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Boolean Centre of Pre A*-Algebra



A. Satyanarayana, J. Venkateswara Rao, U. Suryakumar and P. Nirmala Kumari
 
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ABSTRACT

This manuscript illustrates the essential congruence θx on a Pre A*-algebra A and arrive at a variety of properties of these. Also it bear out certain properties of the operations Γx(p, q) and Φx. It has been confirmed that θ is a factor congruence on A if and only if θ = θx for some xεB(A). Further it was proved that the centre B (A) of a Pre A*-algebra A with 1 is isomorphic with the Boolean centre (A) of A.

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  How to cite this article:

A. Satyanarayana, J. Venkateswara Rao, U. Suryakumar and P. Nirmala Kumari, 2012. Boolean Centre of Pre A*-Algebra. Asian Journal of Mathematics & Statistics, 5: 150-157.

DOI: 10.3923/ajms.2012.150.157

URL: https://scialert.net/abstract/?doi=ajms.2012.150.157
 
Received: January 09, 2012; Accepted: March 25, 2012; Published: May 18, 2012



INTRODUCTION

In an outline study, Manes (1989) introduced the concept of Ada (Algebra of disjoint alternatives) (A, ∧, ∨, (-)'(-)π, 0, 1, 2), (where A is a nonempty set; ∧ and ∨ are two binary operations on A; (-)' and (-)π are unary operations on A and 0, 1, 2 are distinguished elements in A). Which is however different from the definition of the Ada of his later paper (Manes, 1993). While the Ada of the earlier draft seems to be based on extending the If-Then -Else concept more on the basis of Boolean algebra and the later concept is based on C-algebra (A, ∧, ∨, ') introduced by Guzman and Squier (1990).

Koteswara Rao (1994) firstly introduced the concept A*-Algebra (A, ∧, ∨, (-) (-)π, 0, 1, 2) and studied its equivalence with Ada, C-algebra and Ada’s connection with 3- Ring. Further he made an effort on the If-Then-Else structure over A*-algebra and introduced the concept of Ideal of A*-algebra. Venkateswara Rao (2000) introduced the concept of Pre A*-algebra (A, ∧, ∨, ') as the variety generated by the 3-element algebra A = {0, 1, 2} which is an algebraic form of three valued conditional logic. Satyanarayana et al. (2010) generated semilattice structure on Pre A*-Algebras. Venkateswara Rao and Srinivasa Rao (2009) defined a partial ordering on a Pre A*-algebra A and studied its properties. Satyanarayana et al. (2010) derive necessary and sufficient conditions for pre A*-algebra A to become a Boolean algebra in terms of the partial ordering. Srinivasa Rao (2009) studied the structural compatibility of Pre A*-algebra with Boolean algebra.

This study perceives a fundamental congruence θx on a Pre A*-algebra and confer its various properties. Also it establishes certain properties of operations Γx (p, q) and Φx. It has been proved that θ is a factor congruence on A if and only if θ = θx for some xεB(A). Further it has been derived that the centre B(A) of a Pre A*-algebra A with 1 is isomorphic with the Boolean centre Image for - Boolean Centre of Pre A*-Algebra (A) of A.

PRELIMINARIES

Definition 1: Boolean algebra is an algebra (B, ∧, ∨, (-)', 0, 1, 2) with two binary operations, one unary operation (called complementation) and two nullary operations which satisfies:

(i) (B, ∧, ∨) is a distributive lattice
(ii) x ∧ 0 = 0, x ∨ 1 = 1
(iii) x ∧ x' = 0, x ∨ x' = 1

We can prove that x" = x, (x∨y)' = x'∧y', (x∧y)' = x'∨y', for all x, yεB.
Here, we concentrate on the algebraic structure of Pre A*-algebra and state some results which will be used in the later text.

Definition 2: An algebra (A, ∧, ∨, (-)~) where A is non-empty set with 1, ∧, ∨ are binary operations and (-) is a unary operation satisfying:

(a) Image for - Boolean Centre of Pre A*-Algebra
(b) Image for - Boolean Centre of Pre A*-Algebra
(c) Image for - Boolean Centre of Pre A*-Algebra
(d) Image for - Boolean Centre of Pre A*-Algebra
(e) Image for - Boolean Centre of Pre A*-Algebra
(f) Image for - Boolean Centre of Pre A*-Algebra
(g) Image for - Boolean Centre of Pre A*-Algebra is called a Pre A*-algebra

Example 1: 2 = {0, 1, 2} with operations ∧, ∨, (-)~ defined below is a Pre A*-algebra.

Image for - Boolean Centre of Pre A*-Algebra

Note 1: The elements 0, 1, 2 in the above example satisfy the following laws:

(a) 2~ = 2
(b) 1 ∧ x = x for all x ε 3
(c) 0 ∨ x = x for all x ∈ 3
(d) 2 ∧ x = 2 ∨ x = 2 for all x ε 3.

Example 2: 2 = {0, 1} with operations ∧, ∨, (-)~ defined below is a Pre A*-algebra.

Image for - Boolean Centre of Pre A*-Algebra

Note 2:

(i) (2, ∧, ∨, (-)) is a Boolean algebra. So, every Boolean algebra is a Pre A* algebra
(ii) The identities (a) and (d) imply that the varieties of Pre A*-algebras satisfies all the dual statements of (b) to (g) of definition 2

Note 3: Let A be a Pre A*-algebra then A is Boolean algebra if and only if x∨(x∧y) = x, x∧(x∨y) = x (absorption laws holds).

Lemma 1: Every Pre A*-algebra satisfies the following laws.

(a) Image for - Boolean Centre of Pre A*-Algebra
(b) Image for - Boolean Centre of Pre A*-Algebra
(c) Image for - Boolean Centre of Pre A*-Algebra
(d) Image for - Boolean Centre of Pre A*-Algebra

Definition 3: Let A be a Pre A*-algebra. An element xεA is called central element of A if x∨x = 1 and the set {x ∈ A/x∨x = 1} of all central elements of A is called the centre of A and it is denoted by B (A). Note that if A is a Pre A*-algebra with 1 then 1, 0∈B (A). If the centre of Pre A*-algebra coincides with {0, 1} then we say that A has trivial centre.

Theorem 1: Venkateswara Rao and Srinivasa Rao (2009): Let A be a Pre A*-algebra with 1, then B (A) is a Boolean algebra with the induced operations ∧, ∨, (-).

Lemma 2: Venkateswara Rao and Srinivasa Rao (2009): Let A be a Pre A*-algebra with 1:

(a) If y ε B(A) then Image for - Boolean Centre of Pre A*-Algebra
(b) x∧(x∨y) = x∨(x∧y) = x if and only if x, y εB(A)

Congruences on Pre A*-algebra
Definition 4: Srinivasa Rao (2009):
Let A be a Pre A*-algebra and θ be binary relation on A. Then θ is said to be an equivalence relation on A if θ satisfies the following:

(i) Reflexive: (x, x) ε θ, for all xεA
(ii) Symmetric: (x, y) ε θ ⇒(y, x) ε θ, for all x, y ε A
(iii) Transitive: (x, y) ε θ and (y, z) ε θ⇒(x, z) ε θ, f or all x, y, z ε A

We write x θ y to indicate (x, y)εθ.

Definition 5: Srinivasa Rao (2009): A relation θ on a Pre A*- algebra (A, ∧, ∨, (-)) is called congruence relation if:

(i) θ is an equivalence relation
(ii) θ is closed under ∧, ∨, (-)~

Lemma 3: Srinivasa Rao (2009): Let (A, ∧, ∨, (-)~) be a Pre A*-algebra and let a ε A. Then the relation θa = {(x, y)ε AxA/a∧x = a∧y} is:

(a) a congruence relation
(b) Image for - Boolean Centre of Pre A*-Algebra
(c) Image for - Boolean Centre of Pre A*-Algebra
(d) Image for - Boolean Centre of Pre A*-Algebra

We will write x θay to indicate (x, y)εθa.

Lemma 4: Srinivasa Rao (2009): Let A be a Pre A*-algebra with 1 and x, yεB(A). Then

(1) θx ∩ θy = θx∨y
(2) θx o θy ⊆ θx∧y
(3) θx ∨ θy ⊆ θx∧y

Theorem 2: Srinivasa Rao (2009): Let Abe a Pre A*-algebra, then A ={θa/aεA} is a Pre A*-algebra, is called quotient Pre A*-Algebra, whose operations are defined as:

(i) θa∧θb = θa∧b
(ii) θa∨θb = θa∨b
(iii) a)~ = θa~

Lemma 5: Let A be a Pre A*-algebra and x, y ε A. Then, θx ⊆ θy if and only if y = x ∧ y.

Proof: Suppose that θx ⊆ θy.
Since x∧y = x∧x∧y, we have (y, x∧y) ε θx and therefore, (y, x∧y) εθy.
(Supposition)
⇒ y∧y = y∧x∧y
⇒ y = x∧y
Conversely suppose that y = x∧y
Let (p, q)ε θx ⇒x∧p = x∧q
Now y∧p = x∧ y∧p (Supposition)
= y∧x∧p
= y∧x∧q
= y∧q (Supposition)
Therefore, (p, q)εθy and hence, θx⊆θy.

Lemma 6: Let A be a Pre A*-algebra and x,yεA. Then θx⊆θy and θy⊆θx∧y.

Proof: We know that x∧(x∧ y) = x∧y, by above Lemma we have θx⊆θx∧y.
Also, we know that y∧(x∧y) = x∧y, by above Lemma we have θy⊆θx∧y.

Lemma 7: Let A be a Pre A*-algebra and x, yεB(A). Then θx∨y ⊆θx.

Proof: Let (p, q) ε θx∨y. Then (x∨y)∧p = (x∨y)∧q
Now x∧p = x∧(x∨y)∧p (By Lemma 2)
= x∧(x∨y)∧q
= x∧q
Therefore (p, q) εθx and hence, θx∨y⊆θx.
Note that it is similar way θx∨y⊆θy.

Theorem 3: Let A be a Pre A*-algebra with 1 and xεA. Then θx is the smallest congruence on A containing (1, x).

Proof: We know that θx is a congruence of A and clearly (1,x) ε θx.
Let θ be a congruence on A and (1,x) εθ.
Suppose that (p, q)ε θx ⇒x∧p = x∧q
Since (1,x)ε θ, we have (1∧p, x∧p) and (1∧q, x∧q) εθ; that is (p, x∧p) and (q, x∧q) εθ. Therefore (p, q) εθ and hence θx ⊆ θ.

Theorem 4: Let A be a Pre A*-algebra with 1 and x, yεA then the following are equivalent:

(1) x, yεB(A)
(2) θx∨y⊆θy
(3) θx∨y⊆θx∩θy
(4) θx∨y = θx∩θy

Proof: (1)⇒(2) Suppose that x, yεB(A)
Let (p ,q) εθ. Then (x∨y) ∧p = (x∨y)∧q
Now y∧p= y∧(y∨x) ∧p (Since x, yεB(A) by Lemma 2)
= y∧(x∨y) ∧q = y∧q
Therefore, (p, q) εθy and hence, θx∨y⊆θy.
(2)⇒(3) Suppose that θx∨y⊆θx.
By symmetry we have, θx∨y⊆θx.
Therefore, θx∨y⊆θx∩θy.
(3)⇒(4) Suppose that θx∨y⊆θx∩θy.
We know that, θx∩θy⊆θx∨y (By Lemma 3(c))
Therefore, θx∨yx⊆θy
(4)⇒(1) Suppose that θx∨yx∩θy
Then (1, x∨y) ε θx∨y = θx∩θy (Supposition)
Therefore (1, x∨y) ε θx and (1, x∨y) εθy ⇒ x∧1= x∧(x∨y) and y∧1 = y∧(x∨y)
⇒ x = x∧(x∨y) and y= y∧(x∨y)
⇒ x, yεB(A) (By Lemma 2)

Definition 6: Srinivasa Rao (2009): Let A be a Pre -A* algebra with 1. For any x, p, q ε A, define Γx (p, q) = (x∧p)∨(x~∧q).
Now we prove certain properties of Γx(p, q) which will be used later text.

Lemma 8: Let A be a Pre -A* algebra and x, p, q εB(A). Then:

(1) Γx(p, q) = (x~∨p)∧(x∨q)
(2) Γx(p, q)~ = Γx(p~ ,q~)

Proof:
(1) Γx(p, q) = (x∧p)∨(x~∧q)
  = [(x∧p)∨x~]∧[(x∧p)∨q] (By distributive law)
  = [x~∨p]∧[(x∨q)∧(x~∨ p∨q)] (Definition 2 (g) and Lemma 1(d))
  = [(x~∨p)∧(x∨q)∧(x~∨ p)] ∧[(x~∨p)∧(x∨q)∧q)] (By distributive law)
  = [(x~∨p)∧(x∨q)]∧[(x~∨p)∧ (x∨q) ∧q)]
  = (x~∨p)∧(x∨q) (Lemma 2 (b))

(2) By (1) we have Γx (p, q) = (x~∨p) ∧(x∨q).
  Now Γx(p, q)~ = [(x~∨p) ∧(x∨q)]~
  = (x~∨p) ~∨(x∨q)~
  = (x ∧ p~)∨(x~∧q~)
  = Γx(p~, q~)

Lemma 9: Let A be a Pre -A* algebra and x, p, q ε A. Then Γx (p, q) = Γx~ (q, p).

Proof: Γx (p, q) = (x∧p)∨(x~∧q)
= (x~~∧p)∨(x~∧q)
= (x~∧q) ∨(x~~∧p)
= Γx~(q, p).

Definition 7: Let A be a Pre -A* algebra and xεA. Define:
Φx = {(p, q) ε AxA / Γx(p, q) = p}.

Theorem 5: Let A be a Pre -A* algebra and xεB(A). Then:

(1) Φx⊆θx~
(2) Φx is transitive relation on A.

Proof:

(1) Let p, q ε A and (p, q) εΦx, that is Γx(p, q)= p
  ⇒(x∧p)∨(x~∧q) = p
  Now x~∧p = x~∧{(x∧p∨(x~∧q))}
  = {x~∧x∧p} ∨{x~∨x~∧q}
  = (0∧p) ∨(x~∧q)
  = 0∨(x~∧q) (provided p ≠ 2)
  = x~∧q
  If p=2 then x~∧p =2 to get the required result q should be 2.
  Therefore x~∧p = x~∧q , hence (p, q) ε θx~
  Thus, Φx⊆θx~

(2) Let p, q, r εA and (p, q), (q, r) ε Φx, that is Γx(p, q) = p and Γx(q, r) = q.
  Then by (1) we have (p, q), (q, r)ε θx~
  ⇒ x~∧p = x~∧q and x~∧q = x~∧r
  Now Γx(p, r) = (x∧p)∨(x~∧r)
  = (x∧p)∨(x~∧q)
  = Γx(p, q)
  = p

Therefore (p, r) ε Φx and hence Φx is transitive relation on A.

Theorem 6: Let A be a Pre A*-algebra induced by a Boolean algebra and θ be congruence on A. Then θx is a factor congruence on A if and only if θ = θx for some xεA.

Proof: Suppose that θ = θx for some xεB(A).
Then x~εB(A) and θx∩θx~x∨x~ = θ1 = ΔA
and θx o θ x~ = θx∧x~ = θ0 = AxA
Thus, θx is a factor congruence on A
Conversely suppose that θ is a factor congruence on A
Then there exist a congruence β on A such that θ ∩ β = ΔA and θ o β = AxA.
Now we show that θ = θx
Suppose that (p,q) εθx then x∧p= x∧ q.
Since (x,1) εθ we have (x∧p, 1∧p), (x∧q, 1∧ q) ε that is (x∧p, p), (x∧q, q) εθ which imply that (p,q) εθ.
Hence, θx⊆θ
Suppose (p,q) εθ. Then (x∧p, x∧q) εθ.
Since (0,x) εβ we have (0∧p, x∧ p), (0∧q, x∧q) εβ that is (0, x∧p), (0, x∧q) εβ which implies that (x∧p, x∧q) εβ.
Therefore (x∧p, x∧q) εβ∩ θ = ΔA and hence x∧p = x∧q ⇒(p, q)εθx
Hence, θ⊆θx
Thus θ = θx.
Hence, θ is a factor congruence on A if and only if θ = θx for some xεB(A).

Definition 8: Let A be a Pre A*-algebra and αεCon(A). Then α is called factor congruence if there exist βεCon(A) such that α∩β = ΔA and αoβ= AxA. In this case β is called direct complement of α.

Definition 9: A congruence β on Pre A*-algebra A is called balanced if (β∨θ)∩(β∨θ~) = β for any direct factor congruences θ and any of its direct complement θ~ on A.

Theorem 7: Let A be a Pre A*-algebra with 1 and xεB(A). Then θx is balanced.

Proof: Let θx is a congruence on Pre A*-algebra A. Let Ψ be a factor congruence A and Ψ~ be its complement. Then by theorem (b) there exist y, zεB(A) such that Ψ = θy and Ψ~ = θz.
Now (θx∨Ψ)∩(θx∨Ψ~) = (θx∨θy)∩ (θx∨θz)
x∨y∩θx∨z
(x∨y)∨(x∨z)
x∨(y∨z)
= θx∨θy∨z
= θx ∨ (θy∩θz)
= θx ∨(Ψ∩Ψ~)
= θx∨ΔA (Since Ψ and Ψ~ are complements)
= θx
Hence, θx is balanced.

Therefore, the set of balanced congruence which admit a balanced complement is precisely the set Image for - Boolean Centre of Pre A*-Algebra (A) = {θx/xεB(A)} and hence, Image for - Boolean Centre of Pre A*-Algebra (A) is the Boolean centre of A.

Theorem 8: Let A be a Pre A*-algebra with 1. Then the Boolean centre Image for - Boolean Centre of Pre A*-Algebra (A) = {θx / xεB(A)} is a Boolean algebra and the map x→θx~ is an isomorphism of B(A) into Image for - Boolean Centre of Pre A*-Algebra (A).

Proof: It follows from Lemma 4, theorem 6 and theorem 7.

CONCLUSION

This manuscript point ups the essential congruence θx on a Pre A*-algebra and reach your destination at a variety of properties of these. Also it corroborate certain properties of the operations Γx(p, q) and Φx. It has been long-established that θ is a factor congruence on A if and only if θ = θx for some x εB(A). If A is a Pre A*-algebra with 1 and xεA, then obtained that θx is the smallest congruence on A containing the ordered pair (1, x). Additionally it was ensuing that the centre B(A) of a Pre A*-algebra A with 1 is isomorphic with the Boolean centre Image for - Boolean Centre of Pre A*-Algebra (A) of A.

REFERENCES

1:  Guzman, F. and C.C. Squier, 1990. The algebra of conditional logic. Algebra Univ., 27: 88-110.
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2:  Koteswara Rao, P., 1994. A*-Algebras and if-then-else structures. Ph.D. Thesis, Acharya Nagarjuna University, Andhra Pradesh, India.

3:  Manes, E.G., 1989. The equational theory of disjoint alternatives. Personal Communication to Prof. N.V. Subrahmanyam.

4:  Manes, E.G., 1993. Adas and the equational theory of if-then-else. Algebra Universalis, 30: 373-394.
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5:  Rao, J.V., 2000. On A*-algebras. Ph.D. Thesis, Nagarjuna University, A.P., India.

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8:  Srinivasa Rao, K., 2009. Structural compatibility of pre A*-algebra with boolean algebra: A novel approach. Doctoral Thesis, Nagarjuna University, A.P., India.

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