mp-residuated lattices

Subject Areas : Statistics

Saeed Rasouli ¹ , Dariush Heidari ²

1 - Department of Mathematics, Persian Gulf University,Bushehr, 75169, IRAN
2 - Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran

Received: 2019-06-17 Accepted : 2021-02-17 Published : 2021-03-01

Keywords: ω-پالایه, mp-مشبکه‌ مانده‌دار, پالایه اول کمین, پالایه بخش‌یاب, مشبکه‌ مانده‌دار,

Abstract :

In this paper, the notion of mp-residuated lattice, as a subclass of residuated lattices in which every prime filter contains a unique minimal prime filter, is introduced and investigated. For a residuated lattice A, the notion of ω-filter is introduced and it is shown that Ω(A), the set of ω-filters of A, is a bounded distributive lattice. Also, it is observed that γ(A), the set of coannulets of A, is a sublattice of Ω(A). Then for each prime filter P of A, the notion of the divisor filter D(P) as an important tool in investigating of minimal prime filters of A is introduced and it is proved that a prime filter P is minimal prime if and only if P=D(P). Finally, by the notion of ω-filters, as an extension of divisor filters, a fundamental characterization of mp-residuated lattices is given and it is shown that a residuated lattice is mp if and only if the set of its ω-filters is a sublattice of the lattice of its filters.

References:

1. Birkhoff, G. (1940), Lattice theory, Vol. 25, American Mathematical Soc.

 

1. Grätzer, G. and E. T. Schmidt. 1957. On a problem of M. H. Stone. Acta Mathematica Hungarica 8(3-4): 455–460.

 

1. Cornish, W. H. 1972. Normal lattices. J. Austral. Math. Soc. 14(2): 200–215.

 

1. Wallman, H. 1938. Lattices and topological spaces. Ann. Math. 39(2): 112–126.

 

1. Johnstone, P. T. 1982. Stone Spaces. Cambridge Stud. Adv. Math., 3, Cambridge University Press, Cambridge.

 

1. Pawar, Y. S. 1993. Characterizations of normal lattices. Indian J. Pure Appl.Math. 24(11):651–656.

 

1. Zaanen, A.C. 1983. Riesz spaces II (Vol. 30). Elsevier.

 

1. Krull W.,” Axiomatische Begründung der allgemeinen Ideal theorie”, Sitzungsberichte der physikalisch medizinischen Societad der Erlangen, 56 (1924) 47-63.

 

1. Dilworth R. P, “Abstract residuation over lattices”, Bull. Amer. Math. Soc, 44 (1938) 262-268.

 

10.Dilworth R. P, “Non-commutative residuated lattices”, Trans. Amer. Math. Soc,. 46 (1939) 426-444.

 

11.Ward M., “Residuation in structures over which a multiplication is defined”, Duke Math. Journal, 3 (1937) 627-636.

 

12.Ward M., “Structure Residuation”, Annals of Mathematics, 2nd Ser., 39(3) (1938) 558-568.

 

13.Ward M., “Residuated distributive lattices”, Duke Math. J., 6 (1940) 641-651.

 

14.Ward M., Dilworth R. P, “Residuated Lattices”, Proceedings of the National Academy of Sciences, 24 (1938) 162-164.

 

15.Ward M., Dilworth R. P, “Residuated lattices”, Transactions of the American Mathematical Society, 45 (1939) 335-354.

 

16.Höhle U., “Commutative residuated monoids”, in: U. Höhle, P. Klement (Eds.), Non-classical Logics and Their Aplications to Fuzzy Subsets, Kluwer Academic Publishers, (1995).

 

17.Okada M., Terui K., “The finite model property for various fragments of intuitionistic linear logic”, Journal of Symbolic Logic, 64 (1999) 790-802.

 

18.Blok W. J., Pigozzi D., “Algebraizable Logics”, Mem. Am. Math. Soc., vol. 396, Amer. Math. Soc., Providence, (1989).

 

19.Idziak P. M., “Lattice operations in BCK-algebras”, Mathematica Japonica, 29 (1984) 839-846.

 

20.Flondor P., Georgescu G., Iorgulescu A., “Pseudo-t-norms and pseudo-BL algebras”, Soft Computing, 5(2001) 355-371.

 

21.Galatos N., Jipsen P., Kowalski T., Ono H., “Residuated lattices: an algebraic glimpse at substructural logics”, Elsevier (2007).

 

22.Rasouli S., Davvaz B., “An investigation on Boolean prime filters in BL-algebras”, Soft Computing, 19(2015) 2743–2750.

 

23.Rasouli S., Radfar A., “PMTL filters, R  filters and PBL filters in residuated lattices”, Journal of Multiple Valued Logic and Soft Computing, 29(6) (2017) 551–576.

 

24.Rasouli S., “Heyting, Boolean and pseudo-MV filters in residuated lattices”, Journal of Multiple Valued Logic and Soft Computing 31(4) (2018) 287–322.

 

25.Rasouli S., “Generalized co-annihilators in residuated lattices”, Annals of the University of Craiova, Mathematics and Computer Science Series, 45(2) (2018) 1–18.

 

26.Jipsen P., Tsinakis C., “A survey of residuated lattices”, Ordered Algebraic Structures, 7 (2002) 19-56.

 

27.Rasouli, S. 2019. The going up and going down theorems in residuated lattices. Soft computing DOI: 10.1007/s00500-019-03780-3.

 

28. rätzer G., “Lattice theory” San Francisco: W. H. Freeman and Company, (1979).