# Generalized Cut Functions andn-Ary Block Codes on UP-Algebras

Authors: Ali N. A. Koam; Azeem Haider; Moin A. Ansari
Journal: Journal of Mathematics
 (2022)

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## Abstract

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


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## Body

## 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 sayC into code words by encoding each letter of C 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 establishedn-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.Section2 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 structureU,∗,∅ of type 2,0 with a single binary operation ∗ that satisfies the following identities: for any x,y,z∈U,(UP-1):y∗z∗x∗y∗x∗z=∅(UP-2):∅∗x=x(UP-3):x∗∅=∅(UP-4):x∗y=y∗x=∅ implies x=y
For a commutative UP-algebrasU we have the condition for commutativity as x∗x∗y=y∗y∗x.
We define a partial order relation in a UP-algebraU as y≤x if and only if x∗y=∅. If U,∗,∅ and V,∘,∅ are two UP-algebras, then a map f:U⟶V with the property fx∗y=fx∘fy, for all x,y∈U, is called a UP-algebra morphism. If f is one-one and onto map, then f is simply called isomorphism of U.Example 1.
LetU=∅,a,b,c be a set in which ∗ is defined by the following Cayley table
We observe here thatU=∅,a,b,c is a UP-algebra.Example 2.
LetU=an|n=1,2,3,…,9 and define a binary operation ∗ on U as ai∗aj=ak,∀ai,aj,ak∈U where k=lcmi,j/j. Then U,∗,a1 is a UP-algebra. The following table represents this operation:Lemma 1 (see [10]).
In a UP-algebraU the following properties hold for any a,b,c∈U:(UP-5)a∗a=∅(UP-6)a∗b=∅ and b∗c=∅⇒a∗c=∅(UP-7)a∗b=∅⇒c∗a∗c∗b=∅(UP-8)a∗b=∅⇒b∗c∗a∗c=∅(UP-9)a∗b∗a=∅(UP-10)b∗a∗a=∅⇔a=b∗a(UP-11)a∗b∗b=∅Lemma 2.
LetU=A,∗,∅ be UP-algebras, then define a binary relation ≤ on U as follows: for all a,b,c∈A(UP-12)a≤a(UP-13)∅≤a(UP-14)b∗a≤a(UP-15)a≤b and b≤a⇒a=b(UP-16)b≤a and c≤b⇒c≤a(UP-17)b≤a⇒c∗b≤c∗a(UP-18)b≤a⇒a∗c≤b∗c(UP-19)a∗b∗a∗c≤b∗cDefinition 2 (see [9]).
A nonempty subsetA of a UP-algebra U is called a UP-ideal of U if it satisfies the following conditions:(1)
∅∈A(2)
a∗b∗c∈A, b∈A implies a∗c∈A, for all a,b,c∈UProposition 1.
An algebraU,∗,∅ of type 2,0 is a UP-algebra if and only if the given conditions are satisfied:(1)
c∗a∗b∗c∗b∗a=∅ for all a,b,c∈U(2)
b∗∅∗a=a for all a,b∈U(3)
For alla,b,c∈U such that a∗b=∅,b∗a=∅⇒a=bProof.
IfU,∗,∅ 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) asb∗1∗a=1∗a=a.
Indirectly we considerU,∗,∅ satisfies given conditions, then (UP-1) and (UP-4) follows from (1) and (2), respectively. Next, replace b by a, a by 1 and c by 1 in (1) and using (3) we get, ∅∗∅∗a∗∅∗a∗∅=∅⇒a∗∅∗a∗∅=∅⇒a∗∅=∅ which shows (UP-3). Further, using a∗∅=∅ in (2) we get, ∅∗a=a for all a∈U. Hence U,∗,∅ is a UP-algebra.
LetU,∗,∅ be a finite UP-algebra with n elements and U be a finite nonempty set. A map f:U⟶U is called a UP-function. Let Un=0,1,2,…,n−1 be a finite set. In the following, we will consider UP-algebra U and the set U, where U=l0,l1,…,ln−1, U=u0,u1,…,um−1m≤n. A generalized cut function of f is a map flj:U⟶Un,lj∈U, such that fljui=u if and only if lj∗fui=lu, for all lj,lk∈U,ui∈U,andi,j,u∈0,1,2,…,n−1.
For such each UP-functionf:U⟶U, it is easy to define an n-ary block code with codewords having length m. For this purpose, we suppose that for each element l∈U the generalized cut function fl:U⟶Un. For every such function, there will be corresponding a codeword wr, having symbols taken from the set Un. So, we get wl=w0,w1,…,wm−1, with wi=j,j∈Kn, if and only if flxi=j, that means l∗fxi=lj. We denote this new code by UX. Hence, it is easy to associate an n-ary block code for every such UP-algebra.Example 3.
We take the UP-algebraU=1,2,3,4 having ∘ where ∘ is defined by the following table:
We can easily show thatU=U4=1,2,3,4. We consider the generalized cut function f:U⟶U,f1=2,f2=a,f3=b,f4=c and fl:U4⟶U4,l∈U. In this way r=1, returns the codeword w1=0000. For l=a, we get the codeword 1001. In fact, fa1=2, since a∘f1=a∘1=a=f1;faa=1 since a∘f2=a∘a=1=f0;fab=1 and a∘f2=a∘b=1=f1;fac=1, also a∘f4=a∘c=a=f2.
The following result investigates about the existence of the converse part whether it is true or not.


## 3. Main Results

We consider a finite setUn′=1,2,…,n−1 and its n-ary codewords U=v1,v2,…,vm, of length h,h≥n−2, ascending ordered after lexicographic order. We consider vi=vi1vi2,…,vih,vij∈Un′,j∈1,2,…,h, with vij descending ordered such that vivik≤u,i∈1,2,…,m,u∈1,2,…,minn−1,h and vij=1 in the rest.Definition 3.
LetU=v1,v2,…,vm be an n-ary code. Further we suppose that vi=vi1,vi2…,vih,vij∈Un′. j∈1,2,…,h,q≥n−2, as above. We now associate a matrix A=αsts,t∈0,1,…,l−1,A∈AlUn, to this code where l=m+h+1. Let l=m+h+1. We define αss=0,αs0=s,α0s=0,s∈0,1,2,…,l−1. For 1≤s≤h, let αst=1, if t<s, and αst=0, if t≥s. For s>h, we put αst=vit, for t∈1,2,…,h and αsh+j=1, for h+j<s. We suppose that αst=0, for t≥s.
Here,A is the lower triangular matrix, and it is known as the matrix associated with the n-ary block code U=v1,v2,…,vm.Definition 4.
ConsiderA∈AiUn is associated to the n-ary block code U=k1,k2,…,km defined on Un′. Suppose that Ul=0,1,…,l−1 is a nonempty set. The multiplication i∘j=αij is defined on Ul.Theorem 1.
The setU∅,∘,0 is a UP-algebra.Proof.
We see here that Proposition1 (2), (3) are well defined. From Definition 1, we need to show that b∗c∗c∗a∗c∗a=1, for all a,b,c∈0,1,…,l−1. For the elements a,b,c we have 3 situations here that are given as follows:Case 1:c=0,b≠0. We get b∗a≤a, which implies a∗b∗a=0.Case 2:b=0,c≠0. We need to show that c∗c∗a∗a=0. Thus for a=0, it is obvious and for c=0, we obtain 0∗a∗0∗a=0∗a∗a=0. For c≠0,a≥l−m,c∈1,2,…,h, we have a∗c∗a=wxwac≤c, therefore c∗c∗a∗a=0. For c≠0,a≥l−m,c≥h+1, we obtain c∗c∗a∗a∗c=0. Next, if c∗a=1, since 1∗a≤n−1<h+1≤c, in returns we get c∗c∗a∗a=c∗1∗a=0. If c∗a=0, then c∗c∗a∗a=c∗0∗a=c∗a=0. For a<l−m,c≤h+1, we have c∗c∗a∗a=0, since c∗a=1,1∗a=1 and c∗1=0. For a<l−m,c>h+1, it results c∗c∗a∗a=0, since c∗a=0, it yield c∗0∗a=c∗a=0.Case 3:c≠0,b≠0. Here, we have to prove that c∗a∗b∗k∗b∗a=0. Hence, it is shown for a=0. Furthermore, let a≠0. For a≥l−m and b,c<l−m,b<k, we get n−1≥b∗a≥c∗a, hence c∗a∗b∗a=1. We also get b∗c=1, hence c∗a∗c∗b∗b∗a=1∗1=0. For a≥l−m and b,c<l−m,c<b, we get n−1≥c∗a≥b∗a, then c∗a∗b∗a=0. It results that c∗a∗b∗c∗b∗a=0. For a≥l−m and b,c≥l−m,b<c, we can get that b∗a=1 and c∗a=1, so c∗a∗b∗a=0. We also obtain b∗a=1,c∗a=0, and b∗c=1, since b<c. It yield c∗a∗b∗k∗b∗a=1∗0∗1=1∗1=0. Or, we can have b∗a=0,c∗a=0,; hence c∗a∗b∗k∗b∗a=0. For a≥l−m and b,k≥l−m,c<b, we can have b∗a=1 and c∗a=1, hence c∗a∗b∗a=0. Or, we have c∗a=1,b∗a=0, and b∗c=0, as a result we get zero. We also can have b∗a=0,c∗a=0,; hence the relation is 0. For a≤l−m and c<l−m<b, if b∗a=0, it shows that the asked relation is 0. If b∗a=1, then c∗a∗b∗k∗b∗a=0∗c∗a∗1=β∗1, with β≥1, and c<a.
Fora≥l−m and b<l−m<c, we have that b∗a=1. If c∗a=1, we obtain 0. If c∗a=0, hence we find c∗a∗b∗c∗b∗a=b∗c∗0∗1=b∗c∗1=0, since b∗c≥1.
Fora<l−m and b,c<l−m,b<c, we have b∗a=1,c∗a=1,; therefore, we obtain the result as zero. For a<l−m and b,c<l−m,c≤b, we can obtain c∗a∗b∗c∗b∗a=1∗0∗1=0. Or, we find b∗a=0; thus, we can say that obtained result is 0. For a<l−m and b,c<l−m,b<c, since b<c, it returns b∗c=1. We can get b∗a=1,c∗a=0 and b∗c=1, therefore c∗a∗b∗c∗b∗a=0∗1∗1=1∗1=0. For a<l−m and c<l−m≤b, we can get c∗a∗b∗k∗b∗a=1∗0∗1=0. Or, if b∗a=0; thus we obtain 0 and that is required. For a<l−m and b,c≥l−m,b<c, we have b∗a=0; then, we get zero. For a<l−m and b,c≥l−m,b>c, it results b∗a=0, hence the asked relation is 0. □
Note.(1)
We find that a UP-algebraUl,∗,0 from Theorem 1 is extracted by using the matrix A, which is uniquely determined by an n-ary code, say U, given as per Definition 1; thus, we can say that Ur,∗,0 is a uniquely determined algebra.(2)
By Theorem1, we suppose that Cl,∗,0 is the resulted UP-algebra, with Ul=0,1,2,…,l−1. If U=a0=1,a1,a2,…al−1 with multiplication “∗” given by the relation ai∗aj=ac if and only if a∗b=c, for a,b,c∈0,1,2,…,l−1, then U,∗,1 is a UP-algebra.(3)
If we suppose thatCh=0,1,2,…h−1, the map f:Ch⟶U,fa=ai, returns a code UX, that can be associated to the above UP-algebra U,∗,1, that contains the code U as a subset.
We considerU as an n-ary block code. Then, from Theorem 1 and above Note, we can have a UP-algebra U in such a way that the obtained n-ary block code UX contains the n-ary block code U as of its subset. Suppose that U is a binary block code with m code words of length h. By using the abovementioned notations, consider X is the associated UP-algebra and W=1,w1,…,wr is the associated n-ary block codes that contains the code U. Next consider wa=a1a2,…,ah and wb=b1b2,…,bh are two codewords that belong to W. Here, we define an order relation ≤c on W by the following logic wa≤wb if and only if bi≤ai, for all i∈1,2,…,h. On U=W, with the order relation ≤c, we define the following multiplication:(1)
a∘1=1 and a∘a=1,∀a∈U(2)
b∘a=1 if a≤cb,∀a,b∈U(3)
b∘a=a if b≤ca,∀a,b∈U
This order relations give UP-algebra structure. It is clear thatwl≤c⋯≤cw1≤c1.Proposition 2.
V=1,wl−m,wl−m+1,…,wl gives an UP-algebra ideal in the U.Proof.
ConsideringV=1,wl−m,wl−m+1,…,wl. We will show that b∈V,a∈U, and b∘a∈V, implies a∈V. By using multiplication rule in the UP-algebra U and chosen n-ary codes, we get for a∈U−V,b∘a=a∈U−V. If a,b∈V, then b∘a=a∈V or b∘a=1∈V.Example 4.
ConsiderK5=0,1,2,3,4,n=5,q=4,m=3,l=8,V=w1,w2,w3, with w1=3211,w2=4221,w3=4321.
Entries of the matrixA associated with the n-ary code U, are aij=0,∀i≤j, ai1=i−1,a62=3,a63=2,a72=4,a73=2,a74=2,a82=4,a83=3,a84=2 and aij=1 for the rest of i and j.
The corresponding UP-algebra,U,∘,1, where U=a0=1,a1,a2,a3,a4,a5,a6,a7, is shown with the following multiplication table.
ConsideringU=1,2,3,4. The map f:U⟶U,f1=a1,f2=a2,f3=a3,f4=a4 gives us the following block code U′=0000,1000,1100,1110,1111,3211,4221,4321, that contains U as a subset.(1)0000−1000−1100−1110−1111−3211−4221−4321.
Clearly it is a noncommutative UP-algebra asa6∘a7∘a7=a1∘a7=a4 and a7∘a6∘a6=1∘a6=a6. This clarifies that U is not an implicative UP-algebra. Also we note that it is not a positive implicative UP-algebra. Since a6∘a7∘a6=a1∘a6=a3≠a6 and a3∘a6∘a7=a3∘a1=1≠a3∘a6a3∘a7=a1∘a2=a1.Example 5.
ConsiderK4=0,1,2,3,n=4,q=5,m=3,l=9,U=w1,w2,w3, with w1=21111,w2=32111,w3=33111.
Entries of the matrixA associated with the n-ary code U, are aij=0∀,i≤j, ai1=i−1,a72=2,a82=3,a92=3,a83=2,a93=3 and aij=1 for the rest of i and j.
The corresponding UP-algebraX,∘,1, where X=a0=1,a1,a2,a3,a4,a5,a6,a7 is shown with the following multiplication table.
LetK=1,2,3,4,5. Then, f:K⟶X,f1=a1,f2=a2,f3=a3,f4=a4,f5=a5 returns the given block code U′=00000,10000,11000,11100,11110,21111,32211,33111, where U is contained in it as a subset. The diagram of this generated code is given as



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