IFI-ideals of lattice implication algebras

Abstract

The notion of IFI−ideal is introduced in lattice implication algebras. Firstly, the equivalent conditions of IF−ideals and IFI−ideals are given in lattice implication algebras. Then the proposition of IFI−ideal is investigated in lattice implication algebras. Next, the relations between IFI−ideal and IF−ideal, between IFI−ideal and IFI−filter, between IFI−ideal and fuzzy impilcative ideals, between IFI−ideal and implicative ideals are discussed in lattice implication algebras. Moreover, the extension theorem of IFI−ideals is obtained, and Ψ(L) which is composed of all IFI−ideals constitutes a closure system. Finally, we prove that ∀α ∈ [0, 1], A = (µ 0,α , µ 0,α ) is an IFI−ideal of lattice implication algebra L if and only if L is a lattice H implication algebra.

Research of Unary Lattice Implication Algebra Equations on Interval Sets

“…In the paper, the concepts are redefined on interval sets, which are filter, prime filter, LI-ideal, dual atom and convex sub-lattice. Meanwhile, three basic unary lattice implication algebra equations [6][7][8][9][10][11][12] on the interval sets are researched. The necessary and sufficient conditions for existence of solutions for the equations are presented.…”