Derivations of MV‐Algebras

Abstract

We introduce the notion of derivation for an MV-algebra and discuss some related properties. Using the notion of an isotone derivation, we give some characterizations of a derivation of an MV-algebra. Moreover, we define an additive derivation of an MV-algebra and investigate some of its properties. Also, we prove that an additive derivation of a linearly ordered MV-algebral is an isotone.

A note on derivations on basic algebras

“…In the last decade, there have been several papers about derivations on MV-algebras and other related algebras; see e.g., Alshehri (2001), Ghorbani et al (2013), Yazarli (2013). The concept has been inspired by derivations on rings, thus a derivation on an MV-algebra A is a map d : A → A satisfying d(x ⊕ y) = d(x) ⊕ d(y) and d(x y) = (d(x) y) ⊕ (x d(y)) for all x, y ∈ A, although by a derivation is often meant a map satisfying the latter condition only, while a map that satisfies both conditions is called an additive derivation.…”

Study of MV-algebras via derivations

“…Proof. Now, let d be a regular difference derivation on MV-algebra such that e = d (1). Further, let L 1 = [0, e], L 2 = [0, e * ].…”

“…Inspired by this, several authors have studied generalized derivations in BCI-algebras [2,23]. In the past few years, Xin [19,20] introduced the concept of derivations in a lattice, where operations + and • are interpreted as lattice operations ∨ and ∧, respectively, and characterized modular lattices and distributive lattices by isotone derivations; Alshehri [1] introduced the notion of (additive) derivations for an MV-algebra, where operations + and • are interpreted as ⊕ and , and discussed some related properties; Sang and Yong [13,21] investigate derivation and generalized derivation in lattice implication algebra and characterized the fixed set by these derivations; He [11] investigated derivations in residuated lattices and characterize Heyting algebras in terms of derivations; Zhu [22] introduced some derivations in linguistic truth-valued lattice implication algebras and discussed the relationship between them; Wang [18] investigated derivations in commutative multiplicative semilattices and characterize quantales in terms of derivations; Liang [14] introduced the notion of derivations of EQ-algebra and gave some characterizations of them.…”

Derivations in Lukasiewicz semirings

“…Starting with 2010, several authors extended the study of derivations to certain algebras forming an algebraic semantic for non-classical logics including the logic of quantum mechanics. In this context let us mention the papers [1] and [10]. It was shown in [5] that every MV-algebra can be converted into a socalled Łukasiewicz semiring.…”