n-Normal residuated lattices

Abstract

The notion of n-normal residuated lattice, as a class of residuated lattices in which every prime filter contains at most n minimal prime filters, is introduced and studied. Before that, the notion of ω-filter is introduced and it is observed that the set of ω-filters in a residuated lattice forms a distributive lattice on its own, which includes the set of coannulets as a sublattice. The class of n-normal residuated lattices is characterized in terms of their prime filters, minimal prime filters, coannulets and ωfilters. 1 1 2010 Mathematics Subject Classification: 06F99,06D20

Gelfand residuated lattices

“…For an ideal I of ℓ(A), set ω(I) = {a ∈ A|a ∨ x = 1, for some x ∈ I}, and Ω(A) = {ω(I)|I ∈ ℓ(A)}. Using Proposition 3.4 of Rasouli and Kondo (2019), it follows that Ω(A) ⊆ F(A), and so elements of Ω(A) are called ω-filters of A. For an ω-filter F of A, I F denoted an ideal of ℓ(A), which satisfies F = ω(I F ).…”

“…For a prime filter p of A, ω( ṗ) is called the D-part of p and denoted by D(p). For the basic facts concerning ω-filters of a residuated lattice we refer to Rasouli and Kondo (2019).…”

“…Proposition 2.15. (Rasouli and Kondo, 2019) Let A be residuated lattice. The following assertions hold:…”

Quasicomplemented residuated lattices

“…The set of minimal prime filters of A is denoted by M in(A). For the basic facts concerning minimal prime filters of a residuated lattice belonging to a filter we refer to Rasouli and Kondo (2018). Theorem 2.5.…”

Galois connection of stabilizers in residuated lattices

“…After him, they were investigated by M. Ward and R. P. Dilworth in [37], as the main tool in the abstract study of ideal lattices in ring theory. The properties of residuated lattices were presented in [13,20,[26][27][28][29][30][31][32][33][34][35]. For a survey of residuated lattices we refer to [19].…”