Multiplicative and implicative derivations on residuated multilattices

(f, g)-derivation in residuated multilattices

“…minimal element of the set of upper bounds of X and a Multi-infimun is a maxinal element of the set of lower bounds of X. The set of Multi-suprema of X is denoted by Multisup(X) and the set of Multi-infima of X is denoted by Multiinf(X) according to ( [4], [8]). According to [4] a multillatice is said to be full if a b = ∅ and a b = ∅ for all a, b ∈ M .…”

“…An example of bounded full multilattices which is not a lattice is the multillatice with the following Hasse diagram: [8]). A pocrim is a poset (M, ≤) with the maximun element and two binary operation , → such that:…”

“…Definition 1.6. ( [8], [4]). A residuated multilattice M := (M ; , →, ) is a pocrim (M ; ≤, , →, ) whose underlying poset(M, ≤) is an ordered multilattice.…”

“…Proposition 1.9. ( [4], [8]). Let M be a residuated multilattice, for any a, b, c ∈ M , we have the following properties:…”

“…The notion of derivation, which comes from mathematical analysis, is also useful to study some structural properties of various kinds of algebra. The concept of derivation has been introduced in the commutative rings 1957 [3]; BCI-algebra 2004 [7], lattice 2001 [1], MV-algebra 2010 [5], BL-algebra 2017 [1], residuated lattice 2016 [6] and recently greater interest has been developped in residuated multilattice, introduced by Maffeu et al [8]. The notion was further explored in the form of f -derivations on residuated multilattice by the same author.…”