The generalizations of fuzzy monoids and vague monoids

Since the unit interval with a t-norm or a t-conorm is a special monoid, a natural question is to consider the fuzzification of monoid. In this paper, we mainly present the fuzzy monoid and vague monoid with the help of aggregation functions. Firstly, the fuzzy submonoid about aggregation function is proposed, which can further be fuzzified to the fuzzy t-subnorm and t-subconorm to deal with imprecision. In particular, the corresponding concepts and properties can be obtained directly when aggregation function takes uninorm and nullnorm, respectively. Further, the concept of fuzzy submonoid is extended to lattice structure. Finally, the notion of vague monoid about aggregation operator is constructed and further the special cases under uninorms and nullnorms are considered.


Introduction
Aggregation operation is an important tool in fuzzy mathematics, which can fuse a given set of data into a representative value.There are many types of aggregation operations, which require different choices in different situations.In fuzzy logic, we usually use the triangular norms (t-norms for short) and the triangular conorms (t-conorms for short), as two kinds of fuzzy logic operations, which play a crucial role in fuzzy sets theory [16,17].To further enrich the properties of the aggregation operations, Yager and Rybalov [15] proposed the concept of uninorm.The identity element of uninorm can take any number in the unit interval, not just zero and one in the case of t-norms and t-conorms.In addition, Calvo et al. [4] introduced the notion of nullnorm in 2001.Both of uninorms and nullnorms are a generalization of t-norms and t-conorms, and on the other side, there exists close connection between them.
The fuzzification or relaxation of logical connectives can be effective in solving practical problems such as imprecision, lack of accuracy, or the presence of noise.Considering that many scholars are very interested in fuzzified algebraic structures, including groups, rings, actions [2,12].
Boixader and Recasens [3] fuzzified the concept of submonoid and further explored the fuzzy t-subnorm and fuzzy t-subconorm, which can deal with imprecision effectively.Refer to the method of fuzzy monoid in [3], we will study the fuzzification of submonoid by generalizing the t-norm and t-conorm to aggregation operators, named A-fuzzy submonoid of set M, where A is any aggregation operation.Further, the corresponding conclusions are given when the aggregation function A takes the uninorm U and nullnorm F. The second part of this paper, we explore the concept of A-vague binary operations and A-vague monoids.And then, further the homomorphic relationship between A-vague monoids and A-fuzzy monoids is constructed.
The rest of this paper is arranged as follows.In section 2, we review some basic concepts and results related to aggregation functions, t-norms, t-conorms, uninorms, nullnorms and fuzzy monoids.In section 3, we give the definition of fuzzy submonoid which generates by aggregation operators, and use it on uninorms and nullnorms.And then, we apply the concept of fuzzy submonoid to monoid ([0, 1], T ) and ([0, 1], S ), where T and S are t-norm and tconorm respectively.Further, what we call U-fuzzy t-subnorms and U-fuzzy t-subconorms (resp.F-fuzzy t-subnorms and F-fuzzy t-subconorms) can be obtained later, and the above conclusions can be generalized to lattice.In section 4, the vague binary operators and vague monoids are introduced and the concept of homomorphic mapping between vague monoids and fuzzy monoids are given.Next, we obtain the corresponding conclusions when aggregation function A takes the uninorm U and nullnorm F. Finally, conclusions on our research are given in section 5.

Preliminaries
In this section, we briefly recall some fundamental concepts and definitions about aggregation operations, such as t-norms, t-conorms, uninorms and nullnorms, which shall be needed in the sequel.
is a n-ary aggregation operation, which satisfies the following conditions: For more details about aggregate functions, please refer to [1,6,9].
Example 2.2.[8] The following shows several common t-conorms: • Maximum: S M (x, y) = max(x, y); • Probabilistic sum: S P (x, y) = x + y − xy; • Łukasiewicz t-conorm: S L (x, y) = min(x + y, 1); where f [−1] is the pseudo inverse of f , denoted by f is an additive generator of T and two additive generators of the same t-norm differ only by a positive constant multiple.

Proposition 2.2. [8] A continuous t-conorm S is Archimedean if and only if there exists a continuous and strictly decreasing function g
where g [−1] is the pseudo inverse of g, denoted by • U min , U max : those given by minimum and maximum in A(e), respectively.
• U rep : those that have an additive generator.
Theorem 2.1.[5] Let U : [0, 1] 2 → [0, 1] be a uninorm with identity element e ∈ (0, 1).Then the sections x ֒→ U(x, 1) and x ֒→ U(x, 0) are continuous in each point except perhaps for e if and only if U is given by one of the following formulas.
(2) If U(0, 1) = 1, then U max has the same structure by changing minimum by maximum in A(e).
being commutative in the points (x, y) such that y = g(x) with x = g(g(x)).
The function h is usually called an additive generator of U.
Theorem 2.4.[7,14] Let U be a uninorm continuous in (0, 1) 2 with identity element e ∈ (0, 1).Then either one of the following cases is satisfied.(a) There exist u ∈ [0, e], λ ∈ [0, u], two continuous t-norms T 1 and T 2 and a representable uninorm R such that U can be represented as elsewhere. (1) , two continuous t-conorms S 1 and S 2 and a representable uninorm R such that U can be represented as elsewhere. ( U cos is used to denote the class of all uninorms continuous in (0, 1) 2 .A uninorm as in (1) will be defined by U ≡ T 1 , λ, T 2 , u, (R, e) cos,min , U cos,min for short.Similarly, a uninorm as in (2) will be defined by U ≡ (R, e), v, S 1 , ω, S 2 cos,max , U cos,max for short.Definition 2.6.[4,10] A nullnorm is a binary function F : [0, 1] 2 → [0, 1], for all x, y, z ∈ [0, 1], which satisfies the following conditions: In general, k is always given by F(0, 1).Denote X as the universe, then a fuzzy subset A of X can be represented as µ A : X → [0, 1], where for any x ∈ X, µ A (x) is denoted as the membership function of x in fuzzy set A. The family of all fuzzy sets of X will be defined as F (X). Further, let L be a nonempty poset, then (L; ∧, ∨) is called a lattice, if x ∧ y and x ∨ y always exist for all x, y ∈ L, where the join and meet operations are denoted by "∨" and "∧" in L, respectively.Then the concept of fuzzy subset on L will be given.Definition 2.7.Let (L, ∧, ∨) be a lattice.A fuzzy subset A of L can be represented as follows: where for any x ∈ L, µ A (x) is denoted as the membership function of x in fuzzy set A. The family of all fuzzy sets of L will be defined as F (L). Definition 2.8.[3] A monoid (H, * ) consists of a set H with a binary operation * : H 2 → H, which has identity element and associativity.Definition 2.9.[3,12] Let (H, * ) be a monoid with identity element e, T a t-norm and σ a fuzzy subset of H. Then σ is a T -fuzzy submonoid of H is equivalent to the following conditions.

A-fuzzy submonoid
Refer to the Definition 2.9, we propose the concept of fuzzy submonoid based on aggregation operator A.
Definition 3.1.Let A be an aggregation operator, (M, •) a monoid with identity element e and σ a fuzzy subset of M. Then σ is a A-fuzzy submonoid of M if and only if σ satisfies the following two conditions: (1) Proposition 3.1.Let (M, •) be a monoid and σ an A-fuzzy submonoid of M. Then the core H of σ (i.e., the set of elements x in M such that σ(x) = 1) is a submonoid of M.
In fuzzy logic, the unit interval with a t-norm or a t-conorm is the most important monoid.Thus, we will consider fuzzy submonoids of a given t-norm or t-conorm.Definition 3.2.Let A and T be an aggregation operator and a t-norm, respectively.An A-fuzzy submonoid of ([0, 1], T ) will be called an A-fuzzy t-subnorm of T .Definition 3.3.Let A and S be an aggregation operator and a t-conorm, respectively.An A-fuzzy submonoid of ([0, 1], S ) will be called an A-fuzzy t-subconorm of S .
The following propositions give the more general conclusions of Example 3.1.

U-fuzzy submonoid
In particular, when the aggregation operations take uninorms, we have the following conclusions.
Definition 3.4.Let U be a uninorm, (M, •) a monoid with identity element e and σ a fuzzy subset of M. Then σ is a U-fuzzy submonoid of M if and only if σ satisfies the following conditions: (1) Proposition 3.4.Let U be a uninorm, (M, •) a monoid and σ a U-fuzzy submonoid of M. Then the core H of σ (i.e., the set of elements x of M such that σ(x) = 1) is a submonoid of M.
Proof.The identity element of H obviously existed and the associativity is inherited.For all x, y ∈ H, we have that Therefore, x • y ∈ H. Definition 3.6.Let U and T be a uninorm and a t-norm, respectively.A U-fuzzy submonoid of ([0, 1], T ) will be called a U-fuzzy t-subnorm of T .
Definition 3.7.Let U and S be a uninorm and a t-conorm, respectively.A U-fuzzy submonoid of ([0, 1], S ) will be called a U-fuzzy t-subconorm of S .
The following theorem gives the necessary and sufficient conditions for σ to be a U-fuzzy submonoid of (M, •).
Theorem 3.1.For a monoid M with identity element e, if a uninorm U is disjunctive, then σ is U-fuzzy submonoid of (M, •) if and only if σ ≡ 1.
Further, if we take monoid as t-norm and t-conorm, the following corollaries hold.where σ is fuzzy subset.Then σ is U-fuzzy t-subnorm of ([0, 1], T M ) if and only if σ is decreasing on B and σ(1) = 1.
Proof.It can be proven in a similar way as Proposition 3.5.
Example 3.3.In particular, the fuzzy subset σ is denoted as: According to Proposition 3.5, Not all fuzzy subset σ of a monoid M can find corresponding uninorm U such that σ is a U-fuzzy submonoid of M. Then the following propositions hold.Proposition 3.7.Let σ(x) = x be a fuzzy subset, T be a t-norm.There is no uninorm U such that σ is a U-fuzzy t-subnorm of ([0, 1], T ).
Proof.Assume that there exists uninorm U with identity element e that σ is a U-fuzzy t-subnorm of ([0, 1], T ), then
Furthermore, it holds that which contradicts with the case of x = 1 − e, y < 1 − e.

Proposition 3.9. Let U be representable uninorms and T be continuous Archimedean t-norms. h and t are additive generators of U and T , respectively. A fuzzy subset σ on [0, 1] is a U-fuzzy t-subnorm of ([0, 1], T ) if and only if the mapping
is subadditive.
Further, it can be denoted as Let t(x) = a and t(y) = b,

subadditive mapping and representable uninorms U and continuous Archimedean t-norms T have additive generators h and t, respectively. Then
Example 3.4.Let T L be the Łukasiewicz t-norm with additive generator t, where t(x) = 1 − x.U p is a uninorm with additive generator h, and is subadditive.

F-fuzzy submonoid
Similar to section 3.2, when the aggregation operations take nullnorms, we have the following conclusions.
Definition 3.9.Let (M, •) be a monoid and σ be an F-fuzzy submonoid of M. Then the core H of σ(i.e., the set of elements x of M, such that σ(x) = 1) is a submonoid of M.
Proof.The identity element of H obviously existed and the associativity is inherited.Let x, y ∈ H.
Let S and T be a t-conorm and a t-norm, respectively.The nullnorm F = S , k, T with absorbing element k as follows Example 3.6.Let F L be a nullnorm where Definition 3.11.Let F and T be a nullnorm and a t-norm, respectively.An F-fuzzy submonoid of ([0, 1], T ) will be called an F-fuzzy t-subnorm of T .Definition 3.12.Let F and S be a uninorm and a t-conorm, respectively.An F-fuzzy submonoid of ([0, 1], S ) will be called an F-fuzzy t-subconorm of S .It is easy to get the following corollary when M takes the t-norm T and t-conorm S .
Corollary 3.6.If σ is an F-fuzzy t-subnorm of T where F is a nullnorm with absorbing element k and T is a t-norm, then σ(x) ≥ k for any x ∈ [0, 1].
Corollary 3.7.If σ is an F-fuzzy t-subconorm of S where F is a nullnorm with absorbing element k and S is a t-conorm, then σ(x) ≥ k for any x ∈ [0, 1].
Let S and T be a t-conorm and a t-norm, F M denoted the nullnorms F with absorbing element k as follows Proposition 3.12.A fuzzy subset σ is an F M -fuzzy t-subnorm of ([0, 1], T M ) where F M is a nullnorm with absorbing element k if and only if Proof.Firstly, if σ is an F M -fuzzy t-subnorm of ([0, 1], T min ), then we obviously have σ(1) = 1 and σ(x) ≥ k for any Then the discussion will be divided into three cases: ( ≤ σ(T M (x, y)).
Then the dual conclusions about F M -fuzzy t-subnorm of ([0, 1], S M ) can be obtained.
Proposition 3.13.A fuzzy subset σ is an F M -fuzzy t-subnorm of ([0, 1], S M ) where F M is a nullnorm with absorbing element k if and only if Proof.It can be proven in a similar way as Proposition 3.12.

Lattice value fuzzy submonoid
Let (L, ∨, ∧, ≤) be a bounded lattice with maximum element 1 and minimum element 0. A map A : M → L will be called L-fuzzy set of M and the family of all L-fuzzy set of M is denoted F L (M).
We can generalize the definition of fuzzy t-subnorm and t-subconorm onto lattice valued fuzzy sets.
Proposition 3.14.Let σ be a ∧-fuzzy submonoid.The core of σ (i.e., the set of elements x of M such that σ(x) = 1) is a submonoid of M.
By using the same generalization method, the operation ∨ also has above definitions and propositions.
Proof.Let e and e ′ be two identity elements of M.Then, since e is an identity element, we have that •(e, e ′ , e ′ ) = 1.
Since e ′ is an identity element, we have that •(e, e ′ , e) = 1.
From Definition 4.Under the condition of regular A-indistinguishability operators, the definition of the fuzzy mapping • can be given and further the A-vague monoid is obtained.(1) The fuzzy mapping Proof.(1) Firstly, we prove that • satisfies the properties of Definition 4.3.It follows that for all x, y, z, x ′ , y ′ , z ′ ∈ M.
(2) In the following, let's verify that (M, •) has associativity and identity element.By the associativity of • and the regularity of E, for all x, y, z, d, m, q, w ∈ M, we have that Suppose that e be the identity element of M, it holds that then e is the identity element of •.Hence, (M, •) is an A-vague monoid.
Conversely, a monoid (M, •) can be obtained from a vague monoid (M, •).Proof.First to verify the associativity, Further, we can obtain that 1 = A(•(x, e, x), •(x, e, x • e)) ≤ E(x • e, x), Hence, E(x •e, x) = 1 and E(e• x, x) = 1, that is x •e = x and e• x = x, then e is the identity element of (M, •).Further, if A is an aggregation operator and (M, •) is a monoid, then there exist bijective maps between their A-vague monoids and regular A-indistinguishability operators and they can represent each other: Regarding commutativity, the connection between (M, •) and (M, •) is given.
From E separating points and E(x • y, y The relation between T -vague monoid and T -fuzzy monoid have been studied in [3].In the following, the homomorphisms between them is extended to A-vague monoid and A-fuzzy monoid. is the identity element of Proposition 4.6.Let (M, •) and (N, •) be two A-vague monoids with respect to A-indistinguishability operators E and F respectively such that E ≤ F. Then the identity map id : A crisp monoid (M, •) is an A-vague monoid when x • y = z then •(x, y, z) = 1 and 0 otherwise for all x, y, z ∈ M.

Vague monoid by uninorms and nullnorms
Since uninorm and nullnorm are special binary aggregation operators.Then we can obtain same conclusions and just set out them as follow.
If E(x, y) = 1 implies x = y, then it is said that E separates points.
Under the condition of regular F-indistinguishability operators, the definition of the fuzzy mapping • can be given and further the F-vague monoid is obtained.

Concluding remarks
Comparing the difference between using t-norms and aggregation functions as defining criteria, we can get the following points.
• Notice that t-norm is a special aggregation function, substituting aggregate functions for t-norm can further generalize related properties.In the meantime, special aggregation functions such as uninorm and nullnorm can be brought in to expand the scope of this research.
• Despite losing control of the boundary conditions, the diversity of aggregation functions allows us to have more flexible options for dealing with related problems.
In the future, we intend to consider adding some properties such as archimedean and strictness to make more interesting and meaningful conclusions.
is the U p -fuzzy t-subnorm of T L and U p as shown in Figure.1.

Figure 1 :
Figure 1: Three-dimensional image of U p (x, y) = h −1 (h(x) + h(y)) is the U p -fuzzy t-subnorm of T L and U p as shown in Figure.2.

3 , 1 =Definition 4 . 6 .
A(•(e, e ′ , e ′ ), •(e, e ′ , e)) ≤ E(e, e ′ ).So E(e, e ′ ) = 1 and e = e ′ because E separates points.Let • be a binary operation on M and E an A-indistinguishability operator on M. Then E is regular with respect to • if and only if for all x, y, z ∈ M, E(x, y) ≤ E(x • z, y • z) and E(x, y) ≤ E(z • x, z • y).

Proposition 4 . 3 .
Let E be a regular A-indistinguishability operator on M with respect to a binary operation • on M.

Proposition 4 . 4 .
Let (M, •) be a vague monoid with respect to an A-indistinguishability operator E separating points.Then (M, •) is a monoid where x • y is the unique z ∈ M such that •(x, y, z) = 1.

Definition 4 . 7 .
Let (M, •) be a vague monoid with respect to an A-indistinguishability operator E separating points.Then (M, •) is the monoid associated to the vague monoid (M, •).

Proposition 4 . 5 .
Let E be an A-indistinguishability operator separating points, (M, •) an A-vague monoid and (M, •) its associated monoids (x • y = z if and only

Definition 4 . 8 .
Let (M, •) and (N, •) be two A-vague monoids with respect to the A-indistinguishability operators E and F, respectively.A map f :M → N is a homomorphism from M onto N if and only if •(x, y, z) ≤ •( f (x), f (y), f (z)) for all x, y, z ∈ M.Lemma 4.1.Let (M, •) and (N, •) be two A-vague monoids with respect to A-indistinguishability operators E and F respectively and f : M → N a homomorphism from M onto N. If e is the identity element of M, then f (e) is the identity element of N. Proof.Since 1

Proposition 4 . 16 .
Let (M, •) be a vague monoid with respect to an F-indistinguishability operator E separating points.Then (M, •) is a monoid where x • y is the unique z ∈ M such that •(x, y, z) = 1.Definition 4.21.Let (M, •) be a vague monoid with respect to an F-indistinguishability operator E separating points.Then (M, •) is the monoid associated to the vague monoid (M, •).Further, if F is a nullnorm and (M, •) is a monoid, then there exist bijective maps between their F-vague monoids and regular F-indistinguishability operators and they can represent each other:• •(x, y, z) = E(x • y, z); • E(x, y) = •(x, e, y).Regarding commutativity, the connection between (M, •) and (M, •) is given.Proposition 4.17.Let E be an F-indistinguishability operator separating points, (M, •) an F-vague operation and (M, •) its associated operation (x • y = z if and only if •(x, y, z) = 1).Then (M, •) is commutative if and only if (M, •) is commutative.