Some applications of fuzzy sets in residuated lattices

In this paper, based on ideals, we investigate residuated lattices from fuzzy set theory and lattice theory point of view. Ideals are important concepts in the theory of algebraic structures used for formal fuzzy logic and first, we investigate the lattice of fuzzy ideals in residuated lattices. Then we present applications of fuzzy sets in Coding Theory and we study connections between fuzzy sets associated to ideals and Hadamard codes.


Introduction
The notion of residuated lattice, introduced in [17] by Ward and Dilworth, provides an algebraic framework for fuzzy logic.
Managing certain and uncertain information is a priority of artificial intelligence, in an attempt to imitate human thinking.To make this possible, in [18], Zadeh introduced the concept of fuzzy subset of a nonempty set.
In this paper, we study some applications of fuzzy sets in residuated lattices.
In [15], this concept is applied to these algebras and the fuzzy ideals are introduced.In Section 3, we investigate more properties of fuzzy ideals and we study their lattice structure, which is a Heyting algebra.
In Section 4 we found connections between the fuzzy sets associated to ideals in particular residuated lattices and Hadamard codes.
In this paper, L will be denoted a residuated lattice, unless otherwise stated.A Heyting algebra ( [1]) is a lattice (L, ∨, ∧) with 0 such that for every a, b ∈ L, there exists an element a → b ∈ L (called the pseudocomplement of a with respect to b) where a → b = sup{x ∈ L : a ∧ x ≤ b}.Heyting algebras are divisible residuated lattices.
For x, y ∈ L, we define x ⊞ y = x * → y * * and x ⊎ y = x * → y, where x * = x → 0. We remark that ⊞ is associative and commutative and ⊎ is only associative.
An ideal in residuated lattices is a generalization of the similar notion from MV-algebras, see [6].This concept is introduced in [15] using the operator ⊎ which is not commutative.An equivalent definition is given in [3] using ⊞.We remark that ⊞ is associative and commutative and ⊎ is only associative.Definition 1 ( [3]) An ideal residuated lattice L is a subset I = ∅ of L such that: (i 1 ) For x ≤ i, x ∈ L, i ∈ I =⇒ x ∈ I; (i 2 ) i, j ∈ I =⇒ i ⊞ j ∈ I.
Let A be a non-empty set.If [0, 1] is the real unit interval, a fuzzzy subset of A is a function µ : A −→ [0, 1], see [18].If µ is not a constant map, then µ is a proper fuzzy subset of A.
Let B ⊂ A be a non-empty subset of A. The map µ (the characteristic function) is a fuzzy subset.The notion of fuzzy ideal in residuated lattices is introduced in [15] and some characterizations are obtained.

Definition 2 ([15]
) A fuzzy ideal of a residuated lattice L is a fuzzy subset µ of L such that: Two equivalent definitions for fuzzy ideals are given in [15]: A fuzzy ideal of L is a fuzzy subset µ of L such that: We denote by I(L) the set of ideals and by F I(L) the set of fuzzy ideals of the residuated lattice L.
There are two important fuzzy subsets in a residuated lattice L : For The fuzzy subset µ I is a generalization of the characteristic function of I, denoted ϕ I.Moreover, in [15] is proved that I ∈ I(L) iff µ I ∈ F I(L).

Lemma 3 ([15]
) For µ ∈ F I(L), the following hold: For µ 1 and µ 2 two fuzzy subsets of L is define the order relation Moreover, for a family {µ i : i ∈ I} of fuzzy ideals of L we define for every x ∈ L, see [18].
Obviously, ∩ i∈I µ i ∈ F I(L) but, in general ∪ i∈I µ i is not a fuzzy ideal of L, see [14].We recall (see [1]) that a complete lattice (A, ∨, ∧) is called Brouwerian if it satisfies the identity a ∧ (

Remark 4 ([1]
) Let A be a set of real numbers.We say that l ∈ R is the supremum of A if: 1. l is an upper bound for A; 2. l is the least upper bound: for every ǫ > 0 there is a ǫ ∈ A such that a ǫ > l − ǫ, i.e., l < a ǫ + ǫ.
Remark 5 If a, b are real numbers such that a, b ∈ [0, 1] and a > b − ǫ, for every ǫ > 0, then a ≥ b.Indeed, if we suppose that a < b, then there is ǫ 0 > 0 such that b − a > ǫ 0 > 0, which is a contradiction with hypothesis.
Proof.If (µ i ) i∈I is a family of fuzzy ideals of L, then the infimum of this family is Obviously, the lattice (F I(L), ⊂ ) is complete.
Also, by Lemma 6, since x x 1 ⊞ ... ⊞ x n we have that y Thus, we deduce that We conclude that min(µ(x), Thus, , for every k = 1, ..., n, using the fact that x y 1 ⊞ ... ⊞ y n , we obtain But ǫ is arbitrary, so from Remark 5, By [1] and Theorem 11 we deduce that: Moreover, 4 Applications of fuzzy sets in Coding Theory

Symmetric difference of ideals in a finite commutative and unitary ring
In this section, we will present an application of a fuzzy sets on some special cases of residuated algebras, namely Boolean algebras.
Let A be a non-empty set and B ⊂ A be a non-empty subset of A. The map For two nonempty sets, A, B, we define the symmetric difference of the sets Proposition 14 We consider A and B two nonempty sets.
(i) We have µ A∆B = 0 if and only if A = B; (ii) ([13], p. 215).The following relation is true (iii) Let A i , i ∈ {1, 2, ..., n} be n nonempty sets.The following relation is true Remark 15 Let (R, +, •) be a unitary and a commutative ring and I 1 , I 2 , ..., I s be ideals in R.
(i) For i = j, we have I i ∆I j is not an ideal in R. Indeed, 0 / ∈ I i ∆I j , therefore I i ∆I j is not an ideal in R; The vector c B is called the codeword attached to the set B. We can represent c B as a string c B = c 1 c 2 ...c n .

Linear codes
We consider p a prime number and F p n a finite field of characteristic p. F p n is a vector space over the field Z p .A linear code C of length n and dimension k is a vector subspace of the vector space F p n .If p = 2, we call this code a binary linear code.The elements of C are called codewords.The weight of a codeword is the number of its elements that are nonzero and the distance between two codewords is the Hamming distance between them, that means represents the number of elements in which they differ.The distance d of the linear code is the minimum weight of its nonzero codewords, or equivalently, the minimum distance between distinct codewords.A linear code of length n, dimension k, and distance d is called an [n, k, d] code (or, more precisely, [n, k, d] p code).The rate of a code is k n , that means it is an amount such that for each k bits of transmitted information, the code generates n bits of data, in which n − k are redundant.Since C is a vector subspace of dimension k, it is generated by bases of k vectors.The elements of such a basis can be represented as a rows of a matrix G, named generating matrix associated to the code C.This matrix is a matrix of k × n type.(see [Gu;10]).The codes of the type [2 t , t,

Connections between Boolean algebras and Hadamard codes
In the following, we present a particular case of residuated lattices, named MValgebras.
We remark that the concept of ideal in residuated lattices is a generalization for the notion of ideal in MV-algebras.
Definition 23 ([6], p. 13) An ideal P of the MV-algebra (X, ⊕, ′ , θ) is a prime ideal in X if and only if for all x, y ∈ P we have (x ′ ⊕ y) ′ ∈ P or (y ′ ⊕ x) ′ ∈ P .Definition 26 Let (W, •, , 1) be a Wajsberg algebra and P ⊆ W be a nonempty subset.P is called a prime ideal in W if and only if for all x, y ∈ P we have Definition 27 The algebra (B, ∨∧, ∂, 0, 1), equipped with two binary operations ∨ and ∧ and a unary operation ∂, is called a Boolean algebra if and only if (B, ∨∧) is a distributive and a complemented lattice with x ∧ ∂x = 0, for all elements x ∈ B. The elements 0 and 1 are the least and the greatest elements from the algebra B.
Remark 28 (i) Boolean algebras represent a particular case of MV-algebras.
Remark 29 (i) If X is an MV-algebra and I is an ideal (prime ideal) in X, therefore on the Wajsberg algebra structure, obtained as in Remark 3.7.
ii), we have that the same set I is an ideal (prime ideal) in X as Wajsberg algebra.The converse is also true.
(ii) Finite MV-algebras of order 2 t are Boolean algebras.
(iii) Between ideals in a Boolean algebra and ideals in the associated Boolean ring it is a bijective correspondence, that means, if I is an ideal in a Boolean algebra, the same set I, with the corresponded multiplications, is an ideal in the associated Boolean ring.The converse is also true.
We consider (R, +, •) a finite, commutative, unitary ring and I, J be two ideals.Let c I and c J be the codewords attached to these sets, as in Definition 16.
Proposition 30 With the above notations, we have that: (i) To the set I∆J correspond the codeword c I + c J = c I ⊕ c J , where ⊕ is the XOR-operation; (ii) If I 1 , I 2 , ..., I q are ideals in the ring R and c I1 , c I2 , ..., c Iq are the attached codewords, therefore the vectors c I1 , c I2 , ..., c Iq are linearly independent vectors.
With the above notations, we consider a matrix M C , with rows the codewords associated to the ideals I 1 , I 2 , ..., I q , Since these rows are linearly independent vectors, the matrix M C can be considered as a generating matrix for a code, called the code associated to the ideals I 1 , I 2 , ..., I q , denoted C I1I2,...Iq .
Theorem 31 Let (B, ∨∧, ∂, 0, 1) be a finite Boolean algebra of order 2 n .The following statements are true: (i) The algebra B has n ideals of order 2 n−1 ; (ii) The code associated to above ideals generate a Hadamard code of the type Proof.(i).It is clear, since ideals in the Boolean algebra structure are ideals in the associated Boolean ring and vice-versa.(ii).Let I 1 , I 2 , ..., I n be the ideals of order 2 n−1 .With the above notations, we consider a matrix M C , with rows the codewords associated to these ideals, Due to the correspondence between the ideals in the Boolean algebra structure, the ideals in the associated Boolean ring and Proposition 30, we have that the rows of the matrix M C are linearly independent vectors.Since I 1 , I 2 , ..., I n are the ideals of order 2 n−1 , the associated codewords have 2 n−1 nonzero elements, therefore the Hamming distance is d H = 2 n−1 .From here, we have that M C is a generating matrix for the code C I1I2,...In , which is a Hadamard code of the type [2 n , n, Remark 32 A generating matrix M C of a Hadamard code C of the type [2 n , n, 2 n−1 ] 2 , n ≥ 2, has 2 n−1 n elements equal with 1.If the matrix has the following form: on the row i we have the first 2 n−i elements equal to 1, the next 2 n−i elements equal to 0, and so on, for i ≥ 1, we call this form the Boolean form of the generating matrix of the Hadamard code C and we denote it M B .
Remark 33 (i).If G, a r × s matrix over a field K, is a generating matrix for a linear code C, then any matrix which is row equivalent to G is also a generating matrix for the code C. Two row equivalent matrices of the same type have the same row space.The row space of a matrix is the set of all possible linear combinations of its row vectors, that means it is a vector subspace of the space K s , with dimension the rank of the matrix G, rankG.From here, we have that two matrices are row equivalent if and only if one can be deduced to the other by a sequence of elementary row operations.
(ii).If G is a generating matrix for a linear code C, then, from the above notations, we have that M C and M B are row equivalent, therefore these matrices generate the same Hadamard code C of the type [2 n , n, Theorem 34 With the above notations, let M B be the Boolean form of a generating matrix of the Hadamard code of the type [2 n , n, We can construct a Boolean algebra B of order 2 n which has n ideals of order 2 n−1 , with associated codewords being the rows of a matrix M B .
Proof.We consider the set B i = {0 i , 1 i }, with 0 i ≤ i 1 i , i ∈ {1, 2, ..., n}.On B i we define the following multiplication: It is clear that (B i , • i , ′ , 1 i ), where 0 ′ i = 1 i and 1 ′ i = 0 i , is a Wajsberg algebra of order 2. On B i we have the following partial order relation x i ≤ i y i if and only if x i • i y i = 1 i .
Therefore, on the Cartesian product B = B 1 × B 2 × ... × B n we define a component-wise multiplication, denoted ⋄.From here, we have that (B, ⋄, ′ , 1), where (x 1 , x 2 , ..., x n ) ′ = (x ′ 1 , x ′ 2 , ..., x ′ n ) and 1 = (1, 1, ..., 1), is a Wajsberg algebra of order 2 n .We write and denote the elements of B in the lexicographic order.The element (0 1 , 0 2 , ..., 0 n ), denoted (0, 0, ..., 0) or 0 it is the first element in B. With 1 we denote (1, 1, ..., 1) = (1 1 , 1 2 , ..., 1 n ) which is the last element in B. From Definition 3.8, on B we have the following partial order relation It is clear that on B we have that x ≤ B y if and only if x i ≤ i y i , for i ∈ {1, 2, ..., n}.From the Wajsberg algebra structure we obtain the M V -algebra structure on B, which is a Boolean algebra structure, with the multiplication x ⊕ y = x ′ ⋄ y (⊕ which is the component-wise XOR-sum).The ideals of order 2 n−1 in this Boolean algebra of order 2 n are generated by the maximal elements in respect to the order relation ≤ B .These elements have n − 1 "nonzero" components.First maximal element, in the lexicographic order, is m 1 = (0, 1, 1, ..., 1).This element generates an ideal of order 2 n−1 , containing all elements x j equal or less than m 1 in respect to the order relation ≤ B .Indeed, all these elements x j are maximum n − 2 nonzero components and x ji ≤ i m 1i , i ∈ {1, 2, ..., n}, j ∈ {1, 2, ..., 2 n−1 }, with the first component always zero.We denote with J 1 the set all elements equal or less than m 1 .It results that J 1 with the multiplication ⊕ is isomorphic to the vector space Z n−1 2 , therefore J 1 is an ideal in B. The codeword corresponding to this ideal is (1, 1, ..., 1, 0, 0, ..., 0) in which the first 2 n−1 positions are equal with 1 and the next 2 n−1 are 0 and represent the first row of the matrix M B .The next maximal element in lexicographic order is m 2 = (1, 0, 1, ..., 1) , with zero on the second position and 1 in the rest.This element generates an ideal J 2 of order 2 n−1 , containing all elements x j equal or less than m 2 in respect to the order relation ≤ B .All these elements x j are maximum n − 2 nonzero components and x ji ≤ i m 1i , i ∈ {1, 2, ..., n}, j ∈ {1, 2, ..., 2 n−1 }, with the second component always zero.With the same reason as above, we have that J 2 , with the multiplication ⊕, is isomorphic to the vector space Z n−1 2 , therefore J 2 is an ideal in B. The codeword corresponding to this ideal is (1, 1, ..., 1, 0, 0, ..., 0, 1, 1, ..., 0, ...), with the first 2 n−2 positions equal with 1, the next 2 n−2 are 0 and so on.This codeword represent the second row of the matrix M B , etc.

Example 35
In [8], the authors described all Wajsberg algebras of order less or equal with 9.In the following, we provide some examples of codes associated to these algebras.
Case n = 4.We have two types of Wajsberg algebras of order 4. First type is a totally ordered set which has no proper ideals and the second type is a partially ordered Wajsberg algebra, W = {0, a, b, 1}.This algebra has the multiplication given by the following table: This algebra has two proper ideals I = {0, a} and J = {0, b}.The associated M V -algebra of this algebra is a Boolean algebra.We consider c I = (1, 1, 0, 0) and c J = (1, 0, 1, 0) the codewords attached to the ideals I and J.The matrix is the generating matrix for the Hadamard code of the type 2 2 , 2, 2 .As in Remark 17, this matrix has as columns all 2-bits vectors over Z 2 : {11, 10, 01, 00}.

Conclusions
In this paper, based on ideals, we investigate residuated lattices from fuzzy set theory and lattice theory point of view.Also we found connections between the fuzzy sets associated to ideals in a Boolean algebras and Hadamard codes.As a further research, we will study other connections between fuzzy sets and some type of algebras of logic.
Definition 24 ([10], p. 56) Let (W, •, , 1) be a Wajsberg algebra and let I ⊆ W be a nonempty subset.I is called an ideal in W if and only if the following conditions are fulfilled: Definition 25 (i) θ ∈ I, where θ = 1; (ii) x ∈ I and y ≤ x implies y ∈ I; (iii) If x, y ∈ I, then x • y ∈ I.

• 0 a b c d e f 1 a f 1 f 1 f 1 f 1 b e e 1 1
e e 1 1 c d e f 1 d e f 1 d c c c c 1 1 1 1 e b c b c f 1 f 1 f a a c c e e 1 1 1 0 a b c d e f 1 .

• 0 a b c d e f g 1 d d e e g 1 1 g 1 1 e c d e f g 1 f g 1 f b a b e e e 1 1 1 g a b b d e e g 1 1 1 0 a b c d e f g 1 .
and x, y ∈ I 1 ∆I 2 ∆...∆I n , supposing that x ∈ I j and y ∈ I k , we have that xy ∈ I j and xy ∈ I k , therefore xy ∈ I j ∩ I k .We obtain that µ I1∆I2∆...∆In (xy) = µ Ij (xy) + µ I k (xy) − 2µ I j µ I k (xy) = 0, then xy / ∈ I 1 ∆I 2 ∆...∆I n and I 1 ∆I 2 ∆...∆I n is not an ideal in R.
Definition 16If A = {a 1 , a 2 , ..., a n } is a finite set with n elements and B is a nonempty subset of A, we consider the vector c B = (c i ) i∈{1,2,...,n} , where are 10]led Hadamard codes.Hadamard codes are a class of error-correcting codes (see [KK; 12], p. 183).Named after french mathematician Jacques Hadamard, these codes are used for error detection and correction when transmitting messages are over noisy or unreliable channels.Usually, Hadamard codes are constructed by using Hadamard matrices of Sylvester's type, but there are Hadamard codes using arbitrary Hadamard matrix not necessarily of the above type (see[CR;  20]).As we can see, Hadamard codes have a good distance property, but the rate is of a low level (see[Gu;10]).17([11],Definition16).The generating matrix of a Hadamard code of the type [2 t , t, 2 τ −1 ] 2 , t ≥ 2, has as columns all t-bits vectors over Z 2 (vectors of length t).
⊕, ′ , θ) be an MV-algebra.The nonempty subset I ⊆ X is called an ideal in X if and only if the following conditions are satisfied: 1) is a Wajsberg algebra.