Generalized Cut Functions and -Ary Block Codes on UP-Algebras

Abstract

In this paper, the work is comprised of -ary block codes for UP-algebras and their interrelated properties. -ary block codes for a known UP-algebra is constructed and further it is shown that for each -ary block code , it is easy to associate a UP-algebra in such a way that the newly constructed -ary block codes generated by , i.e., , contain the code as a subset. We define a UP-algebra valued function on a set say , then we prove that for every -ary block-code , a generalized UP-valued cut function exists that determines . We have also proved that the UP-algebras associated to an -ary block code are not unique up to isomorphism.

1. Introduction

Logical algebras like BCI/BCK, BE, KU-algebras, and many others with their fuzzy, intuitionistic, and more related concepts have been interesting topics of study for researchers in recent years and have been widely considered as a strong tool for information systems and many other branches of computer sciences including fuzzy informatics with rough and soft concepts. Imai and Iseki [1] introduced BCK/BCI algebras as a generalization of the concept of set-theoretic difference and proportional calculi. BCI/BCK algebras form an important class of logical algebras. They have numerous applications to different domains of mathematics, e.g., sets theory, semigroup theory, group theory, derivational algebras, etc. As per the requirement to establish certain rational logic systems as a logical foundation for uncertain information processing, different types of logical systems are felt to be established. For this reason, researchers introduced and studied many types of logical algebras by using the concepts of BCI/BCK algebras.

A block code is related to channel coding that is one of the main types of it. Block code adds redundancy to a message so that, at the receiver end, one can easily decode the message with a minimum number of errors, where it is already provided that the information rate would not exceed the channel capacity. The task of a block code is to encode the strings that are formed by an alphabet set say into code words by encoding each letter of separately. As per the importance block of codes, they can be source codes used in data compression or channel codes used for detection and correction of channel errors [2]. Codes based on a family of algorithms were constructed by Lempel and Ziv [3], which are applicable for real-world problems and sequences. A detailed terminology based on codes and decoding through graphs is discussed in [4]. Ali et al. introduced the concept of n-ary block codes related to KU-algebras in [5].

Many researchers have made their studies based on block codes in the past few years considering different branches and different directions. One of them is logical algebra. Surdive et al. studied coding theory in hyper BCK-algebras [6]. Jun and Song [7] defined and studied codes based on BCK-algebras. Further Fu and Xin [8] introduced the concept of block codes in lattices.

Iampan introduced the concept of UP-algebras [9]. Iampan contributed on different aspects related to UP-algebras in [10]. Senapati et al. [11] represented UP-algebras in an intervalued intuitionistic fuzzy environment. Moin et al. [12] introduced graphs of UP-algebras and studied related results. The binary block codes associated to UP-algebras were discussed by Moin et al. [13]. Wajsberg algebras arising from binary block codes were studied by Flaut and Vasile [14].

In this paper, we have introduced and investigated generalized UP-valued cut functions and their several properties. Also, we have established n-ary block-codes for UP-algebras by using the notion of generalized UP-valued cut functions. We show that every finite UP-algebra determines a block-code.

Section 2 contains preliminaries and related definitions with some examples. Section 3 is based on the main results.

2. Preliminaries

This section is comprises with the concepts of UP-algebras, UP-subalgebras, UP-ideals, UP-valued function (cut function), and other important terminologies with examples and some related results.

Definition 1 (see [9]). A UP-algebra is a structure of type with a single binary operation that satisfies the following identities: for any , (UP-1):  (UP-2):  (UP-3):  (UP-4): implies For a commutative UP-algebras we have the condition for commutativity as .
We define a partial order relation in a UP-algebra as if and only if . If and are two UP-algebras, then a map with the property , for all , is called a UP-algebra morphism. If is one-one and onto map, then is simply called isomorphism of .

Example 1. Let be a set in which is defined by the following Cayley table

We observe here that is a UP-algebra.

Example 2. Let and define a binary operation on as where . Then is a UP-algebra. The following table represents this operation:

Lemma 1 (see [10]). In a UP-algebra the following properties hold for any : (UP-5)  (UP-6) and  (UP-7)  (UP-8)  (UP-9)  (UP-10)  (UP-11)

Lemma 2. Let be UP-algebras, then define a binary relation on as follows: for all  (UP-12)  (UP-13)  (UP-14)  (UP-15) and  (UP-16) and  (UP-17)  (UP-18)  (UP-19)

Definition 2 (see [9]). A nonempty subset of a UP-algebra is called a UP-ideal of if it satisfies the following conditions:(1)(2), implies , for all

Proposition 1. An algebra of type is a UP-algebra if and only if the given conditions are satisfied:(1) for all (2) for all (3)For all such that

Proof. If is a UP-algebra. Then, (1) follows from (UP-1).
Next, (3) follows from (UP-4).
By using (UP-2) and (UP-3) we get (2) as .
Indirectly we consider satisfies given conditions, then (UP-1) and (UP-4) follows from (1) and (2), respectively. Next, replace by , by 1 and by 1 in (1) and using (3) we get, which shows (UP-3). Further, using in (2) we get, for all . Hence is a UP-algebra.
Let be a finite UP-algebra with elements and be a finite nonempty set. A map is called a UP-function. Let be a finite set. In the following, we will consider UP-algebra and the set , where , . A generalized cut function of is a map , such that if and only if , for all .
For such each UP-function , it is easy to define an -ary block code with codewords having length . For this purpose, we suppose that for each element the generalized cut function . For every such function, there will be corresponding a codeword , having symbols taken from the set . So, we get , with , if and only if , that means . We denote this new code by . Hence, it is easy to associate an -ary block code for every such UP-algebra.

Example 3. We take the UP-algebra having where is defined by the following table:

We can easily show that . We consider the generalized cut function and . In this way , returns the codeword . For , we get the codeword 1001. In fact, , since since and , also .
The following result investigates about the existence of the converse part whether it is true or not.

3. Main Results

We consider a finite set and its -ary codewords , of length , ascending ordered after lexicographic order. We consider , with descending ordered such that and in the rest.

Definition 3. Let be an -ary code. Further we suppose that . , as above. We now associate a matrix , to this code where . Let . We define . For , let , if , and , if . For , we put , for and , for . We suppose that , for .
Here, is the lower triangular matrix, and it is known as the matrix associated with the -ary block code .

Definition 4. Consider is associated to the -ary block code defined on . Suppose that is a nonempty set. The multiplication is defined on .

Theorem 1. The set is a UP-algebra.

Proof. We see here that Proposition 1 (2), (3) are well defined. From Definition 1, we need to show that , for all . For the elements we have 3 situations here that are given as follows: Case 1: . We get , which implies . Case 2: . We need to show that . Thus for , it is obvious and for , we obtain . For , we have , therefore . For , we obtain . Next, if , since , in returns we get . If , then . For , we have , since and . For , it results , since , it yield . Case 3: . Here, we have to prove that . Hence, it is shown for . Furthermore, let . For and , we get , hence . We also get , hence . For and , we get , then . It results that . For and , we can get that and , so . We also obtain , and , since . It yield . Or, we can have ,; hence . For and , we can have and , hence . Or, we have , and , as a result we get zero. We also can have ; hence the relation is 0. For and , if , it shows that the asked relation is 0. If , then , with , and .For and , we have that . If , we obtain 0. If , hence we find , since .
For and , we have ; therefore, we obtain the result as zero. For and , we can obtain . Or, we find ; thus, we can say that obtained result is 0. For and , since , it returns . We can get and , therefore . For and , we can get . Or, if ; thus we obtain 0 and that is required. For and , we have ; then, we get zero. For and , it results , hence the asked relation is 0. □
Note.(1)We find that a UP-algebra from Theorem 1 is extracted by using the matrix , which is uniquely determined by an -ary code, say , given as per Definition 1; thus, we can say that is a uniquely determined algebra.(2)By Theorem 1, we suppose that is the resulted UP-algebra, with . If with multiplication “” given by the relation if and only if , for , then is a UP-algebra.(3)If we suppose that , the map , returns a code , that can be associated to the above UP-algebra , that contains the code as a subset.We consider as an -ary block code. Then, from Theorem 1 and above Note, we can have a UP-algebra in such a way that the obtained -ary block code contains the -ary block code as of its subset. Suppose that is a binary block code with code words of length . By using the abovementioned notations, consider is the associated UP-algebra and is the associated -ary block codes that contains the code . Next consider and are two codewords that belong to . Here, we define an order relation on by the following logic if and only if , for all . On , with the order relation , we define the following multiplication:(1) and (2) if (3) if This order relations give UP-algebra structure. It is clear that .

Proposition 2. gives an UP-algebra ideal in the .

Proof. Considering . We will show that , and , implies . By using multiplication rule in the UP-algebra and chosen -ary codes, we get for . If , then or .

Example 4. Consider , with .
Entries of the matrix associated with the -ary code , are , and for the rest of and .
The corresponding UP-algebra, , where , is shown with the following multiplication table.

Considering . The map gives us the following block code , that contains as a subset.Clearly it is a noncommutative UP-algebra as and . This clarifies that is not an implicative UP-algebra. Also we note that it is not a positive implicative UP-algebra. Since and .

Example 5. Consider , with .
Entries of the matrix associated with the -ary code , are , and for the rest of and .
The corresponding UP-algebra , where is shown with the following multiplication table.

Let . Then, returns the given block code , where is contained in it as a subset. The diagram of this generated code is given as

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.