A New Characterization for n–Fold Positive Implicative BL–Logics

Fuzzyn-Fold Filters of Pseudoresiduated Lattices

“…(5) 1 → x = x; x → x = 1; x → 1 = 1; x ≤ y → x, x ≤ x, x = x; (6) x ⊗ x = 0; x ⊗ y = 0 iff x ≤ ȳ; (7) x ≤ y implies x ⊗ z ≤ y ⊗ z, z → x ≤ z → y, y → z ≤ x → z, ȳ ≤ x; (8) x ⊗ y = x → ȳ; (9) x ∨ y = 1 implies x ⊗ y = x ∧ y and x n ∨ y n = 1 for every n ≥ 1; (10) x ⊗ (y ∨ z) = (x ⊗ y) ∨ (x ⊗ z); (11) (x ∨ y) → z = (x → z) ∧ (y → z); (x → z) ∨ (y → z) ≤ (x ∧ y) → z; (12) (x ∨ y) ⊗ (x ∨ z) ≤ x ∨ (y ⊗ z), hence (x ∨ y) mn ≤ x n ∨ y m ; (13) x ∨ y ≤ ((x → y) → y) ∧ ((y → x) → x); (14) x → y ≤ (y → z) → (x → z); (15)…”

“…Since Hájek introduced his Basic Fuzzy logics, (BL-logics) in short in 1998 [1], as logics of continuous t-norms, a multitude research papers related to algebraic counterparts of BL-logics, has been published. In [2], [3], [9] and [13] the authors defined the notion of n-fold implicative filters, n-fold positive implicative filters, n-fold boolean filters, n-fold fantastic filters, n-fold obstinate filters, n-fold normal filters in BL-algebras and studied the relation among many type of n-fold filters in BL-algebra. The aim of this paper is to extend this research to residuated lattices with the connection of the results obtaining in [14], [11], [7].…”

Folding Theory Applied to Residuated Lattices

“…Remark 24. A filter satisfying Proposition 23(iii) is also said to be -fold positive implicative [5].…”

“…Since Hájek introduced his basic fuzzy logics, (BL-logics in short) in 1998 [1], as logics of continuous -norms, a multitude of research papers related to the algebraic counterparts of BLlogics, has been published. In [2][3][4][5], the authors defined the notions of -fold (implicative, positive implicative, Boolean, fantastic, obstinate, and normal) filters in BL-algebras and studied the relation among them.…”

“…Then ( , ∧, ∨, ⊗, → , 0, 1) is a residuated lattice which is not a BL-algebra, since ( → ) ∨ ( → ) = ∨ = ̸ = {1}, 2 = {1, }, 3 = {1, , }, 4 = {1, , } are filters of . Let = {0, , , , 1} be a lattice such that 0 < < < 1, 0 < < < 1, where and are not comparable.Define the operations ⊗ and → by the two tables shown in(5). Then ( , ∧, ∨, ⊗, → , 0, 1) is a residuated lattice which is not a MTL-algebra, since ( → ) ∨ ( → ) = ∨ = {1} is the only proper filter of .…”