$M_K(T)$-Spaces: Some Order-Theoretical Properties and Applications to Distributive Logics

Cristian Brunetta; Víctor Fernández

doi:10.52737/18291163-2024.16.13-1-21

Authors

Cristian Brunetta National university of San Juan https://orcid.org/0000-0002-9511-6051
Víctor Fernández National university of San Juan https://orcid.org/0000-0001-9903-5892

DOI:

https://doi.org/10.52737/18291163-2024.16.13-1-21

Keywords:

Lattices of Closure Spaces, Recovery of $M_K(T)$-Spaces, Distributive Logics

Abstract

We present some results involving several ordered structures related to the construction of $M_K(T)$-spaces. These are special closure spaces defined on the basis of a previously given one ${K}$ and were developed by Fern\'andez and Brunetta, 2023. Specifically, we show that for every $T \in {K}$, the poset $\mathbb{RE}_{{K}}(T)$ of weak-relative closure spaces is a sublattice of $\mathbb{CSP}(X)$ (the family of all the spaces with support $X$). On the other hand, it will be shown that the poset $\mathbb{M}({K})$ of all the $M_{{K}}(T)$-spaces does not verify this property, but it is a complete lattice itself. Also, we show in which way some order-theoretic properties are related to the recovery of closure spaces. Finally, we show some applications of $M_{{K}}(T)$-spaces to the class of distributive logics.

References

T.M. Al-Shami, Some results related to supra topological spaces. J. Adv. Stud. Topology, 7 (2016), no. 4, pp. 283-294. DOI: https://doi.org/10.20454/jast.2016.1166

K. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst., 20 (1986), no. 1, pp. 87-96. DOI: https://doi.org/10.1016/S0165-0114(86)80034-3

D. Brown and R. Suszko, Abstract logics. Diss Math., 102 (1973), pp. 9-41.

S. Burris and H. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York, 1981. DOI: https://doi.org/10.1007/978-1-4613-8130-3

D. Çoker, A note on intuitionistic sets and intuitionistic points. Turk. J. Math., 20 (1996), no. 3, pp. 343-351.

D. Çoker, An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets Syst., 88 (1997), no. 1, pp. 81-89. DOI: https://doi.org/10.1016/S0165-0114(96)00076-0

Á. Császár, Generalized topology, generalized continuity. Acta Math. Hung., 96 (2002), no. 4, pp. 351-357. DOI: https://doi.org/10.1023/A:1019713018007

J.Dugundji, Topology, Allyn And Bacon, Boston, 1966.

V. Fernández and C. Brunetta, A note on closure spaces determined by intersections. Bol. da Soc. Parana. de Mat., 41, (2023), pp. 1-10. DOI: https://doi.org/10.5269/bspm.52790

J.M. Font, Abstract algebraic logic. An introductory textbook, College Publications, London, 2016.

J.M. Font and V. Verdú, Algebraic logic for classical conjunction and disjunction. Stud. Log., 50 (1991), no. 3--4, pp. 391-419. DOI: https://doi.org/10.1007/BF00370680

J.M. Font and V. Verdú, The lattice of distributive closure operators over an algebra. Stud. Log., 52 (1993), no. 1, pp. 1-13. DOI: https://doi.org/10.1007/BF01053060

P. Hájek, Metamathematics of fuzzy logic, Kluwer Academic Publishers, Amsterdam, 1998. DOI: https://doi.org/10.1007/978-94-011-5300-3

A.S. Mashhour, A.A. Allam, F.S. Mahmoud and F.H. Kheder, On supra topological spaces. Indian J. Pure Appl. Math., 14 (1983), no. 4, pp. 502-510.

F. Nakaoka and N. Oda, Some applications of minimal open sets. Int. J. Math. Math. Sci., 27 (2001), no. 8, pp. 471-476. DOI: https://doi.org/10.1155/S0161171201006482

B. Roy and R. Sen, On maximal μ-open and minimal μ-closed sets via generalized topology. Acta Math. Hung., 136 (2012), no. 4, pp. 233-239. DOI: https://doi.org/10.1007/s10474-012-0201-z

M. Schroeder, On classification of closure spaces: search for the methods and criteria. In: Algebraic Systems and Theoretical Computer Science (Akihiro Yamamura ed.), Research Institute for Mathematical Sciences, Kyoto University, Kyoto 2012, pp.145-154.

T. Witczak, Infra-topologies revisited: logic and clarification of basic notions. Commun. Korean Math. Soc., 37 (2022), no. 1, pp. 279-292.