Prime, minimal prime and maximal ideals spaces in residuated lattices

Fuzzy Prime Ideal Theorem in Residuated Lattices

“…Definition 6 (see [10,20]). Let I be a proper ideal of a residuated lattice A. I is said to be a prime ideal, if, for all I 1 , I 2 ∈ I(A), if I 1 ∩ I 2 ⊆ I, then I 1 ⊆ I and I 2 ⊆ I.…”

“…Let I be a proper ideal of a residuated lattice A. I is said to be a prime ideal, if, for all I 1 , I 2 ∈ I(A), if I 1 ∩ I 2 ⊆ I, then I 1 ⊆ I and I 2 ⊆ I. Proposition 1 (see [10,20]). Let P be a subset of residuated lattice A. P is a prime ideal if and only if, for all a, b ∈ A, if a ″ ∧ b ″ ∈ P, then a ∈ P or b ∈ P.…”

“…e notion of ideal has been introduced in several algebraic structures such as BL-algebras [9] and residuated lattice [5,8]. Piciu [10] gives and proves the prime ideal theorem in residuated lattices. Dealing with certain and uncertain information is an important task of the artificial intelligence, in order to make computer simulate human being.…”

“…e notion of fuzzy ideal has been studied in several structures such as rings [12], lattices [13,14], MV-algebras [15], BLalgebras [16], and residuated lattices [6,8,17]. However, recent work of Piciu [10] gives the prime ideal theorem in residuated lattices. But that theorem is not yet investigated in fuzzy logic.…”

Pseudo-irreducible ideals of residuated lattices

“…In 2021, the notion of minimal prime ideals was introduced in residuated lattices and related properties were investigated. Also, new equivalent characterizations and properties for prime and maximal ideals were obtained and the relation between these ideals and minimal prime ideals was discussed for De Morgan residuated lattices ( [14]).…”