On codes based on BCK -algebras

1 Introduction

BCI/BCK-algebras are two classes of logical algebras which were first introduced in mathematics in 1966 by Y. Imai and K. Iseki, through the paper [1], as a generalization of the concept of set-theoretic difference and propositional calculi. BCK-algebras is a proper subclass of the class of BCI-algebras. These algebras have many applications, one of the recent applications of BCK-algebras was given in the Coding Theory (see [2, 3]).

2 Preliminaries

An algebra (X, ∗ , θ) of type (2, 0) is called a BCI-algebra if the following conditions are fulfilled:

((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = θ, for all x, y, z∈ X ;

If a BCI-algebra X satisfies the following identity: (5) θ ∗ x = θ,  for all x ∈ X,  then X is called a BCK-algebra (see [4]).

The partial order relation on a BCI / BCK-algebra is defined such that x ≤ y if and only if x ∗ y = θ .

A BCI / BCK-algebra X is called commutative if x ∗ (x ∗ y) = y ∗ (y ∗ x) , for all x, y ∈ X and implicative if x ∗ (y ∗ x) = x, for all x, y ∈ X .

If (X, ∗ , θ) and (Y, ∘ , θ) are two BCI / BCK-algebras, a map f : X → Y with the property f (x ∗ y) = f (x) ∘ f (y) , for all x, y ∈ X, is called a BCI / BCK -algebras morphism. If f is a bijective map, then f is an isomorphism of BCI / BCK-algebras (see [4]).

Hereafter in this paper, X always denotes a finite BCI / BCK-algebra.

In the following, we will use some notations and results given in the paper [3].

Definition 2.1. A mapping A ˜ : A → X is called a BCK-function on A, where A is a nonempty set and X is a BCK-algebra.

Definition 2.2. A cut function of A ˜ is defined to be a mapping A ˜ q : A → { 0 , 1 } with q ∈ X, such that  for each x ∈ A we have A ˜ q ( x ) = 1 , if and only if q ∗ A ˜ ( x ) = θ .

Definition 2.3. Let A = {1, 2, …, n} be a set and let X be a BCK-algebra. In [3], to each BCK-function A ˜ : A → X was associated a binary block-code of length n. A codeword in a binary block-code V is an element of the form v _x  = x ₁ x ₂ … x _n such that x _i  = j if and only if A ˜ x ( i ) = j , for i ∈ A and j ∈ {0, 1}.

Definition 2.4. Let V a binary block-code and let v _x  = x ₁ x ₂ … x _n and v _y  = y ₁ y ₂ … y _n be two codewords belonging to V. We define an order relation ⩽ _c on V as follows:v _x  ⩽  _c v _y if and only if y _i  ⩽ x _i , for all i ∈ {1, 2, …, n}.

3 Main results

Definition 3.1. Let (S, ⩽) be a partially ordered set. For q ∈ S, we define a mapping S _q  : S → {0, 1} such that for each b ∈ S, we have S _q  (b) =1 if and only if q ⩽ b .  Using this map, a codeword v _x  = x ₁ x ₂ ⋯ x _n of a binary block-code V can be determined as follow:x _i  = j if and only if S _x  (i) = j, for i ∈ S and j ∈ {0, 1}.

In the following, we will give some examples of binary block codes obtained from a partially ordered set.

Example 3.2. Let S = {0, 1, 2, 3, 4} be a set with a partial order over S as in Fig. 1(a).

Therefore, using above definition, we obtain the table below.

V = {11111, 01111, 00101, 00011, 00001} .

Example 3.3. Let S = {0, 1, 2, 3, 4} be a set with a partial order over S, as we can see in the Fig. 2(a).

Therefore, using Definition 3.1, we obtain the table below.

Example 3.4. Let S = {A, B, C, D} be a set with a partial order over S, as in the Fig. 3(a).

Therefore, using Definition 3.1, we obtain the table below.

Using Definition 2.3, in the following, we will provide examples of binary block-codes arising from BCK-algebras.

Example 3.5. Let X = {0, 1, 2, 3, 4} be a BCK-algebra with the following Cayley table and the order on X represented in the Fig. 4(a).

Let A ˜ : X → X be a BCK-function on X given by

A ˜ = ( 0 1 2 3 4 0 1 2 3 4 ) .

Therefore we obtain the following table.

Example 3.6. Let X = {0, 1, 2, 3, 4} be a BCK-algebra with the following Cayley table and the order on X represented in the See Fig. 5(a).

Let A ˜ : X → X be a BCK-function on X given by A ˜ = ( 0 1 2 3 4 0 1 2 3 4 ) .

Therefore we obtain the following table.

From this table, we obtain the following binary block code V = {11111, 01010, 00101, 00010, 00001} ,   the same as in the Example 3.3. See Fig. 5(b).

Remark 3.7. On a partial ordered set with a minimum element θ we can define a BCK-algebra structure(see [2], (2.1)) From the obtained block-codes by the aforesaid methods, it is obvious that the code obtained in Example 3.2 is the same with the code obtained in Example 3.5 and the code obtained in Example 3.3 is the same with the code obtained in Example 3.6. The explanation is that we use only the order of BCK-algebra, not its algebraic properties. From above examples, it is obvious that the method presented in paper [3] dose not depend on algebraic properties of BCK-algebra and from the above examples, we remark that there is a one-to-one correspondence between the ordering relation ⩽ and order relation ⩽ _c  . Nevertheless, as we can see in the following, binary block codes can be a useful tool which helps us to provide some order relations with which we can find examples of BCK/BCI algebras with some properties. They also can be useful to find ideals and closed ideals in a BCK/BCI algebra.

Let X be a BCK-algebra and V be a linear binary block-code with n codewords of length n . We consider the matrix M V = ( m i , j ) i , j ∈ { 1 , 2 , . . . , n } ∈ M n ( { 0 , 1 } ) with the rows consisting of the codewords of V . This matrix is called the matrix associated to the code V .We consider the codewords in V lexicographically ordered in the ascending sense. With this remark, for V = {w ₁, . . . . , w _n } , we denote lines in M _V with L _{w 1} , . . . , L _{w n}  . Obviously, w 1 = 00 . . . 0 ︸ n - time . On V, we define the following multiplication “∗”

w_i* w_j=w_k if and only if } L_wi+L_wj=L_wk.

Proposition 3.8. With this multiplication, (V,*,θ), where θ =w₁, becomes an abelian group.

Remark 3.9. The above group is a BCI-algebra.

Let C be a binary block code. In Theorem 3.9, from [2], we find a BCK -algebra Xsuch that the obtained binary block-code V_X contains the binary block-code Cas a subset.Let Cbe a binary block code with mcodewords of length q.With the above notations, let Xbe the associated BCK-algebra and W ={θ,w₁,...,w_m+q} the associated binary block code which include the code C.We consider the codewords θ,w₁,w₂,...,w_m+qlexicographically ordered, θ≥_lexw₁ ≥_{lexw 2≥ lex}...≥_{lexw m+q}.Let M∈ M_m+q+1({0,1})be the associated matrix with the rows θ,w₁,...,w_m+q,in this order and L_wiand C_wjthe lines and columns in the matrix M. We consider the sub-matrix M^′ of the matrix Mwith the rows L_w1,...,L_{w mand} the columns C_wm+1,...,C_wm+q, which is the matrix associated to the code C.

Proposition 3.12. With the above notations, we have that {θ, w_m+1,...,w_m +q} determines a closed right ideal in the algebra X.

Proof. Let Y= {θ,w_m+1,...,w_m+q}.Due to the following multiplication { θ * x = θ and x * x = θ , ∀ x ∈ X ; x * y = θ if x ≤ y , x , y ∈ X ; , x * y = x , otherwise , in the obtained BCK-algebra, we can have only the following two possibilities: w_i* w_j=θ or w_i* w_j=w_i.Therefore Yis a right-ideal in X.

Remark 3.13. From Proposition 3.12, we obtain that to each binary block code we can associate a BCK-algebra in which this code determines a right ideal.Let Abe a nonempty set and Xbe a BCK-algebra.

Proposition 3.14. Let C be a binary block code with m codewords of length q and let X be the associated BCK-algebra, as the above. Therefore, there are the sets A and B⊆ X, the BCK-function f:A→ X and a cut function f_r such that the code C can be written under the form C={f_r:A→ {0,1} /  f_r (x) =1, if and only if r* f (x) =θ,∀ x∈ A,r∈ B}.

Remark 3.15. i) Let S={1,2,...,n}be a set with nelements. We know that (P (S),Δ,∩) is a Boolean ring, where {{P(S)}}is the power set of the set S,Δ is symmetric difference of the sets and ∩ is the intersection of two sets. Let 𝔉={f:S→ {0,1} /ffunction}.To each f∈ 𝔉 corresponds a binary block codeword and to each binary block codeword c^′ corresponds an element from {{P(S)}}. Indeed, to each binary codeword c=(i₁,...,i_n),we will associate the set I_c={j₁,j₂,...,j_k}∈ P (S) such that i_j1=i_j2=...= i_jk=1.ii) Using the above established correspondence, if C={c₁,c₂,...,c_m}is a linear binary block code and Q={I_c1,I_c2,...,I_cm}⊆ P (S), where I_ci is the associated subset for the codeword c₁, then Qis a sub ring in the Boolean ring (P (S),Δ,∩).It results a bijective map between the sub rings of the Boolean ring (P (S),Δ,∩) and linear binary block codes with codewords of length n.

From Proposition 3.12, we remark that {θ,w₆,w₇, w₈,w₉}is a right ideal in the BCK-algebra X.From Proposition 3.14, for A={w₆,w₇,w₈,w₉ }and B={w₂,w₃,w₄,w₅ },we recover the initial code C.

Example 3.17. For the same linear binary block codeC={0000,0001,0010,0011},let Q={∅,{4}, {3},{3,4}}as in Remark 3.15 ii). It is clear that Qis a sub-ring in the Boolean ring (P ({1,2,3,4}),Δ,∩) and Ccan be considered as a sub ring of this Boolean ring.

Remark 3.18. Proposition 3.14 improved Theorem 3.9 from [2], since we can always obtain the code Cand, from Proposition 3.12, we have that the code Cgenerate a right ideal in the algebra X.

Remark 3.19. The obtained results of the above remarks and propositions can be illustrated by partially ordered sets. We know that we can find the matrix M∈ M_m+q+1({0,1})associated to the code C. For set S={1,2,...,m+q+1},we can find the sets A, B⊆ Sand the function f:A→ S, such that the code Ccan be recovered by writing it under the form: C={f_r:A→ {0,1} /  f_r (b) =1,if and only if r≤ b,∀ b∈ A,r∈ B}.Here, A={m+2,...,m+q+1}and B={2,..., m+1}.

From above proposition, we have A={6,7,8,9}and B={2,3,4,5},therefore, from Aand B, we can recover the initial code C.

4 Conclusions

From the above examples, we saw that the codes can be used in the study of BCK/BCIalgebras as an important tool in providing orders with which we can build algebras with some asked properties. Will be very interesting to study in a further paper the reverse problem, namely, how and if the properties of BCK/BCI-algebras can influence the properties of the associated binary block codes.

Acknowledgments

The authors are very grateful to the referees for their constructive suggestions and valuable comments which improved the standard of this paper.