n ‐fold filters in BL‐algebras

Abstract

Key words n-fold (positive) implicative basic logic, n-fold (positive) implicative BL-algebra, n-fold (positive) implicative filter, Gödel algebra, Boolean algebra. MSC (2000) 06F35, 03G25In this paper we introduce n-fold (positive) implicative basis logic and the related algebras called n-fold (positive) implicative BL-algebras. Also we define n-fold (positive) implicative filters and we prove some relations between these filters and construct quotient algebras via these filters.

n-Fold implicative basic logic is Gödel logic

“…Their main result in this respect, Theorem 4.10 in (Haveshki and Eslami 2008), conveys that L is an n-fold implicative BL-algebra iff {1} is an n-fold implicative filter of L.…”

“…Next consider n-fold positive implicative filters F of a BL-algebra L, where n is a fixed natural number, see Theorem 6.3 in Haveshki and Eslami (2008). They are such filters F that, for all x 2 L; ðx n Þ Ã !…”

“…(i) L is Gödel algebra, (ii) any filter F is an n-fold implicative filter of L, (iii) {1} is an n-fold implicative filter of L. (iv) any filter F is an implicative filter of L, (v) {1} is an implicative filter of L. n-fold implicative filters and n-fold implicative algebras are introduced by Haveshki and Eslami mainly for logic reasons, see Definition 3.1 in Haveshki and Eslami (2008); an n-fold implicative basic logic is Hájak's BL logic with an additional axiom ' a nþ1 , a n for all well formed formulae a:…”

“…Haveshki and Eslami (2008) studied n-fold implicative filters, n a fixed natural number, [see (Haveshki and Eslami 2008), Theorem 4.6]; they are such filters F of a BL-algebra L that for all x, y, z [ L holds…”

“…Haveshki and Eslami (2008) defined n-fold implicative basic logic and n-fold positive implicative basic logic, n a natural number, and studied these extensions of Basic fuzzy logic by means of particular filters and algebraic structures. To be precise, an n-fold implicative basic logic is Hájek's BL logic with an additional axiom ' a nþ1 , a n for all well formed formulae a;…”