# Some Results about Weak UP-algebras

Authors: Muhammad Anwar Chaudhry; Asfand Fahad; Muhammad Imran Qureshi; Urwa Riasat
Journal: Journal of Mathematics
 (2022)

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## Abstract

We introduce a new class of algebras, weak UP-algebras, briefly written as WUP-algebras, as a wider class than the classes of UP-algebras and KU-algebras. We investigate fundamental aspects including maximal elements, branches and the subalgebra consisting of maximal elements of a WUP-algebra. We also study regular congruences on a WUP-algebra as well as the corresponding quotient WUP-algebras. We prove that the congruence generated from the branches of a WUP-algebra is regular and the corresponding quotient algebraχ/q is isomorphic to the WUP-algebra Maxχ, the subalgebra of χ consisting of its maximal elements.


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## Body

## 1. Introduction

A few abstract algebras, such as BCK-algebras and BCI-algebras, have their roots in implicational logic, see [1, 2]. Many facets of daily life have benefited substantially from recent advances in artificial intelligence. Logical systems are the fundamental artificial intelligence techniques that aid in decision-making. The evolution and introduction of BCK-algebras and BCI-algebras are driven by the essential axioms of the implicational calculus, see [1, 2]. Taking into account the important linkages between these algebras and related logics, translation techniques have been created to link the theorems and formulae of a logic and the corresponding algebra. As a result, those involved in the fields of artificial intelligence, logical systems, and algebraic structures continue to be interested in the study of abstract algebras (and their generalizations) that have been inspired by logical systems. As a result, the BCH-algebra class, a new class of algebras, has been introduced [3]. Recently, another algebra class, known as gBCH-algebra class, has been introduced and studied in [4, 5]. The significant feature of this new class is its ability to cover all other mentioned classes. In the context of this class, further study through introduction of generalized d-algebras and isomorphism theorems in generalized d-algebras has been developed [6]. For more on such other algebras such as SU-algebra [7], BCC algebra [8] and the significance of these algebras we refer to [9–11]. Although the immediate applications played vital role in getting attention to the above mentioned algebras, however, the development of theoretical aspects of these algebras can not be ignored. Particularly, when they provide different (or a generalized) framework. Consequently, the study on these algebras continued and recently the notion of a KU-algebra [12] was introduced and studied further in [13]. Later, a generalized class of KU-algebras, known as UP-algebras [14] was defined and investigated. The range of the notions studied for these algebras is very vast, such as: the ideals and congruences [12], cubic KU-ideals [13], α,β-fuzzy KU- Ideals [15], interval-valued fuzzy KU ideals and θ,δ-fuzzy KU-ideals [16, 17], anti fuzzy and intuitionistic fuzzy KU-ideals [18, 19] in KU-algebras. In UP-algebras, the fuzzy sets and the generalized fuzzy sets were introduced and studied in [20, 21]. Moreover, the derivations of UP-algebras have been studied in [22, 23]. In 2019, the isomorphism theorems for UP-algebras were proved [24] and the results were extended later on. For more on the recent related study/applications of these algebras in multiple directions, we refer [25, 26] and references therein. Keeping the above motivation in view, we introduce a general class of algebras, known as weak UP-algebras (WUP-algebras) in such a way that the classes of KU-algebras and UP-algebras become the subclasses of the class of WUP-algebras. Further, we give necessary examples to show that the UP-algebra (consequently) class is a proper sub-class of the WUP-algebra class. Consequently, all investigations of this paper are valid for the KU-algebra class and the UP-algebra class. In particular, we study some fundamental aspects including maximal elements, branches and the subalgebra consisting of maximal elements of a WUP-algebra. We also define regular congruences on a WUP-algebra and study the corresponding quotient algebras. We prove that the congruence q generated from the maximal elements of a WUP-algebra χ is regular and the corresponding quotient algebra χ/q is isomorphic to Maxχ, the subalgebra consisting of maximal elements of χ.


## 2. Weak UP-Algebras (WUP-Algebras)

In this section, we introduce the notion of a Weak UP-algebra and prove that the classes of KU-algebras and UP-algebras are proper subclasses of the class of WUP-algebras. Before proceeding further, we recall the notions of a KU-algebra [12] and an UP-algebra [14].By an algebraχ,.,0 of type 2,0, we mean a non-empty set χ together with a binary operation “.” and a distinguished element 0 satisfying some specified axioms. Further, for x.y, we shall write xy and .,..,. will mean “.” between the bracketed expressions.Defintion 1.
(KU-algebra). An algebraχ=χ,.,0 of type 2,0 is called a KU-algebra if it satisfies the following conditions for any x,y,z∈χ:
KU−1yxxzyz=0,
KU−20x=x,
KU−3x0=0,
KU−4 xy = 0 and yx = 0 imply x = y.Defintion 2.
(UP-algebra). An algebraχ=χ,.,0 of type 2,0 is called a UP-algebra if it satisfies the following conditions for any x,y,z∈χ:
UP−1yzxyxz=0,
UP−20x=x,
UP−3x0=0,
UP−4 xy = 0 and yx = 0 imply x = y.
It is shown in [14] that every KU-algebra is an UP-algebra. Moreover, it is described in [14] that the algebra of the following Example 1 is an UP-algebra but not a KU-algebra. Consequently, the class of UP-algebras is wider than the class of KU-algebras.Example 1.
(see [14]). Letχ=0,1,2,3,4and the binary operation defined as (see Table1):
We now give our notion of a weak UP-algebra (WUP-algebra).Table 1
UP-algebra which is not a KU-algebra.
.01234001234100000202000302202402240.Defintion 3.
(Weak UP-algebra). An algebraχ=χ,.,0 of type 2,0 is called a weak UP-algebra (WUP-algebra) if it satisfies the following conditions for any x,y,z∈χ:
WUP−1yzxyxz=0,
WUP−20x=x,
WUP−3x00y=0 implies x00=y,
WUP−4 xy = 0 and yx = 0 imply x = y.Defintion 4.
(Subalgebra of a WUP-algebra). A subsetY of a WUP-algebra χ is called a subalgebra of χ if Y is closed under the operation of χ.
Now, we prove the following proposition.Proposition 1.
Every UP-algebraχ is a WUP-algebra.Proof.
It is sufficient to prove WUP-3. Letx,y∈χ be such that x00y=0. The condition UP-3 gives x0=0 and hence x00y=0 becomes 0y=0. Consequently, by UP-2 y=0. On the other hand, x000=000=00=0 and WUP-3 gives x00=0. Thus, y=x00. Hence χ is a WUP-algebra.
On a WUP-algebraχ=χ,.,0, we define the binary relation ≤ by: x≤y if and only if xy=0. The following example shows that the converse of Proposition 1 is not true, in general.Example 2.
Letχ=0,1,2 and the binary operation be defined as in the following table: (see Table 2):
Routine calculations show thatχ is a WUP-algebra but not a UP-algebra because 20=2≠0.Table 2
A WUP-algebra which is not a UP-algebra.
.012001210022220Example 3.
Letχ=0,a,b,c,d,e,f,g,h and the binary operation be defined as in the following table: (see Table 3):
Routine calculations show thatχ is a WUP-algebra but not a UP-algebra because g0=f≠0.Table 3
A WUP-algebra which is not a UP-algebra.
.0abcdefgh00abcdefGha00bbdefFhb0a0addfghc0000ddfFhdddEe0bhHfedddd00hHhffgFghh0AhgffFfhh00dhhhHhFfdd0Remark 1.
From Proposition1 and Examples 2 and 3, we deduce that every UP-algebra is a WUP-algebra but converse is not true. Thus the class of WUP-algebras contains both classes of KU-algebras and UP-algebras. Consequently, we feel that the investigation of the properties of WUP-algebras will add some important results to the mathematical literature.
Now, we prove some important properties of WUP-algebras.Theorem 1.
For a WUP-algebraχ and for any x,y,z in χ, we have:(1)
xx=0,(2)
xy=0 and yz=0 imply xz=0,(3)
xy=0 implies zxzy=0,(4)
xy=0 implies yzxz=0,(5)
xy0yx=0.Proof.
The proofs of the parts1−4 of this theorem are exactly the same as in Proposition 1.7, parts 1−4 (page 39 of [14]). So, we only prove (5). For this, consider xy0yx=0xy0yx=0 (by WUP-1).Remark 2.
(1)
In Theorem1 (5), if we substitute x for y, then we get xx0xx=0, that is, xx00=0 which gives x≤x00 for all x∈χ.(2)
In an UP-algebra,y0=0, so Theorem 1 (5) gives x0yx=0 or xyx=0, which yields proposition 1.75 of [14].(3)
In the sequel, we denote the elementx00 of a WUP-algebra χ by x0.(4)
From Theorem1, we conclude that the relation ≤ on a WUP-algebra χ is reflexive as well as transitive.Defintion 5.
(UP-Part). Letχ be a WUP-algebra. The UP-part of χ, denoted by UPPχ, is defined as y:y∈χsuch thaty0=0.Example 4.
In WUP-algebras of Examples2-3, their UP-parts are 0,1 and 0,a,b,c, respectively.Defintion 6.
(Maximal Element). Letχ be a WUP-algebra. The element x0=x00 corresponding to x∈χ is called a maximal element of χ. We also call it the maximal element corresponding to x and denote it by maxx. That is maxx=x0.
We denote the set of all maximal elements of a WUP-algebra byMaxχ. That is, Maxχ=x0:x0=x00,x∈χ.
Now, we define the notion of a branch of a WUP-algebra.Defintion 7.
(Branch of a WUP-algebra). Let x0 be a maximal element of a WUP-algebra χ. Then the branch of χ determined by x0, denoted by Bx0, is the subset y:y∈χandy≤x0 =  y:y∈χandyx0=0 of χ.Remark 3.
We note that(1)B0=y:y∈χandy≤0=y:y∈χandy0=0=UPPχ.Example 5.
In Example2, the elements 0,2 are maximal elements and the branches are B0=0,1 and B2=2, whereas in Example 3, 0,e,g,h are maximal elements and the branches are B0=0,a,b,c=UPPχ,Be=e,d, Bg=g,fandBh=h.
In the following theorem, we prove a sufficient condition for two elementsx and y of a WUP-algebra to have same corresponding maximal elements.Theorem 2.
Letχ be a WUP-algebra. If x,y∈χ are such that x≤y, then x0=y0.Proof.
Sincex≤y, so by applying Theorem 1 (4) twice we get x00≤y00. That is, x0≤y0, which gives x0y0=0. Now, by WUP-3, we get x0=y0.Proposition 2.
Letχ be a WUP-algebra. If x0∈Maxχ, then x0=x00.Proof.
Letx0∈Maxχ, we consider x0x00=x0x000=0x0x00x0x0=0, by WUP-1. Thus x0x00=0, which by WUP-3 gives x0=x00.Theorem 3.
Letχ be a WUP-algebra. If x,y∈χ then xy0=x0y0 and Maxχ is a subalgebra of χ.Proof.
Letx0,y0∈Maxχ. Then there exist x,y∈χ such that x≤x0 and y≤y0. By using Theorem 1 (3) and (4), we get xy≤xy0 and x0y0≤xy0. Using Theorem 2, we get xy0=xy00 and x0y00=xy00. Thus xy0=x0y00. Since xy≤xy0 and xy0=x0y00, therefore xy≤x0y00. Now, by Theorem 2 and Proposition 2, we get xy0=x0y000=x0y0. Hence xy0=x0y0. Since xy0∈Maxχ, so x0y0∈Maxχ. Thus Maxχ is a subalgebra of χ.Theorem 4.
Letχ be a WUP-algebra. Then for all x,y∈χ, xy∈B0=UPPχ if and only if x and y belong to the same branch of χ.Proof.
Letx,y∈Bx0. Then x≤x0 and y≤x0. So by Theorem 1 (3), we get xy≤xx0=0. Thus xy≤0, which implies xy∈B0. Conversely, let x,y∈χ and xy∈B0. Thus xy≤0. By Theorem 2, we get xy0=00=000=00=0. This gives x0y0=0, which by WUP-3 gives x0=y0. So x,y belong to the same branch of χ.Proposition 3.
Letχ be a WUP-algebra. Let x∈Bx0, y∈By0. Then xy∈Bx0y0Proof.
Clearly,xy0∈Mχ. Also xy≤xy0=x0y0. Since Maxχ is a subalgebra of χ, so x0,y0∈Maxχ implies x0y0∈Maxχ. Hence, xy≤x0y0 gives xy∈Bx0y0 □Proposition 4.
Letχ be a WUP-algebra. Then its UPP-part B0 is a subalgebra of χ.Proof.
Letx,y∈B0. So by above proposition, we get xy∈B00=B0. Hence B0=UPPχ is a subalgebra of χ.
In the next theorem, we show that the collection of branches produced by all the maximal elements in a WUP-algebraχ partitions χ:Theorem 5.
Letχ be a WUP-algebra. Let Mχ be the collection of all maximal elements of χ. Then(a)
χ=∪x0∈MχBx0.(b)
Bx0∩By0=ϕ for any x0,y0∈Mχ and x0≠y0.Proof.
(a)
ObviouslyBx0 is a non-empty subset of χ because x0≤x0 gives x0∈Bx0. Consequently, ∪x0∈MχBx0⊆χ. Also for each x∈χ, there is x0=x00 such that x≤x0, so x∈Bx0⊆∪x0∈MχBx0. Hence χ⊆∪x0∈MχBx0. Combining, we get χ=∪x0∈MχBx0.(b)
Letx0,y0∈Mχ and x0≠y0. Let z∈Bx0∩By0. So, z≤x0 and z≤y0. That is zx0=0 and zy0=0. Now, WUP-1 gives x0y0zx0zy0=0. So, x0y000=0. That is, x0y00=0. So, xy00=0. Now, WUP-3 give xy00=0. That is, x0y0=0. Again, WUP-3 gives x0=y0, a contradiction. Thus, Bx0∩By0=ϕ.Remark 4.
(1)
Every WUP-algebraχ is the union of disjoint branches of χ, determined by its maximal elements.(2)
Everyx∈χ belongs to a unique branch Bx0 determined by its corresponding maximal element x0. That is, for every x∈χ, there is a unique maximal element x0 such that x≤x0.


## 3. Regular Congruences on WUP-Algebras

In this section, we discuss regular congruences on WUP-algebras and the corresponding quotient WUP-algebras.Defintion 8.
Letχ be a WUP-algebra. A relation q on χ is called a congruence on χ if for any x,y,z,p,r∈χ:, the following conditions hold:(1)
xqx,(2)
xqy implies yqx,(3)
xqy, yqz imply xqz(4)
xqp, yqr imply xyqpr.Defintion 9.
(Regular Congruence). A congruenceq on a WUP-algebra χ is called a regular congruence if 5xyq0 implies xqy, x,y∈χ.Example 6.
Letχ be a WUP-algebra. We define a relation q by: x,y∈χ, xqy if and only if x0=y0. Then the relation q is a regular congruence on χ.
Verification.(1)
Letx∈χ, since x0=x0 therefore xqx.(2)
Letx,y∈χ and xqy, therefore x0=y0. So y0=x0. Thus yqx.(3)
Letx,y,z∈χ, xqy and yqz. Then x0=y0 and y0=z0. Thus x0=z0 and hence xqz.(4)
Letx,y,p,r∈χ with xqp and yqr. Hence x0=p0 and y0=r0. Now, consider xy0=x0y0=p0r0=pr0. So, xyqpr. Thus q is a congruence on χ. We, now show that it is a regular congruence.(5)
Letx,y∈χ such that xyq0. Thus xy0=00=0, which gives x0y0=0. Now, by WUP-3, we get x0=y0. This gives xqy. Hence q is a regular congruence of χ.Defintion 10.
Letχ be a WUP-algebra and let q be a congruence on χ. Let x∈χ. We define:(2)xq=y:y∈χandyqx.
and callxq, the class determined by x. Then the set(3)χ/q=xq:x∈χ.
is called the quotient set ofχ with respect to the congruence q.Theorem 6.
Letχ be a WUP-algebra and q a regular congruence on χ. Let χ/q be the quotient set of χ. Let the binary operation ∗ in χ/q be defined as:(4)xq∗yq=xyq,for all xq,yq∈χ/q. Then χ/q,∗,0q is a WUP-algebra.Proof.
We first show that the operation∗ defined by (1) is well defined. Let x,y,p,r∈χ and xq=pq and yq=rq. By property of equivalence relation q, we have xqp and yqr. Since q is a congruence so xyqpr. Thus xyq=prq. That is xq∗yq=pq∗rq.
Now, we verify WUP-1, WUP-2, WUP-3 and WUP-4.(1)
We, now consideryq∗zq∗xq∗yq∗xq∗zq=yzq∗xyq∗xzq=yzxyxzq=0q.(2)
Letxq,0q∈χ/q, so 0q∗xq=0xq=xq.(3)
Letxq∗0q∗0q∗yq=0q. Then x00yq=0q. So x00yq0. That is, x0yq0. Since q is a regular congruence, so x0qy. Hence x0q=yq. So x00q=yq. Thus, xq∗0q∗0q∗yq=0q implies xq∗0q∗0q=yq.(4)
Letxq∗yq=0q and yq∗xq=0q. Thus xyq=0q and yxq=0. This implies xyq0. Since q is a regular congruence on χ, so xqy. Thus, xq=yq.Remark 5.
We know that, for a WUP = algebraχ, the relation q defined on χ by: xqy if and only if x0=y0, that is, if and only if x,y belong to the same branch of χ, is a regular congruence on χ. Thus from Theorem 6, we have the following corollary.Corollary 1.
Letχ be a WUP-algebra. Let q be the regular congruence on χ defined by: x,y∈χ, xqy if and only if x0=y0. Then χ/q,∗,0q is a WUP-algebra with ∗ defined by (1).Defintion 11.
Letχ1, χ2 be WUP-algebras. A mapping ϕ:χ1⟶χ2 is called:(1)
a homomorphism ifϕxy=ϕxϕy for all x,y∈χ1.(2)
an isomorphism ifϕ is a homomorphism and a bijection.(3)
the WUP-algebrasχ1 and χ2 are isomorphic if there is an isomorphism ψ:χ1⟶χ2.Theorem 7.
Letχ be a WUP-algebra. Let q be the regular congruence on χ defined by: x,y∈χ, xqy if and only if x0=y0. Let χ/q be the corresponding quotient algebra. Let Maxχ be the subalgebra of χ consisting of maximal elements of χ. Then χ/q is isomorphic to Maxχ.Proof.
We defineϕ:χ/q⟶Maxχ by ϕxq=x0 for all xq∈χ/q.(a)
Firstly, we show thatϕ is well-defined. Let xq=yq, so xqy. That is x,y belong to the same branch. So x0=y0. Hence, ϕxq=ϕyq.(b)
Now, we show thatϕ is homomorphism. Let xq,yq∈χ/q. Now, ϕxqyq=ϕxyq=xy0=x0y0=ϕxqϕyq. So, ϕ is a homomorphism.(c)
To show thatϕ is onto, let x0∈Maxχ. Then there exist x0q∈χ/q such that ϕx0q=x0. Hence ϕ is onto.(d)
Now, we show thatϕ is one-one. Let xq,yq∈χ/q be such that ϕxq=ϕyq. So, x0=y0. That is x and y belong to the same branch of χ. Hence xqy. So, xq=yq. Therefore, ϕ is one-one.
Summing up we get,χ/q is isomorphic to Maxχ.Example 7.
We consider the WUP-algebraχ=0,a,b,c,d,e,f,g,h of Example 3. We note that Maxχ=0,d,f,h. We define the congruence q on χ by: x,y∈χ and xqy if and only if x,y belong to the same branch of χ. So(5)q=0,0,0,b,0,a,0,c,b,0,a,0,c,0,b,a,a,b,b,c,c,b,c,a,a,c,e,e,d,d,e,d,d,e,g,f,f,g,h,h.Thus(6)χ/q=0q,dq,fq,hq.
We define a mappingψ:χ/q⟶Maxχ by ψ0q=0,ψdq=d,ψfq=f,ψhq=h. It is easy to verify that ψ is an isomorphism. So χ/q is isomorphic to Maxχ.


## 4. Conclusion

We have introduced a new class of algebras, known as Weak UP-algebras. This class is so general that it contains the class of KU-algebras as well as the class of UP-algebras. We have studied some basic properties, maximal elements, branches, regular congruences, quotient algebras of this class of algebras. We have also established an isomorphism betweenχ/q and Maxχ, where q is the regular congruence defined on χ with the help of its maximal elements and Maxχ is the subalgebra of χ consisting of maximal elements of χ.


## 5. Future Work

It will be interesting to explore other aspects of this wider class of WUP-algebras. In particular, proper commutative WUP-algebras, positive implicative WUP-algebras and implicative WUP-algebras, their properties and interconnection between them. The study of ideals, filters and different types of graphs on all above mentioned classes of algebras will also be interesting.



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