mp-residuated lattices

Subject Areas : Statistics

Saeed Rasouli ¹ (Department of Mathematics, Persian Gulf University, Bushehr, 75169, IRAN)
Dariush Heidari 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.

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