Completion of BCC-algebras

1. Introduction

In 1966, Y. Imai and K. Iséki in [1] introduced a class of algebras of type \((2,0)\) called BCK-algebras which generalizes on one hand the notion of algebra of sets whit the set subtraction as the only fundamental non-nullary operation, on the other hand the notion of implication algebra. K. Iséki posed an interesting problem whether the class of BCK-algebras form a variety. In connection with this problem Y. Komori in [2] introduced a notion of BCC-algebras which is a generalization of notion BCK-algebras and proved that class of all BCC-algebras is not a variety. W. A. Dudek in [3] redefined the notion of BCC-algebras by using a dual form of the ordinary definition. Further study of BCC-algebras was continued [4,5,6].

In 1937, André Weil in [7] introduced the concept of a uniform space as a generalization of the concept of a metric space in which many non-topological invariants can be defined. The study of quasi uniformities started in 1948 with Nachbin's investigations on uniform preordered spaces. Mehrshad and Kouhestani in [8] introduced a quasi-uniformity on a BCC-algebra by a family of ideals and studied some properties of this structure. Now, in this present work, we consider the set \( C \) of all cauchy nets on BCC-algebras \( X \) and define a congruence relation \( \sim \) on this set. Then we consider the quotient BCC-algebra \( \mathcal{C}=\frac{C}{\sim} \) and prove that \( \mathcal{C} \) is a BCC-algebra. We construct a uniformity on \( \mathcal{C} \) and show that this uniformity is a completion of uniform space on \( X \) induced by a family of ideals of \( X. \)

2. Preliminary

BCC-algebras

A BCC-algebra is a non empty set \(X\) with a constant \(0\) and a binary operation \(\ast\) satisfying the following axioms, for all \(x,y,z\in X:\)

• (1) \( ((x\ast y)\ast (z\ast y))\ast (x\ast z)=0,\)
• (2) \(0\ast x=0,\)
• (3) \(x\ast 0=x\)
• (4) \( x\ast y=0 \) and \(y\ast x=0\) imply \(x=y\).

A non empty subset \(S\) of BCC-algebra \(X\) is called subalgebra of \( X\) if it is closed under BCC-operation. For a BCC-algebra \(X,\) we denote \(x\wedge y=y\ast(y\ast x)\) for all \(x,y\in X\). On any BCC-algebra \(X\) one can define the natural order \(\leq\) putting \[x \leq y \Leftrightarrow x\ast y=0.\] It is not difficult to verify that this order is partial and \(0\) is its smallest element. In BCC-algebra \(X,\) following hold: for any \(x,y,z\in X\)

• (5) (\(x\ast y)\ast(z\ast y)\leq x\ast z\),
• (6) \(x\leq y\) implies \(x\ast z \leq y\ast z\) and \(z\ast y \leq z\ast x,\)
• (7) \(x\wedge y \leq x,y\)
• (8) \(x\ast y\leq x\)
• (9) \((x\ast y)\ast z \leq x\ast (y\ast z)\)
• (10) \(x\ast x=0,\)
• (11) \((x\ast y)\ast x=0\) [ see, [6]].

Definition 1.[9] Let \(X\) be a BCC-algebra and \(\emptyset\neq\) \(I\subseteq X\). \(I\) is called an ideal of \(X\) if it satisfies the following conditions:

• (12) \(0\in I\),
• (13) \(x\ast y \in I\) and \(y\in I\) imply \(x\in I\).

If \(I\) is an ideal in BCC-algebra of \(X\), then \(I\) is a subalgebra. Moreover, if \(x\in I\) and \(y\leq x\), then \(y\in I\). An ideal \(I\) is said to be regular ideal if the relation \[x\equiv^I y\Longleftrightarrow x\ast y, y\ast x\in I\] is a congruence relation. In this case we denote \(x/I=\left\{y: x\equiv^I y\right\}\) and \(X/I=\left\{x/I: x\in X\right\}.\) \(X/I\) is a BCC-algebra by \(x/I\ast y/I=(x\ast y)/I.\)

Uniform and quasi uniform space

Let \(A\) be a non-empty set and \(\emptyset\not=\mathcal{F}\subseteq P(A).\) Then \(\mathcal{F}\) is called a \(filter\) on \(P(A)\), if for each \(F_1,F_2\in\mathcal{F}:\)

• (i) \(F_1\in\mathcal{F}\) and \(F_1\subseteq F\) imply \(F\in\mathcal{F},\)
• (ii) \(F_1\cap F_2\in\mathcal{F},\)
• (iii) \(\emptyset\not\in\mathcal{F}\).

A subset \(\mathcal{B}\) of a filter \(\mathcal{F}\) on \(A\) is a \(base\) of \(\mathcal{F}\) iff, every set of \(\mathcal{F}\) contains a set of \(\mathcal{B}\). If \(\mathcal{F}\) is a family of nonempty subsets of \(A\), then we denote generated filter by \(\mathcal{F}\) with \(fil(\mathcal{F}).\)

A \(quasi\)-\(uniformity\) on a set \(A\) is a filter \(Q\) on \(P(X\times X)\) such that

• (i) \(\bigtriangleup=\left\{(x,x)\in A\times A: x\in A\right\}\subseteq q\), for each \(q\in Q,\)
• (ii) For each \(q\in Q\), there is a \(p\in Q\) such that \(p\circ p\subseteq q\) where \[p\circ p=\left\{(x,y)\in A \times A:\exists z\in A\ s.t\ (x,z),(z,y)\in p\right\}.\]

The pair \((A,Q)\) is called a \(quasi\)-\(uniform\) \( space\). If \(Q\) is a quasi-uniformity on a set \(A\), then \(q^{-1}=\left\{q^{-1}:q\in Q \right\}\) is also a quasi-uniformity on \(A\) called the \(conjugate\) of \(Q\). It is well-known that if a quasi-uniformity satisfies condition: \(q\in Q\) implies \(q^{-1}\in Q\), then \(Q\) is a \(uniformity.\) Also \(Q\) is a uniformity on \(A\) provided \[\forall q\in Q\ \exists p\in Q\ s.t\ p^{-1}\circ p\subseteq q.\] Furthermore, \(Q^*=Q \vee Q^{-1}\) is a uniformity on \(A\). A subfamily \(\mathcal{C}\) of quasi-uniformity \(Q\) is said to be a base for \(Q\) iff, each \(q\in Q\) contains some member of \(\mathcal{C}\). The topology \(T(Q)=\left\{G\subseteq X:\forall x\in G\ \exists q\in Q\ s.t\ q(x)\subseteq G\right\}\) is called the topology induced by the quasi-uniformity \(Q\) [ See, [10]].

3. Main results

Let \(X\) be a \(BCC\)-algebra and \(\eta\) be an arbitrary family of ideals of \(X\) which is closed under intersection.

Theorem 1.[8] Let \(X\) be a BCC-algebra. The set \(\mathcal{I}=\left\{I_L:I\in \eta \right\}\) is a base for a quasi uniformity \(\mathcal{U}\) on \(X\), where \(I_L=\left\{(x,y)\in X\times X: y\ast x\in I \right\}\).

Lemma 1.[8] Let \(I\) be a regular ideal of BCC- algebra \(X\). Define \(I_L^{-1}=\left\{(x,y)\in X\times X:(y,x)\in I_L\right\}\) and \(I_L ^{\star}=I_L\cap I_L^{-1}\). Then following holds:

• (i) \(I_L^{-1}=\left\{(x,y)\in X\times X: x\ast y\in I\right\},\)
• (ii) \(I_L^{-1}(x)=\left\{y\in X: x\ast y\in I\right\},\)
• (iii) \(I_L^{-1}(0)=X,\)
• (iv) \(I_L^{\star}=\left\{(x,y)\in X\times X: x\equiv^{I} y \right\},\)
• (v) \(I_L^{\star}(x)=\left\{y\in X: x\equiv^{I} y \right\}=x/I,\)
• (vi) if \(x\in I\), then \(I_L^{\star}(x)=I.\)

Theorem 2.[8] Let \( \mathcal{U}^{\star}=\left\{U\subseteq X\times X: \exists I\in\eta \ \ I_{L}^{\star}\subseteq U\right\}. \) Then the pair \( (X,\mathcal{U}^{\star}) \) is a uniform space. Moreover, \( (X,T(\mathcal{U}^{\star})) \) is a topological BCC-algebra, where \( T(\mathcal{U}^{\star})=\left\{G\subseteq X: \forall x\in G \ \exists I\in \eta \ \ I_{L}^{\star}(x)\subseteq G\right\} \) is the induced topology by \(\mathcal{U}^{\star}\) on \(X.\) Let \( J=\bigcap_{I\in\eta} I. \) Then \( \mathcal{U}^{\star}=\left\{U\subseteq X\times X: J_{L}^{\star}\subseteq U\right\} \) and \( \tau_{J}=\left\{G\subseteq X: \forall x\in G \ \ J_{L}^{\star}(x)\subseteq G\right\}.\)

Proposition 1. \( T(\mathcal{U}^{\star})=\tau_{J}, \) where \( J=\bigcap_{I\in\eta} I. \)

Proof. Let \(x\in G\in T(\mathcal{U}^{\star}). \) Then there exists \( I\in \eta \) such that \( I_{L}^{\star}(x)\subseteq G. \) Since for any \( I\in \eta \) \( J\subseteq I, \) we get \( J_{L}^{\star}\subseteq I_{L}^{\star}. \) Hence \( J_{L}^{\star}(x)\subseteq I_{L}^{\star}(x)\subseteq G \) and so \( G\in \tau_{J}. \) Thus \( T(\mathcal{U}^{\star})\subseteq \tau_{J}. \) Conversely, let \( x\in G\in \tau_{J}. \) Then \( J_{L}^{\star}(x)\subseteq G. \) Since \( \eta \) is closed under intersection, \( J\in \eta \) and so \( J_{L}^{\star}\in \mathcal{U}^{\star}. \) Hence \( G\in T(\mathcal{U}^{\star}). \) Therefore \( \tau_{J}\subseteq T(\mathcal{U}^{\star}). \)

Definition 2.[11]

• (i) A poset \( (D,\leq) \) is called an upward directed set if for any \( i,j\in D \) there exists \( k\in D \) such that \(i\leq k \) and \( j\leq k. \)
• (ii) Let \( (D,\leq) \) be an upward directed set and \( X \) be a BCC-algebra. The mapping \( x:D\rightarrow X \) is called a net in \( X \) and denoted by \( \left\{x_{i}\right\}_{i\in D}. \)

Definition 3. Let \( \left\{x_{i}\right\}_{i\in D} \) be a net in topological space \( (X,\tau_{J}). \) Then

• (i) \( \left\{x_{i}\right\}_{i\in D} \) is called converges to \( x\in X \) if for any neighborhood \( G \) of \( x \) there exists \( i_{0}\in D \) such that \( x_{i}\in G \) for any \( i\geq i_{0}. \) In this case we write \( x_{i}\rightarrow x. \)
• (ii) \( \left\{x_{i}\right\}_{i\in D} \) is called Cauchy if there exists \( i_{0}\in D \) such that \( \frac{x_{i}}{J}=\frac{x_{j}}{J} \) for any \( i,j\geq i_{0}. \)

Proposition 2. Let \( \left\{x_{i}\right\}_{i\in D} \) and \( \left\{y_{i}\right\}_{i\in D} \) be two nets in \( (X,\tau_{J}) .\) Then

• (i) If \( x,y\in X, \) \( x_{i}\rightarrow x \) and \( y_{i}\rightarrow y, \) then \( x_{i}\ast y_{i}\rightarrow x\ast y. \)
• (ii) Each convergent net in \( X \) is a cauchy net.

Proof.

• (i) Let \( x\ast y\in G\in \tau_{J}. \) Then \( J_{L}^{\star}(x\ast y)\subseteq G. \) Since \( x_{i}\rightarrow x \) and \( J_{L}^{\star}(x) \) is a neighbohood of \( x, \) there exists \( i_{0}\in D \) such that \( x_{i}\in J_{L}^{\star}(x) \) for any \( i\geq i_{0}. \) Similarly, there exists \( i_{1}\in D \) such that \( y_{i}\in I_{L}^{\star}(y) \) for any \( i\geq i_{1}. \) Since \( D \) is an upward directed set, there exists \( i_{2}\in D \) such that \( i_{0},i_{1}\leq i_{2}. \) Hence by Lemma 1 \( x_{i}\ast y_{i}\in J_{L}^{\star}(x)\ast J_{L}^{\star}(y)=\frac{x}{J}\ast \frac{y}{J}=\frac{x\ast y}{J}=J_{L}^{\star}(x\ast y)\subseteq G \) for any \( i\geq i_{2} \) and so \( x_{i}\ast y_{i}\rightarrow x\ast y. \)
• (ii) Let \( \left\{x_{i}\right\}_{i\in D} \) be a net in \( X \) and \( x_{i}\rightarrow x\in X. \) Since \( J_{L}^{\star}(x) \) is a neighborhood of \( x, \) there exists \( i_{0}\in D \) such that \( x_{i}\in J_{L}^{\star}(x) \) for any \( i\geq i_{0}. \) Hence \( x_{i}\equiv ^{J} x \) and \( x_{j}\equiv ^{J} x \) for any \( i,j\geq i_{0} \) and so \( x_{i}\equiv ^{J} x_{j} \) for any \( i,j \geq i_{0}. \) Therefore \( \frac{x_{i}}{J}=\frac{x_{j}}{J} \) for any \( i,j\geq i_{0}. \) Thus \( \left\{x_{i}\right\}_{i\in D} \) is a cauchy net in \( X. \)

Definition 4.[11] Let \( (A,Q) \) be a uniform space.

• (i) A net \( \left\{x_{i}\right\}_{i\in D} \) in \( A \) is said to converge to a point \( x\in A \) if for each \( q\in Q \) there exists \( i_{0}\in D \) such that \( (x_{i},x)\in q \) for any \( i\geq i_{0}. \)
• (ii) A net \( \left\{x_{i}\right\}_{i\in D} \) in \( A \) is said to be a Cauchy net if for each \( q\in Q \) there exists \( i_{0}\in D \) such that \( (x_{i},x_{j})\in q \) for any \( i,j\geq i_{0}. \)

Let \( C \) be the set of all Cauchy sequence in \( (X,\mathcal{U}^{\star}). \) define a binary relation on \( C \) in the following way. For each \( \left\{x_{i}\right\}_{i\in D},\left\{y_{j}\right\}_{j\in D}\in C, \) \(\left\{x_{i}\right\}_{i\in D}\sim\left\{y_{j}\right\}_{j\in D}\) if and only if for all \( U\in \mathcal{U}^{\star} \) there exist \( i_{0}, j_{0}\in D\) such that \( (x_{i}, y_{j})\in G \) for any \( i\geq i_{0} \) and \( j\geq j_{0}. \)

Theorem 3. The relation \( \sim \) is a congruence relation on \( C. \)

Proof. Since \( (X,\mathcal{U}^{\star}) \) is a uniform space, \( \bigtriangleup\subseteq U \) for any \( U\in \mathcal{U}^{\star}. \) Hence \( (x_{i},x_{i})\in U \) for any \( i\in D \) and so \( \left\{x_{i}\right\}_{i\in D}\sim \left\{x_{i}\right\}_{i\in D}. \) Let \( \left\{x_{i}\right\}_{i\in D}\sim\left\{y_{j}\right\}_{j\in D}. \) Then for all \( U\in \mathcal{U}^{\star} \) there exist \( i_{0},j_{0}\in D \) such that \( (x_{i},y_{j})\in U \) for any \( i\geq i_{0} \) and \( j\geq j_{0}. \) Since \( U\in \mathcal{U}^{\star}, \) \( U^{-1}\in \mathcal{U}^{\star}. \) By definition of \( U^{-1} \) we have \( (y_{j},x_{i})\in U^{-1} \) for any \( i\geq i_{0} \) and \( j\geq j_{0}. \) Hence \( \left\{y_{j}\right\}_{j\in D}\sim \left\{x_{i}\right\}_{i\in D}.\) Let \( \left\{x_{i}\right\}_{i\in D}\sim\left\{y_{j}\right\}_{j\in D} \) and \( \left\{y_{j}\right\}_{j\in D}\sim\left\{z_{i}\right\}_{i\in D}. \) Let \( U\in \mathcal{U}^{\star}. \) There exists \( V\in \mathcal{U}^{\star} \) such that \( V\circ V\subseteq U. \) Since \( \left\{x_{i}\right\}_{i\in D}\sim\left\{y_{j}\right\}_{j\in D}, \) there exist \( i_{0},j_{0}\in D\) such that \( (x_{i},y_{j})\in V \) for any \( i\geq i_{0}, j\geq j_{0}. \) Similarly, there exist \(k_{0}, l_{0}\in D \) such that \( (y_{j},z_{k})\in V \) for any \( j\geq l_{0}, k\geq k_{0}. \) Since \( D \) is an upward directed set, there exsits \( n\in D \) such that \( j_{0}, l_{0}\leq n. \) If \( j\geq n, \) then \( (x_{i},y_{j})\in V \) and \((y_{j},z_{k})\in V \) for any \( i\geq i_{0} \) and \( k\geq k_{0}. \) Hence \((x_{i},z_{k})\in V\circ V\subseteq U\) for any \( i\geq i_{0} \) and \( k\geq k_{0} \) and so \( \left\{x_{i}\right\}_{i\in D}\sim\left\{z_{k}\right\}_{k\in D}. \) Thus \( \sim \) is an equivalence relation on \( C. \) Finally, we show that \( \sim \) is congruence. Let \( I\in\eta, \) \( \left\{x_{i}\right\}_{i\in D}\sim\left\{y_{j}\right\}_{j\in D} \) and \( \left\{z_{k}\right\}_{k\in D}\sim\left\{w_{l}\right\}_{l\in D}. \) Hence there exist \( i_{0}, j_{0},k_{0} \) and \( l_{0}\in D \) such that \( (x_{i},y_{j})\in I_{L}^{\star} \) for any \( i\geq i_{0},j\geq j_{0} \) and \( (z_{k},w_{l})\in I_{L}^{\star} \) for any \( k\geq k_{0} \) and \( l\geq i_{0}. \) Let \( i\geq i_{0}, j\geq j_{0} \) and \( k\geq k_{0}. \) Then \( y_{j}\in I_{L}^{\star}(x_{i}) \) and \( z_{k}\in L_{L}^{\star}(z_{k}). \) Thus \( y_{j}\ast z_{k}\in I_{L}(x_{i})\ast z_{k}\subseteq I_{L}^{\star}(x_{i})\ast I_{L}^{\star}(z_{k})=I_{L}^{\star}(x_{i}\ast z_{k}) \) and so \( (x_{i}\ast z_{k}, y_{j}\ast z_{k} )\in I_{L}^{\star}. \) Similarly, If \( j\geq j_{0}, k\geq k_{0} \) and \( l\geq l_{0}, \) then \( (y_{j}\ast z_{k},y_{j}\ast w_{l})\in I_{L}^{\star}. \) Thus \( (x_{i}\ast y_{j},z_{k}\ast w_{l})\in I_{L}^{\star}\circ I_{L}^{\star}subseteq I_{L}^{\star} \) for any \( i\geq i_{0},j\geq j_{0}, k\geq k_{0} \) and \( l\geq l_{0}. \) Since for each \( U\in \mathcal{U}^{\star} \) there exists \( I\in \eta \) such that \( I_{L}^{\star}\subseteq U, \) \( (x_{i}\ast y_{j},z_{k}\ast w_{l})\in U \) for \( i\geq i_{0},j\geq j_{0}, k\geq k_{0} \) and \( l\geq l_{0}. \) Hence \( \sim \) is a congruence relation on \( C. \)

Let \(\mathcal{ C}=\frac{C}{\sim}. \) Define a binary operation on \( \mathcal{C} \) as follow:

\[\ast:\mathcal{C}\times \mathcal{C}\rightarrow \mathcal{C}\ \ \ \ \ \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\rightarrow \frac{\left\{x_{i}\ast y_{j}\right\}_{i,j\in D}}{\sim}. \]

Theorem 4. \( \left(\mathcal{C},\ast, \frac{\left\{0\right\}_{i\in D}}{\sim}\right) \) is a BCC-algebra.

Proof. The proof is clear.

Let \( \mathcal{V}=\left\{\hat{U}: U\in \mathcal{U}^{\star}\right\} \) where,

\[\hat{U}=\left\{\left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\in \mathcal{ C}\times \mathcal{ C}: \exists i_{0},j_{0}\in D: \ \forall i\geq i_{0}, j\geq j_{0}, \ (x_{i},y_{j})\in U\right\}. \]

Theorem 5. The pair \( (\mathcal{ C}, \mathcal{V}) \) is a uniform space.

Proof. Let \( \hat{U}\in \mathcal{V} \) and \( \frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\in \mathcal{C}. \) Since \( \left\{x_{i}\right\}_{i\in D}\sim \left\{x_{i}\right\}_{i\in D}, \) there exists \( i_{0}\in D \) such that \( \left(x_{i},x_{i}\right)\in U \) for any \( i\geq i_{0}. \) Hence \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\right)\in \hat{U}. \) Since \( \frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\in \mathcal{C} \) is arbitrary, we get \( \bigtriangleup\subseteq \hat{U}. \) Let \( \hat{U}\in \mathcal{V}. \) Then \( U\in \mathcal{U}^{\star} \) and so \( U^{-1}\in \mathcal{U}^{\star}. \) Hence \( \widehat{U^{-1}}\in\mathcal{V}. \) We show that \( \widehat{U^{-1}}=\left(\hat{U}\right)^{-1}. \) Let \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\in \left(\hat{U}\right)^{-1}. \) Then \( \left(\frac{\left\{y_{j}\right\}_{j\in D}}{\sim},\frac{\left\{x_{i}\right\}_{j\in D}}{\sim}\right)\in \hat{U}. \) Hence there exist \( i_{0},j_{0}\in D \) such that \( \left(y_{j},x_{i}\right)\in U \) for any \( i\geq i_{0} \) and \( j\geq j_{0} \) and so \( \left(x_{i},y_{j}\right)\in U^{-1} \) for any \( i\geq i_{0} \) and \( j\geq j_{0} .\) Therefore \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\in \widehat{U^{-1}} \) and hence \( \left(\hat{U}\right)^{-1}\subseteq \widehat{U^{-1}}. \) Similarly, we have \( \widehat{U^{-1}}\subseteq \left(\hat{U}\right)^{-1}.\) Thus \( \left(\hat{U}\right)^{-1}\in\mathcal{V} \) for any \( \hat{U}\in \mathcal{V}. \) Let \( \hat{U}\in \mathcal{V}. \) Then \( U\in \mathcal{U}^{\star}. \) There exists \( V\in \mathcal{U}^{\star} \) such that \( V\circ V\in U. \) We claim that \( \hat{V}\circ \hat{V}\subseteq \hat{U}. \) Let \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{z_{k}\right\}_{k\in D}}{\sim}\right)\in \hat{V}\circ \hat{V}. \) There exists \( \frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\in \mathcal{C} \) such that \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\in \hat{V}\) and \( \left(\frac{\left\{y_{j}\right\}_{j\in D}}{\sim},\frac{\left\{z_{k}\right\}_{k\in D}}{\sim}\right)\in \hat{V}.\) Hence there exist \( i_{0},j_{0},k_{0} \) and \( l_{0}\in D \) such that \( \left(x_{i},y_{j}\right)\in V \) for any \( i\geq i_{0}, j\geq j_{0} \) and \( \left(y_{j},z_{k}\right)\in V \) for any \( j\geq l_{0}, k\geq k_{0}. \) Since \( D \) is an upward direcred set, there exists \( n\in D \) such that \( n\geq j_{0}, l_{0}. \) If \( j\geq n, \) then \( \left(x_{i},y_{j}\right)\in V \) and \( \left(y_{j},z_{k}\right)\in V \) for any \( i\geq i_{0}, k\geq k_{0}. \) Hence \( \left(x_{i},z_{k}\right)\in V\circ V\subseteq U \) for any \( i\geq i_{0}, k\geq k_{0} \) and so \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{z_{k}\right\}_{k\in D}}{\sim}\right)\in \hat{U}. \) Let \( \hat{U},\hat{V}\in \mathcal{V}. \) Then \( U,V\in \mathcal{U}^{\star} \) and so \( U\cap\in V\in \mathcal{U}^{\star}. \) Hence \( \widehat{U\cap V}\in \mathcal{V}. \) We show that \( \widehat{U\cap V}=\hat{U}\cap \hat{V}. \) Let \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\in \hat{U}\cap \hat{V}. \) Then \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\in \hat{U} \) and \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\in \hat{V}. \) There exist \( i_{0},j_{0},i_{1} \) and \( j_{1}\in D \) such that \( \left(x_{i},y_{j}\right)\in U \) for any \( i\geq i_{0}, j\geq j_{0} \) and \( \left(x_{i},y_{j}\right)\in V \) for any \( i\geq i_{1}, j\geq j_{1}. \) There exist \( i_{2},j_{2}\in D \) such that \( i_{0},i_{1}\leq i_{2} \) and \( j_{0},j_{1}\leq j_{2}. \) Hence \( \left(x_{i},y_{j}\right)\in U \) and \( \left(x_{i},y_{j}\right)\in V \) for any \( i\geq i_{2} \) and \( j\geq j_{2} \) and so \( \left(x_{i},y_{j}\right)\in U\cap V \) for any \( i\geq i_{2} \) and \( j\geq j_{2}. \) Hence \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\in \widehat{U\cap V}\) and so \( \hat{U}\cap \hat{V}\subseteq \widehat{U\cap V}. \) Similarly, we can show that \( \widehat{U\cap V}\subseteq \hat{U}\cap \hat{V}. \) Finally, let \( \hat{U}\in \mathcal{V} \) and \( \hat{U}\subseteq \widetilde{V}\subseteq \mathcal{C}\times \mathcal{C}. \) We have to show that \( \widetilde{V}\in \mathcal{V}. \) Let \( \left(x,y\right)\in U\in \mathcal{U}^{\star}. \) Then \( \left(\frac{\left\{x\right\}_{i\in D}}{\sim},\frac{\left\{y\right\}_{j\in D}}{\sim}\right) \in \hat{U}\) and so \( \left(\frac{\left\{x\right\}_{i\in D}}{\sim},\frac{\left\{y\right\}_{j\in D}}{\sim}\right) \in\widetilde{V}. \) Thus \( \left(x,y\right)\in V \) and so \( U\subseteq V. \) Hence \( V\in \mathcal{U}^{\star} \) and so \( \widetilde{V}\in \mathcal{V}. \)

Theorem 6. \( \left(\mathcal{ C},\ast, T\left(\mathcal{V}\right)\right) \) is a topological BCC-algebra where, \[ T\left(\mathcal{V}\right)=\left\{G\in \mathcal{C}: \forall \ \ \frac{\left\{x\right\}_{i\in D}}{\sim} \ \ \exists\hat{U}\in\mathcal{V} \ \ s.t. \ \ \hat{U}\left(\frac{\left\{x\right\}_{i\in D}}{\sim}\right)\subseteq G\right\}. \]

Proof. Let \( \frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\ast\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\in G\in T\left(\mathcal{V}\right). \) Then there exists \( U\in \mathcal{U}^{\star} \) such that \( \hat{U}\left(\frac{\left\{x_{i}\ast y_{j}\right\}_{i,j\in D}}{\sim}\right) \subseteq G.\) Since \( U\in \mathcal{U}^{\star}, \) there exists \( I\in\eta \) such that \( I_{L}^{\star}\subseteq U. \) Clearly, \( \widehat{I_{L}^{\star}}\left(\frac{\left\{x_{i}\ast y_{j}\right\}_{i,j\in D}}{\sim}\right)\subseteq \hat{U}\left(\frac{\left\{x_{i}\ast y_{j}\right\}_{i,j\in D}}{\sim}.\right) \) We claim that \( \widehat{I_{L}^{\star}}\left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\right)\ast \widehat{I_{L}^{\star}}\left(\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\subseteq \widehat{I_{L}^{\star}}\left(\frac{\left\{x_{i}\ast y_{j}\right\}_{i,j\in D}}{\sim}\right). \) Let \( \frac{\left\{a_{k}\right\}_{k\in D}}{\sim}\in \widehat{I_{L}^{\star}}\left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\right) \) and \( \frac{\left\{b_{l}\right\}_{l\in D}}{\sim}\in \widehat{I_{L}^{\star}}\left(\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right).\) Then \( \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},\frac{\left\{a_{k}\right\}_{k\in D}}{\sim}\right)\in \widehat{I_{L}^{\star}} \) and \( \left(\frac{\left\{y_{j}\right\}_{j\in D}}{\sim},\frac{\left\{b_{l}\right\}_{l\in D}}{\sim}\right)\in \widehat{I_{L}^{\star}}. \) Hence there exist \( i_{0},j_{0},k_{0} \) and \( l_{0}\in D \) such that \( \left(x_{i},a_{k}\right)\in I_{L}^{\star} \) and \( \left(y_{j},b_{l}\right)\in I_{L}^{\star} \) for any \( i\geq i_{0}, j\geq j_{0}, k\geq k_{0} \) and \( l\geq l_{0}. \) Thus \( x_{i}\equiv^{I} a_{k} \) and \( y_{j}\equiv^{I} b_{l} \) and so \( x_{i}\ast y_{j}\equiv^{I} a_{k}\ast b_{l} \) for any \( i\geq i_{0}, j\geq j_{0}, k\geq k_{0} \) and \( l\geq l_{0}. \) Therefore \( \left( x_{i}\ast y_{j},a_{k}\ast b_{l}\right)\in I_{L}^{\star} \) for any \( i\geq i_{0}, j\geq j_{0}, k\geq k_{0} \) and \( l\geq l_{0}\) and so \( \left(\frac{\left\{x_{i}\ast y_{j}\right\}_{i,j\in D}}{\sim},\frac{\left\{a_{k}\ast b_{l}\right\}_{k,l\in D}}{\sim}\right)\in \widehat{I_{L}^{\star}}. \) Hence \( \frac{\left\{a_{k}\ast b_{l}\right\}_{k,l\in D}}{\sim}\in \widehat{I_{L}^{\star}}\left(\frac{\left\{x_{i}\ast y_{j}\right\}_{i,j\in D}}{\sim}\right). \) Thus \( \widehat{I_{L}^{\star}}\left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\right)\ast \widehat{I_{L}^{\star}}\left(\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\subseteq \widehat{I_{L}^{\star}}\left(\frac{\left\{x_{i}\ast y_{j}\right\}_{i,j\in D}}{\sim}\right). \)

Definition 5.[11] The uniform space \( \left(A,Q\right) \) is complete if each cauchy net in \( A \) is convergent.

Definition 6.[11] Let \( \left(A,Q\right) \) be a uniform space. a uniform space \( \left(\hat{A},\hat{Q}\right) \) is said to be a completion of \( \left(A,Q\right) \) if

• (i) \( \left(\hat{A},\hat{Q}\right) \) is a complete uniform space.
• (ii) \( \left(A,Q\right) \) with its topology induced by its uniform structure is homeomorphic to a dense subspace of \( \left(\hat{A},\hat{Q}\right). \)

Theorem 7. The uniform space \( \left(\mathcal{ C}, \mathcal{V}\right) \) is a completion of \( \left(X,\mathcal{U}^{\star} \right). \)

Proof. Let \( i:X\rightarrow \mathcal{C} \) be defined by \( i\left(x\right)=\frac{\left\{x\right\}_{i\in D}}{\sim}. \) Clearly, \( i \) is one to one. We show that \( i\left(X\right) \) is dense in \( \mathcal{C}. \) Let \( \hat{U}\left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\right)\in T\left(\mathcal{V}\right). \) Then \begin{eqnarray*} \hat{U}\left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\right)\cap i\left(X\right)&= & \left\{i\left(x\right): \left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim},i\left(x\right)\right)\in \hat{U}\right\}, \\ &= &\left\{i\left(x\right): \exists i_{0}\in D \ \ \forall i\geq i_{0} \ s.t. \ \ \left(x_{i},x\right) \in U\right\}, \\ &= &\left\{i\left(x\right): \exists i_{0}\in D \ \ \forall i\geq i_{0} \ s.t. \ \ x \in U\left(x_{i}\right)\right\}, \\ &=& \left\{i\left(x\right): x\in \bigcup_{i\in D}\bigcap_{i_{0}\leq i} U\left(x_{i}\right)\right\}, \\&=& i\left(V\right) \end{eqnarray*} where \( V=\bigcup_{i\in D}\bigcap_{i_{0}\leq i} U\left(x_{i}\right). \) Hence \( \hat{U}\left(\frac{\left\{x_{i}\right\}_{i\in D}}{\sim}\right)\cap i\left(X\right)\neq \emptyset \) and so \( i\left(X\right) \) is dense in \( \mathcal{C}. \) It is easy to see that \( i:X\rightarrow i\left(X\right) \) is a homeomorphism. Now we show that the uniform space \( \left(\mathcal{ C}, \mathcal{V}\right) \) is complete. Let \( \left\{\frac{\left\{x_{i}^{\alpha}\right\}_{i\in D}}{\sim}\right\}_{\alpha\in D} \) be a cauchy net in \( \mathcal{C}. \) We have to show that it is convergent. Let \( U\in \mathcal{U}^{\star}. \) Since \(\left\{\frac{\left\{x_{i}^{\alpha}\right\}_{i\in D}}{\sim}\right\}_{\alpha\in D}\) is a cauchy net, there exists \( \gamma\in D \) such that \( \left(\frac{\left\{x_{i}^{\alpha}\right\}_{i\in D}}{\sim},\frac{\left\{x_{i}^{\beta}\right\}_{i\in D}}{\sim}\right)\in\hat{U} \) for any \( \alpha, \beta\geq \gamma. \) Hence there exist \( \alpha_{0},\beta_{0}\in D \) such that \( \left(x_{i}^{\alpha},x_{i}^{\beta}\right)\in U \) for any \( \alpha\geq \alpha_{0} \) and \( \beta\geq \beta_{0}. \) We define the net of \( {\left\{y_{j}\right\}_{j\in D}} \) by \( y_{j}=x_{i}^{\beta_{0}} \) for any \( j\in D. \) Clearly, \( \left(\frac{\left\{x_{i}^{\alpha}\right\}_{i\in D}}{\sim},\frac{\left\{y_{j}\right\}_{j\in D}}{\sim}\right)\in \hat{U} \) for any \( \alpha\geq \alpha_{0}. \) Therefore \( \left\{\frac{\left\{x_{i}^{\alpha}\right\}_{i\in D}}{\sim}\right\}_{\alpha\in D} \) is converges to \( \frac{\left\{y_{j}\right\}_{j\in D}}{\sim}. \)

4. Conclusion

The aim of this paper was to study the concept of completion of a quasi-uniformity on a BCC-algebra. This work can be the basis for further and deeper research of the properties of BCC-algebras.

Conflictofinterests

The author declares no conflict of interest.