# On Subtractive Derivations of Rl-Monoids

Authors: Yu Gao; Bingfang Li
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

This paper is intended to introduce the subtractive derivations and study some of their algebraic properties on Rl-monoids. Also, we give some characterizations of subtractive derivations on the Gödel center. Moreover, Gödel algebras are characterized by a fixed set of subtractive derivations. Finally, we discuss the relationship between subtractive derivations and other derivations for Rl-monoids. These results of the paper can provide the common properties of subtractive derivations in the t-norm-based fuzzy logical algebras.


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## Body

## 1. Introduction

Residuated lattice ordered monoids (Rl-monoids, for short) were introduced by Swamy [1] as a common generalization of Abelian lattice ordered groups and Heyting algebras. Moreover, residuated lattice ordered monoids are in very close connections with algebras of t-norm-based fuzzy logics [2–8]. In particular, BL algebras and MV-algebras can be viewed as particular cases of such algebras. It is worth noting that many properties of BL-algebras are also satisfied in all Rl-monoids. In view of this point, Rl-monoids could be taken as an algebraic semantics of a more general logic than Hájek’s [9] basic fuzzy logic. So, R l-monoids can also play an important role in studying fuzzy logic.The notion of derivations is instrumental in studying properties and structure in fuzzy logical algebraic structure. Posner [10], in 1957, studied different kinds of derivations in a prime ring and some of their basic algebraic properties. After that, Borzooei et al. [11–13] gave some characterizations of p-semisimple BCI-algebras via derivations with respect to BCI-algebras with derivation. In 2008, Xin et al. [14–16] characterized modular lattices and distributive lattices by isotone derivations with respect to lattices with derivations. Moreover, Alshehri et al. [17–19] derived the derivations on MV-algebras and gave some conditions under which an additive derivation is, in fact, isotone for a linearly ordered MV-algebra. In 2013, Lee et al. [20, 21] introduced and studied derivations and f-derivations on lattice implication algebras and discussed the relations between derivations and filters. In 2016, He et al. [22] investigated the kinds of derivations in residuated lattices and characterized Heyting algebras with respect to the above derivations. In 2017, Hua [23] studied derivations in R0-algebras, which are equivalent to NM-algebras, and discussed the relation between filters and fixed point set of these derivations. In 2022, Liu studied some results on implicative derivations on MTL-algebras and gave some characterizations of them by these kinds of derivations. The paper is motivated by the following considerations: the previous research about derivations on t-norm-based fuzzy logical algebras is multiplicative derivation and implicative derivations, which are two maps that satisfy.(1)dx⊛y=dx⊛y⋓x⊛dy,multiplicativederivation,dx⊛y=dx↪y⋓x↪dy,implicativederivation.However, there have been few research studies on derivations defined by⋒ and any other operations on residuated structures so far. But this point is worth exploring since it can be studied in algebraic structures much more thoroughly by other operations. Then, it is interesting to study these kinds of derivations on fuzzy logical algebras.Based on these considerations, we propose a new type of derivation, subtractive derivation, for Rl-monoids and study some of their algebraic properties. The structure of this paper is as follows: In Section 2, we review some basic notions and definitions of Rl-monoids. In Section 3, we introduce subtractive derivation on Rl-monoids and give some of their characterizations. In Section 4, we discuss the relations between the fixed point set of subtractive derivations and the ideals of Rl-monoids. In Section 5, we discuss the relations between subtractive derivations and other derivations, for example, multiplicative derivations and implicative derivations on Rl-monoids.


## 2. Preliminaries

First, some basic notions of Rl-monoids and their related algebraic results are presented.Definition 1.
(see [9]). An algebra M,⊛,↪,⋒,⋓,0,1 is said to be a residuated lattice if(1)
M,⋒,⋓,0,1 is a bounded lattice,(2)
M,⊛,1 is a commutative monoid,(3)
u⊛v≤w iff u≤v↪w, for any u,v,w∈L.
ByM we mean that the universe of a residuated lattice M,⊛,↪,⋒,⋓,0,1. On M, we define(2)u≤viffu↪v=1.
Then,≤ is a binary partial order on M and for u∈M,0≤u≤1.
A residuated latticeM is an Rl-monoid if it satisfies the divisibility equation(3)DIVu⋒v=u⊛u↪v.
AnRl-monoid M is a Gödel algebra if it satisfies(4)IDEu⊛u=u.
We denote the setuu⊛u=u of M by IM.
In everyRl-monoid, we define the operation as follows:(5)u⊟v=u⊛v⋆,where v⋆=v↪0.Proposition 1 (see [1]).
The following hold inRl-monoid M, for all u,v,w∈M:(1)
u⊟0=u,0⊟u=0,u⊟u=0,1⊟u=u⋆,u⊟1=0,(2)
u↪v⇀w=u⊛v↪w=v↪u↪w,(3)
ifu≤v, then v↪w≤u↪w,w↪u≤w↪v,u⊛w≤v⊛w,u⊟w≤v⊟w,w⊟v≤w⊟u,(4)
u⊛v≤u⋒v≤u, v≤u⋓v≤u⊞v,(5)
u⊟v≤u, u⊟v≤v,(6)
u⊛u⋆=0,u⊛0=0,(7)
u≤v iff u⊛v⋆=0 iff u⊟v=0,(8)
u⊟v⊟w=u⊟w⊟v.Definition 2.
(see [24]). A nonempty subset I of an Rl-monoid M is an ideal if it satisfies the following conditions:(1)
ifu≤v and v∈I, then u∈I,(2)
ifu,v∈I, then u⋓v∈I.Definition 3.
(see [15]). A self-map d on an Rl-monoid M is called a lattice derivation if it satisfies, for any u,v∈M,(6)du⋒v=du⋒v⋓u⋒dy.Definition 4.
(see [24]). A self-map d on an Rl-monoid M is called a multiplicative derivation if it satisfies, for any u,v∈M,(7)du⊛v=du⊛v⋓u⊛dv.Proposition 2 (see [22]).
A self-mapda:M⟶M(8)dau=a⊛u.
On anRl-monoid M is a multiplicative derivation.


## 3. Subtractive Derivations of Rl-Monoids

Then, we introduce a new kind of derivations on Rl-monoids and give some characterizations of them.Definition 5.
LetM be an Rl-monoid. A mapping d:L↪L is called a subtractive derivation on M if(9)du⊟v=du⊟v⊛u⊟dv,for any u,v∈M.
We will denote byDM to be the set of all subtractive derivations of M.
Some examples of subtractive derivations on Rl-monoids are presented.Example 1.
LetM be an Rl-monoid. Define a mapping 0d:M⟶M by(10)0du=0,for all u∈M. Then, 0d∈DM. Moreover, defining d1:M⟶M by(11)d1u=u,for all u∈M. Then, d1∈DM.Example 2.
LetM=0,u,v,1 be a chain. Defining operations ⊛ and ↪ as follows (see Table1):
Then,M,⊛,↪,⋒,⋓,0,1 is an Rl-monoid. Now, we define d:M⟶M as follows:(12)dx=0,x=0,u,1,x=v,1.
Then,d∈DM.Table 1
The operations of ⊛, and ↪.
⊛0uv100000u0uuuv0uuv10uv1↪0uv101111u0111v0u1110uv1Example 3.
LetMn be the standard n-valued MV-algebra, and hence an Rl-monoid, for some n≥2.(13)du=1n−1,u=1,0,u∈Mn−1.
Then,d∈DM.Remark 1.
Considering the subtractive derivationd in Example 3, we have du⊛v=du=0≠du⊛v⋓u⊛dv, which implies that d is not a multiplicative derivation on M. Moreover, du⋒v=du≠0⋓u=du⋒v⋓u⋒dv, and hence d not a lattice derivation. This all shows that not every subtractive derivation is a multiplicative or lattice derivation on M.Definition 6.
A subtractive derivationd on an Rl-monoid M is called isotone if u≤v implies du≤dv for any u,v∈M.Example 4.
The subtractive derivations in Example 2, 3 are all isotone.Proposition 3.
Ifd∈DM, then for any u,v∈M,(1)
d0=0,(2)
du=du⊛u,(3)
du≤u,(4)
d is isotone,(5)
du⊟v≤u⊟dv,(6)
du=d1⊛u⊛du⋆⋆⋆,(7)
du⊟v≤du⊟dv≤du⋓dv.Proof.
(1)
d0=d0⊟0=d0⊟0⊛0⊟d0=0.(2)
du=du⊟0=du⊟0⊛u⊟d0=du⊛u.(3)
du=du⊛u≤1⊛u=u.(4)
Ifu≤v, then u=u⋒v=v⊛v↪u, and hence(14)du=du⋒v=dv⊛vu=dv⊟v↪u⋆⋆=dv⊟v↪u∗∗⊛v⊟dv↪u⋆⋆≤dv⊟v↪u⋆⋆≤dv.(5)
It can be directly obtained from (2) and Proposition 1 (3).(6)
du=d1⊟u⋆⋆=d1⊛u⊛du⋆⋆⋆.(7)
Obviously from Definition5 and (3).
We will give some characterizations of subtractive derivations onIM, which is a Gödel algebra, and study some of their basic algebraic properties.Theorem 1.
Letd:M⟶M be a map on an R l-monoid M. Then, the following are equivalent:(1)
d∈DIM,(2)
du⊟v=du⊟v, ∀u,v∈IM.Proof.
1⇒2 if d∈DIM, then we have(15)du⊟v=du⊟v⊛u⊟dv≥du⊟v⊛du⊟v=du⊟v.
Conversely,du⊟v=du⊟v⊛u⊟dv≤du⊟v. So du⊟v=du⊟v, ∀u,v∈IM.
2⇒1 let d be a map on DIM such that du⊟v=du⊟v, ∀u,v∈IM. Then, d0=d0⊟d0=d0⊟d0=0. Furthermore, 0=du⊟u=du⊟u, which implies du≤u, hence by Proposition 3 (6), we have du⊟v≤u⊟dv.
Sodu⊟v=du⊟v⊛u⊟dv=du⊟v⋒u⊟dv=du⊟v, ∀u,v∈IM.Proposition 4.
Letd∈DM. Then, the following hold, ∀u,v∈IM:(1)
du=d1⊛u=d1⋒u,(2)
du⊛v=du⊛dv,(3)
dIM⊆IM,(4)
du⋓v=du⋓dv,(5)
du↪v≤du↪dv,(6)
u∈0,d1 iff du=u,(7)
d1≤u iff du=d1.Proof.
(1) By Proposition3 (3), we have u≤du⋆⋆⋆, and hence du=d1⊛u⊛du⋆⋆⋆=di⊛u⋒du⋆⋆⋆=di⊛u=d1⋒u.(2) By (1), we havedu⊛v=d1⋒u⊛v=d1⋒u⋒v=du⋒dv.(3) Ifu∈IM, then by (2), du=du⊛u=du⊛du, which shows dIM⊆IM.(4) By (1), we havedu⋓v=d1⋒u⋓v=d1⋒u⋓d1⋒v=du⋓dv.(5) By (2), we havedu⊛du↪v=du⊛u↪v=du⋒v≤dv, and hence du↪v≤du⟶dv.(6) and (7) are directly from (1), and hence we omit the proof of them.


## 4. The Fixed Point Set of Subtractive Derivations on Rl-Monoids

LetM be an Rl-monoid. Define FM=u∈M|du=u, which is called the fixed point set of subtractive derivation on an Rl-monoid M.Proposition 5.
Ifd∈DM, then FM⊆IM.Proof.
Ifu∈FM, then by Proposition 3 (2), du=du⊛du, and hence u=u⊛u, which shows u∈IM.
The converse of Proposition5 is not true in general.Example 5.
LetM=0,u,v,1 be a chain. Defining operations ⊛ and ↪ as follows (see Table 2):
Then,M,⊛,↪,⋒,⋓,0,1 is an Rl-monoid. Defining d:M⟶M as follows:(16)dx=0,x=0,1,1,x=u,v.
ButFM=0⊆0,1=IM and d∉DM since du⊟v=du=1≠0=du⊟v⊛u⊟dv.Table 2
The operations of ⊛, and ↪.
⊛0uv100000u0uuuv00uv10uv1↪0uv101111uv111vuv1110uv1Proposition 6.
The identity mapidM∈DM iff M is a Gödel algebra.Proof.
IfidM∈DM, then by Proposition 5, M=FM⊆IM, and hence M=IM, which implies that M is a Gödel algebra.
Conversely, ifM is a Gödel algebra, then idM∈DM. Indeed, idMu⊟v=u⊟v=idMu⊟v, by Theorem 1, idM∈DM.
Proposition6 shows that the identity map on a Gödel algebra is a subtractive derivation. Then, we give some conditions under which a subtractive derivation is identified.Theorem 2.
LetM be a Gödel algebra and d∈DM. Then, the following are equivalent:(1)
d=idM,(2)
u⊟dv=du⊟v,(3)
d is injective.Proof.
1⇒2 Obviously.2⇒1 if d satisfies u⊟dv=du⊟v, then by Theorem 1, du=du⊟0=du⊟0=u⊟d0=u, and hence d=idM.1⇒3 Obviously.3⇒1 if d is injective and for any u∈M, then du⊟u=du⊟du=0=d0, and hence u⊟du=0, which implies u≤du. So du=u by Proposition 3 (3).Proposition 7.
LetM be a Gödel algebra and d∈DM. Then(1)
ifu∈M and v∈FM, then v⊟u∈FM,(2)
ifv∈FM and ∀u∈M, then v⋒u∈FM.Proof.
(1)
ifu∈M and v∈FM, then du=d and by Theorem 1, dv⊟u=dv⊟u=v⊟u, which implies v⊟u∈FM.(2)
Ifv∈FM and ∀u∈M, then by Proposition 4 (2), dv⋒u=dv⊛du=v⋒u, which implies v⋒u∈FM.Proposition 8.
LetM be an Rl-monoid. Define a map ha:M⟶M,hau=u⊟a, ∀x,a∈M, then ha∈DM iff haM⊆IM.Proof.
IfhaM⊆IM, then by Proposition 3 (3),(17)hau⊟v⊛u⊟hav=hau⊛u⊛v⋆⊛hav⋆=hau⋒u⊛v⋓hav⋆=hau⊛v⋆=u⊟a⊛v⋆=u⊟a⊟v=u⊟v⊟a=hau⊟v,which implies ha∈DM.
Conversely, ifha∈DM, then(18)u⊟a=hau=hau⊟0=hau⊟0⊛u⊟ha0=u⊟a⊛u⊟a,∀u∈M,which implies haM⊆IM.Theorem 3.
Ifd∈DM such that d is injective, then FM is a lattice ideal iff M is a Gödel algebra.Proof.
Ifd∈DM such that d is injective and M is a Gödel algebra, then by Theorem 2 (3), d=idM, and hence FM=M, which shows that FM is a lattice ideal.
Conversely, ifFM is a lattice ideal and d∈DM such that d is injective, then(19)dd1⋆=1⊟d1=d1⊟d1⊛1⊟dd1=0=d0,that is d1=1, and hence M=FM⊆IM, which shows that M is a Gödel algebra.Proposition 9.
IfdaM⊆IM, then da∈DM.Proof.
IfdaM⊆IM, then by Proposition 4 (3), ∀u,v∈M,(20)dau⊟v⊛u⊟dav=dau⊛u⊛v⋆⊛dav⋆=dau⋒u⊛v⋓dav⋆=dau⊛v⋆=a⊛u⊛v⋆=a⊛u⊟v=dau⊟v,which implies da∈DM.Corollary 1.
IfM is a Gödel algebra, then da∈DM.Proposition 10.
IfM is a Gödel algebra, then the following hold:(1)
d1∈FM,(2)
dM=FM.Proof.
(1)
It follows from Proposition4 (1).(2)
It is obvious thatdM⊇FM. Conversely, if u∈dM, then there exists v∈M such that u=dv. Since u=dv≤d1 and d1∈FM, by Theorem 2, u∈FM, and hence dM⊆FM.Theorem 4.
IfM is a Gödel algebra and I is a lattice ideal with the greatest element, then there exists d∈DM such that FM=I.Proof.
Ifb=∨a∈Ia∈I and db∈∈DM, then dbu≤b with b∈I, and hence dbM⊆I. By Theorem 3, db∈DM. Moreover, if u∈I, then dbu=u⋒b, and hence u∈FM with respect to db, which implies I⊆FM. Furthermore, FM=dM, and hence FM⊆I and FM=I.


## 5. The Relations between Kinds of Derivations on Rl-Monoids

In this section, we will discuss the relations between subtractive derivations and other derivations on Rl-monoids. In particular, we discuss the relations among subtractive derivations, lattice derivations, and multiplicative derivations on Rl-monoids.Proposition 11.
Every subtractive derivation is multiplicative on a Gödel algebraM.Proof.
It follows from Propositions4 (1) and (3).Proposition 12.
Ifd is a multiplicative derivation on an Rl-monoid M and dM⊆IM, then d∈DM.Proof.
It follows from Propositions2 and Corollary 1.Proposition 13 (see [22]).
Ifd is a multiplicative derivation on an Rl-monoid M and d1∈IM, then the following are equivalent:(1)
d is isotone,(2)
du=d1⊛u.Proof.
It follows from Propositions4 (1) and (3).Proposition 14 (see [22]).
Ifd is a lattice derivation on an Rl-monoid M, then the following are equivalent:(1)
d is isotone;(2)
du=d1⋒u.Proof.
It follows from Propositions4 (1) and (3).Theorem 5.
Ifd is a map such that d1∈IM on an Rl-monoid M, then d is a multiplicative derivation iff it is a lattice derivation.Proof.
It follows from Propositions13 and 14.Proposition 15.
Every subtractive derivation is multiplicative on a Gödel algebraM.Proof.
It follows from Proposition11.Corollary 2.
Ifd is a lattice derivation on an R l-monoid M and dM⊆IM, then d∈DM.Corollary 3.
Subtractive derivations and lattice derivations are equivalent on the Gödel algebra.


## 6. Conclusions

The notion of subtractive derivations is beneficial for discussing structures and properties in fuzzy logic algebraic. In order to provide the common properties of subtractive derivations in thet-norm-based logical algebras, we introduce the subtractive derivations on Rl-monoids and obtain some characterizations of them. We also discuss the relations between the fixed point set of subtractive derivations and other kinds of derivations on Rl-monoids. In the future, we will study some representations of Rl-monoids by the algebraic structures of the set of subtractive derivations.



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