Abstract
This paper is intended to introduce the subtractive derivations and study some of their algebraic properties on R-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 R-monoids. These results of the paper can provide the common properties of subtractive derivations in the t-norm-based fuzzy logical algebras.
1. Introduction
Residuated lattice ordered monoids (R-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 R-monoids. In view of this point, R-monoids could be taken as an algebraic semantics of a more general logic than Hájek’s [9] basic fuzzy logic. So, R -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 -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 -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.
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 R-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 R-monoids. In Section 3, we introduce subtractive derivation on R-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 R-monoids. In Section 5, we discuss the relations between subtractive derivations and other derivations, for example, multiplicative derivations and implicative derivations on R-monoids.
2. Preliminaries
First, some basic notions of R-monoids and their related algebraic results are presented.
Definition 1. (see [9]). An algebra is said to be a residuated lattice if(1) is a bounded lattice,(2) is a commutative monoid,(3) iff , for any .By we mean that the universe of a residuated lattice . On , we defineThen, is a binary partial order on and for .
A residuated lattice is an -monoid if it satisfies the divisibility equationAn -monoid is a Gödel algebra if it satisfiesWe denote the set of by .
In every -monoid, we define the operation as follows:where .
Proposition 1 (see [1]). The following hold in -monoid , for all :(1),(2),(3)if , then ,(4), ,(5), ,(6),(7) iff iff ,(8).
Definition 2. (see [24]). A nonempty subset of an -monoid is an ideal if it satisfies the following conditions:(1)if and , then ,(2)if , then .
Definition 3. (see [15]). A self-map on an -monoid is called a lattice derivation if it satisfies, for any ,
Definition 4. (see [24]). A self-map on an -monoid is called a multiplicative derivation if it satisfies, for any ,
Proposition 2 (see [22]). A self-map On an -monoid is a multiplicative derivation.
3. Subtractive Derivations of R-Monoids
Then, we introduce a new kind of derivations on R-monoids and give some characterizations of them.
Definition 5. Let be an R-monoid. A mapping is called a subtractive derivation on iffor any .
We will denote by to be the set of all subtractive derivations of .
Some examples of subtractive derivations on R-monoids are presented.
Example 1. Let be an R-monoid. Define a mapping byfor all . Then, . Moreover, defining byfor all . Then, .
Example 2. Let be a chain. Defining operations and as follows (see Table1):
Then, is an R-monoid. Now, we define as follows:Then, .
Example 3. Let be the standard -valued MV-algebra, and hence an R-monoid, for some .Then, .
Remark 1. Considering the subtractive derivation in Example 3, we have , which implies that is not a multiplicative derivation on . Moreover, , and hence not a lattice derivation. This all shows that not every subtractive derivation is a multiplicative or lattice derivation on .
Definition 6. A subtractive derivation on an R-monoid is called isotone if implies for any .
Example 4. The subtractive derivations in Example 2, 3 are all isotone.
Proposition 3. If , then for any ,(1),(2),(3),(4) is isotone,(5),(6),(7).
Proof. (1).(2).(3).(4)If , then , and hence(5)It can be directly obtained from (2) and Proposition 1 (3).(6).(7)Obviously from Definition 5 and (3).We will give some characterizations of subtractive derivations on , which is a Gödel algebra, and study some of their basic algebraic properties.
Theorem 1. Let be a map on an R -monoid . Then, the following are equivalent:(1),(2), .
Proof. if , then we haveConversely, . So , .
let be a map on such that , . Then, . Furthermore, , which implies , hence by Proposition 3 (6), we have .
So , .
Proposition 4. Let . Then, the following hold, :(1),(2),(3),(4),(5),(6) iff ,(7) iff .
Proof. (1) By Proposition 3 (3), we have , and hence . (2) By (1), we have . (3) If , then by (2), , which shows . (4) By (1), we have . (5) By (2), we have , and hence . (6) and (7) are directly from (1), and hence we omit the proof of them.
4. The Fixed Point Set of Subtractive Derivations on R-Monoids
Let be an R-monoid. Define , which is called the fixed point set of subtractive derivation on an R-monoid .
Proposition 5. If , then .
Proof. If , then by Proposition 3 (2), , and hence , which shows .
The converse of Proposition 5 is not true in general.
Example 5. Let be a chain. Defining operations and as follows (see Table 2):
Then, is an R-monoid. Defining as follows:But and since .
Proposition 6. The identity map iff is a Gödel algebra.
Proof. If , then by Proposition 5, , and hence , which implies that is a Gödel algebra.
Conversely, if is a Gödel algebra, then . Indeed, , by Theorem 1, .
Proposition 6 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. Let be a Gödel algebra and . Then, the following are equivalent:(1),(2),(3) is injective.
Proof. Obviously. if satisfies , then by Theorem 1, , and hence . Obviously. if is injective and for any , then , and hence , which implies . So by Proposition 3 (3).
Proposition 7. Let be a Gödel algebra and . Then(1)if and , then ,(2)if and , then .
Proof. (1)if and , then and by Theorem 1, , which implies .(2)If and , then by Proposition 4 (2), , which implies
Proposition 8. Let be an R-monoid. Define a map , , then iff .
Proof. If , then by Proposition 3 (3),which implies .
Conversely, if , thenwhich implies .
Theorem 3. If such that is injective, then is a lattice ideal iff is a Gödel algebra.
Proof. If such that is injective and is a Gödel algebra, then by Theorem 2 (3), , and hence , which shows that is a lattice ideal.
Conversely, if is a lattice ideal and such that is injective, thenthat is , and hence , which shows that is a Gödel algebra.
Proposition 9. If , then .
Proof. If , then by Proposition 4 (3), ,which implies .
Corollary 1. If is a Gödel algebra, then .
Proposition 10. If is a Gödel algebra, then the following hold:(1),(2).
Proof. (1)It follows from Proposition 4 (1).(2)It is obvious that . Conversely, if , then there exists such that . Since and , by Theorem 2, , and hence .
Theorem 4. If is a Gödel algebra and is a lattice ideal with the greatest element, then there exists such that .
Proof. If and , then with , and hence . By Theorem 3, . Moreover, if , then , and hence with respect to , which implies . Furthermore, , and hence and .
5. The Relations between Kinds of Derivations on R-Monoids
In this section, we will discuss the relations between subtractive derivations and other derivations on R-monoids. In particular, we discuss the relations among subtractive derivations, lattice derivations, and multiplicative derivations on R-monoids.
Proposition 11. Every subtractive derivation is multiplicative on a Gödel algebra .
Proof. It follows from Propositions 4 (1) and (3).
Proposition 12. If is a multiplicative derivation on an R-monoid and , then .
Proof. It follows from Propositions 2 and Corollary 1.
Proposition 13 (see [22]). If is a multiplicative derivation on an R-monoid and , then the following are equivalent:(1) is isotone,(2).
Proof. It follows from Propositions 4 (1) and (3).
Proposition 14 (see [22]). If is a lattice derivation on an R-monoid , then the following are equivalent:(1) is isotone;(2).
Proof. It follows from Propositions 4 (1) and (3).
Theorem 5. If is a map such that on an R-monoid , then is a multiplicative derivation iff it is a lattice derivation.
Proof. It follows from Propositions 13 and 14.
Proposition 15. Every subtractive derivation is multiplicative on a Gödel algebra .
Proof. It follows from Proposition 11.
Corollary 2. If is a lattice derivation on an R -monoid and , then .
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 the t-norm-based logical algebras, we introduce the subtractive derivations on R-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 R-monoids. In the future, we will study some representations of R-monoids by the algebraic structures of the set of subtractive derivations.
Data Availability
The data supporting the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was funded by the grants of the Natural Science Foundation of Education Committee of Shannxi Province (grant no. 17JK0170) and the Scientific Research Fund Project of Shaanxi Railway Engineering Vocational and Technical College (grant no. KY2017-20).