chelmsford
Zahiri O, Borzooei RA (2016) Topology on BL-algebras. Fuzzy Sets Syst 289:137–150
Acknowledgements
The authors are highly grateful to referees for their valuable comments and suggestions which were helpful in improving this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by A. Di Nola.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rasouli, S., Dehghani, A. Topological residuated lattices. Soft Comput 24, 3179–3192 (2020). https://doi.org/10.1007/s00500-020-04709-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-020-04709-x