Ideals and fuzzy ideals on residuated lattices

Fuzzy Prime Ideal Theorem in Residuated Lattices

“…A is not an MTL-algebra. Theorem 1 (see [8]). For any residuated lattice A � (A, ∧, ∨, ⊙ , ⟶ , 0, 1), the following properties hold for every x, y, z ∈ A:…”

“…Let A � (A, ≤ ) be a lattice and let F be a nonempty subset of A. We say that F is a filter of A, if it satisfies the following conditions: [5,8,20]). Let A � (A, ∧, ∨, ⊗ , ⟶ , 0, 1) be a residuated lattice and let I be a nonempty subset of A.…”

“…Definition 10 (see [8]). Let η be a fuzzy subset of A. η is a fuzzy ideal of A, if it satisfies the following conditions:…”

“…Ward and Dilworth [3] initiated the notion of residuated lattice and it interested other authors [4][5][6][7][8]. 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.…”

“…Since then, a lot of works have been done on fuzzy mathematical structures and most authors used the above original definition of a fuzzy set. 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.…”