The main purpose of this paper is to use a new way to extend fuzzy implications $I$ from a generalized sublattice $M$ to a bounded lattice $L$, such that the extended implications preserve many of the considered properties of fuzzy implications on $M$.Furthermore, as a special case, we investigate the extension of $(S,N)-$implications. Results indicate that the extended implications preserve ma...

A two-sided sequence (cn)n∈Z with values in a complex unital Banach algebra is a cosine sequence if it satisfies cn+m + cn−m = 2cncm for any n,m ∈ Z with c0 equal to the unity of the algebra. A cosine sequence (cn)n∈Z is bounded if supn∈Z ‖cn‖ < ∞. A (bounded) group decomposition for a cosine sequence c = (cn)n∈Z is a representation of c as cn = (b n +b−n)/2 for every n ∈ Z, where b is an inver...

Theorem 1.1. Let G, X, p0, and r be as above. Suppose T is translation invariant, maps L log L to L, and is bounded on L0 . Then T is bounded on L, 1 < p < p0 with an operator norm of O((p − 1) ). This theorem is false without the assumption of translation invariance, since L is not an interpolation space between L log L and L0 . For a concrete counterexample, take E and F be subsets of X of me...

Let L be a bounded lattice, let [a, b] and [c, d] be intervals of L, and let φ : [a, b] → [c, d] be an isomorphism between these two intervals. Let us consider the algebra L↔ φ = 〈L;∧,∨, φ, φ−1〉, which is a lattice with two partial unary operations. We construct a bounded lattice K (in fact, a convex extension of L) such that the congruence lattice of L↔ φ is isomorphic to the congruence lattic...