(f, g)-derivation in residuated multilattices

The primary goal of this paper is to extend the notion of derivation studied in lattices, residuated lattices and residuated multilattices by introducing two-parameter derivations in residuated multilattices. After defining this notion, we illustrate it with several examples and study the properties of some related notions. Finally, we use the set of complemented elements and the set of fixed points to characterize those derivations.


Introduction
A lattice is a poset where every pair of elements has a least upper bound and a greatest lower bound. In order to generalize the theory of lattices, some hyperstructures were introduced. In particular, multilattice is a structure where the property of being a complete lattice, namely the existence of least upper bound (resp greatest lower bound) for every subset, is weakened to the existence of minimal upper (resp maximal lower) bound (Cabrera et al. 2014).
Residuation plays a prominent role in the algebraic study of logical systems, which are usually modeled as partially ordered sets with some operations reflecting the properties of connectives. Focusing on the link between the theory of multilattices and residuation, I.P. Cabrera et al. introduce residuated multilattices (Cabrera et al. 2014).
The notion of derivation, which comes from mathematical analysis, is also useful for studying some structural properties of various algebra. The concept of derivation has been introduced in the commutative rings 1957 (Bell and Kappe 1989); BCI algebra 2004 (Jun and Xin 2004), lattice 2001 (Alsatayhi and Moussavi 2017), MV algebra 2010 (Ceven and Ozturk 2008), BL algebra 2017 (Alsatayhi and Moussavi 2017), residuated lattice 2016 (He et al. 2016;Keubeng et al. 2020) and in residuated multilattice 2019 . This notion was extended to f -derivations in lattice 2008 (Ceven and Ozturk 2008) and residuated multilattice 2019 . The concept was further explored in the form of ( f , g)-derivations in lattices 2008 (Asci and Ceran 2013), BL algebra (Alsatayhi and Moussavi 2017) and in residuated lattices 2020 (Keubeng et al. 2020). In this paper, we extend the study of ( f , g)-derivation to residuated multilattices and find some examples and first properties. We study f -contractive, ( f , g)-contractive derivation, ( f , g)ideal derivation and investigate several of its properties. This paper is organized as follows: In Section 2, we recall some basic notions and properties of residuated multilattice; in Section 3, we introduce the notions of ( f , g)-derivation on residuated multilattice with some useful examples and investigate some properties. The notion of f -contractive derivation, ( f , g)-contractive derivation, ( f , g)-ideal derivation, f -fixed points and ( f , g)-fixed points are also introduced with some examples and related properties.

Definitions and preliminaries
In this section, we recall some definitions and essential properties used in this work.
Let (M, ≤) be a poset, the set of upper bounds of x ∈ M will be denoted ↑ M x = {a ∈ M : a ≥ x} and dually ↓ M x = {a ∈ M : a ≤ x} denotes the set of lower bounds.
tisupremum of X is a minimal element of the set of upper bounds of X , and a multi-infimum is a maximal element of the set of lower bounds of X . The set of multisuprema of X is denoted by Multisup(X), and the set of multi-infima of X is denoted by Multiinf(X) according to Cabrera et al. (2014); . Cabrera et al. (2014) .
• We will write a b to denote Multisup{a, b} and a b to denote Multiinf{a, b} Definition 2.2  . A poset (M, ≤) is an ordered multilattice if it satisfies that: According to Cabrera et al. (2014), a multilattice is said to be full if a b = ∅ and a b = ∅ for all a, b ∈ M. An example of bounded full multilattice which is not a lattice is the multilattice with the following Hasse diagram, depicted in Fig A pocrim is said to be bounded if it has a least element.
According to Cabrera et al. (2014), for a bounded pocrim (M; ≤, , →, ) with a least element ⊥, we set x = x → ⊥ and X = {x : x ∈ X }, x 0 = , x n = x n−1 x for every x ∈ M and integer n > 0. Now we recall the properties of pocrim that will be used.

Remark 2.7 Cabrera et al. (2014)
• A residuated multilattice is called bounded if it has a bottom element ⊥, • Notice that every residuated multilattice is full.
The example below is the smallest residuated multilattice that is not a residuated lattice. According to Maffeu Nzoda (2018), it has seven elements and will be denoted by R M L 7 . The operations and → on M are defined by the following tables: From now, we give some useful properties and definitions in the rest of the paper. Every residuated multilattice M:=(M; , →, ) will be denoted just by M to simplify notation. Let M be a residuated multilattice, for any a, b, c ∈ M, we have the following properties: and the following ones for a bounded multilattice:

(f , g)-derivations on residuated multilattices
This section introduces ( f , g)-derivation on residuated multilattices and investigates some fundamental properties. The following assertions are equivalent.
Proof If d satisfies A 1 then for all x, y ∈ M and by using the commutativity of the law , we have d( ; with the commutativity of , it follows that d satisfies A 2 . The converse is similar.
Lemma 3.1 allows us to give the following definition. • The map d : We apply the ordinal sum to construct a new example of a residuated multilattice using the procedure defined by Maffeu in Maffeu Nzoda (2018). The constructed residuated multilattice, denoted by M, is obtained by gluing the top element of R M L 7 and the bottom element of L together. We rename the elements in the ordinal sum as : ⊥, a 0 , a 1 , a 2 , a 3 , a 4 , b 0 , b 1 , b 2 , b 3 , . We obtain the residuated multilattice where the order is defined as follows: The operations and → on M are defined by the following tables: We define the homomorphisms f , g and the maps d 1 , d 2 on M by: It is easy to verify that d 1 is a The constructed residuated multilattice, denoted by M, is obtained by gluing the top element of L and the bottom element of R M L 7 together. We rename the elements in the ordinal sum as : ⊥, a 0 , a 1 , a 2 , a 3 , b 0 , b 1 , b 2 , b 3 , b 4 , . We obtain the residuated multilattice such the order are defined as follows: The operations and → on M are defined by the following tables: Now, we define the homomorphisms f , g and the map d on M by: It is easy to verify that d is a ( f , g)-derivation.

Proposition 3.7 Let M be a residuated multilattice and d be a ( f , g)-derivation on M.
For all x, y ∈ M, we have the following statements: If d(⊥) = ⊥ and x ≤ y , then d(y) ≤ (g(x)) and (2) By using Proposition 2.9 (M 10 ), we have x ≤ y implies x y = ⊥; thus, Proposition 3.8 Let M be a residuated multilattice and d be Definition 3.9 Let M be a residuated multilattice and d be a In particular, if d is both monotone and f -contractive, we call d an f -ideal ( f , g)-derivation.
Example 3.10 • Let M be a residuated multilattice depicted in Example 3.5. We can observe that d 1 is ( f , g)contractive. • Let M be a residuated multilattice depicted in Example 3.6, and let us define maps f , g, d so that f = I d M , We can observe that d is g-contractive but d is not fcontractive since d(a 0 ) = a 1 , f (a 0 ) = a 0 and a 1 a 0 . • Let M be a residuated multilattice depicted in Example 3.5. It can be easy to verify that d 1 is not ideal ( f , g)derivation on M, because d 1 is not monotone.
Proposition 3.12 Let M be a residuated multilattice and d be a monotone ( f , g)-derivation on M. We have the following statements: for all x, y ∈ M.
Proof (1) Let x, y, z ∈ M. z ≤ x → y implies x z ≤ y by Definition 2.4 2). Then we have d(x z) ≤ d(y) by the monotonicity of d. Proposition 3.14 Let M be a residuated multilattice and d be a ( f , g)-derivation on M. We have the following statements: (1) For all x, y ∈ M and b ∈ d(x) d(y), there exists a ∈ (d(x) f (y)) (g(x) d(y)) such that a ≤ b,

Conclusion and future work
The present paper has initiated the study of ( f , g)-derivations in residuated multilattices with illustrative examples. We have also studied the notion of f -contractive derivation, ( f , g)-contractive derivation, ( f , g)-ideal derivation, the set FIX f ,g d (M) and investigated some fundamental properties. The derivation that we have defined coincides with derivation defined in  when f and g are identity functions. For future work, we plan to study the notion of principal ( f , g)-derivations and investigate the representation of a residuated multilattice in terms of its principal ( f , g)-derivations.
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