In this paper we studied the free A*-algebras, the sub A*-algebra generated by a subset and its characterization, an A*-algebra freely generated by a subset and introduced the concept of A*-field of sets. Also we established some theorems on making A*-field of sets into A*-algebras

Koteswara Rao, P

Srinivasa Rao, T

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Journal of Experimental Sciences 2012, 3(4): 10-12 ISSN: 2218-1768 Available Online: http://jexpsciences.com/    Free A*-algebras  T. Srinivasa Rao1 and P. Koteswara Rao2  1Department of Mathematics, Bapatla Engineering College, Bapatla-522101, A.P, India. 2Department of Commerce and Business Management, Acharya Nagarjuna University, Nagarjuna Nagar-522510, A.P, India.  Abstract  In this paper we studied the free A*-algebras, the sub A*-algebra generated by a subset and its characterization, an A*-algebra freely generated by a subset and introduced the concept of A*-field of sets. Also we established some theorems on making A*-field of sets into A*-algebras.  Keywords: A*-algebra, Free A*-algebra, A* field of sets    INTRODUCTION       In 1994, P. Koteswara Rao introduced the concept of A*- algebras and studied their equivalence with Adas, C – algebras, Stone type representation and introduced the concept of A*- clones and the if-then-else structure over A*- algebras and ideals of A*- algebras. In this paper we introduced the concept of free A*- algebras analogous to the free Boolean algebras.  PRELIMINARIES  Definition: For any non-empty set X, a class M of subsets of X which is closed under finite union of sets, finite intersection of sets, complementation of sets is called a field of sets.  Definition: A Boolean algebra is an algebra (B,∨, ∧, (-)′, 0, 1) with two binary operations, one unary operation(called complementation), and two nullary operations which satisfies:   (1) (B,∨, ∧) is a distributive lattice. (2) x ∧ 0 = 0,   x ∨ 1 = 1 (3) x ∧ x′ = 0, x ∨ x′ = 1  Definition: An indexed set {Bt }t∈T of subalgebras of a Boolean algebra B is said to be independent if a1∧a2∧…∧an ≠ 0 for every finite sequence of non zero elements ai choosen from subalgebras Bt with different indices.  Definition: Let {Bt }t∈T be an indexed set of Boolean algebras. By a Boolean product of {Bt }t∈T we mean any pair {{it }t∈T, B} such that   a) B is a Boolean algebra. b) For every t∈T, it : Bt →B is an isomorphism. c) The indexed set {it(Bt) }t∈T of subalgebras of B is independent. d) The union of all subalgebras{it(Bt) }t∈T generates B.  Definition: Suppose {Xt}t∈T is an indexed set of non-empty sets. Let X be the Cartesian product of Xt’s. Let Mt be the field of subsets of Xt.. For every set A⊆ Xt, let A* be the set of all points x∈X whose tth coordinate xt∈A and let Mt* be the field composed of all sets A* where A∈ Mt. The field M of subsets of X generated by all the classes {Mt*} t∈T is called the field product of {Mt} t∈T.    Definition: An algebra (A, ∧, ∗  ,(-)~, (-)pi ,1)  is an A* - algebra if it  satisfies : For a, b, c∈A i. api ∨( api)~ = 1, ( api) pi = api , where a∨b = (a~∧ b~)∼. ii. api∨ bpi = bpi∨ api iii. (api∨ bpi )∨cpi = api∨ (b pi ∨cpi) iv. (api ∧ bpi )∨ (api ∧ (bpi)~) = api v. (a ∧ b)pi = api∧ bpi ,  (a ∧ b )# = a# ∨ b# ,where a# = (api ∨ a~pi)~ vi. a~pi = (api∨a#)~ , a~# = a# vii. (a∗b)pi =  api, (a∗b)# = (api)~∧(b~pi)~ viii. a = b if and only if  api = bpi, a# = b#. We write 0 for 1~, 2 for 0*1.  Example: 3 ={0,1,2} with the operations defined below is an A* -algebra    Definition : Let (A1, ∧,* ,(-)~, (-)pi ,1) and (A2, ∧, (-)~,(-)pi , *,1) be  two A*-algebras. A Mapping f: A1→ A2 is called an A* - homomorphism if for all   a, b∈A1    I. f(a∧b) =  f(a)∧f(b) II. f(a∨b) =  f(a)∨f(b) III. f(a * b) =  f(a) * f(b) IV. f(api ) = (f(a))pi V. f(a~) = (f(a))~        Received: Feb 10, 2012; Revised: March 15, 2012; Accepted: April 25, 2012.  *Corresponding Author T. Srinivasa Rao  Department of Mathematics, Bapatla Engineering College, Bapatla-522101,  A.P, India. Tel: +91-1234567898; Fax: +91-1234567898 Email: tsnivas@gmail.com  Journal of Experimental Sciences 2012, 3(4): 10-12  11VI.  f(1) = 1 VII. f(0) = 0.  Definition: Let C be a non-empty set. By a free A*-algebra on C we shall mean an A*-algebra F together with a mapping f: C→F such that for every A*-algebra A and every mapping g: C→A there is a unique A*-homomorphism  h: F→A such that hog = f. Now we state some routine theorems without proof.  Theorem: If (F,f) is a free A*-algebra on a non-empty set C then f is injective and Im f generates F.  Theorem:Let (F,f ) be a free A*-algebra on a nom empty set C. Then (Fl, fl ) is also a free A*-algebra on C iff there is a unique A*-isomorphism j: F→ Fl such that jof = fl.  Theorem: Let ( B, ∧, (-)′, 0, 1) be a Boolean algebra. The A(B) = {(a, b) /a, b ∈B, a ∧ b = 0 } becomes an A*- algebra with the following A*- algebraic operations: For a = (xpi, x# ), b = (ypi, y# ) ∈A(B)  I. a ∧ b = (xpi ypi, xpi y# + x# ypi+ x# y#) II. a ∨ b = (xpi ypi + xpi y# + x# ypi,  x# y#) III. a ∗ b = (xpi, ((xpi)′ y#) IV. a~ = ( x# , xpi) V. api = (xpi, (xpi)′) VI. 1 = (1, 0), 0 = (0, 1), 2 = (0, 0).  Theorem : If A is an A*- algebra then  I. B(A(B)) ≅ B. II. A(B(A)) ≅ A.  Theorem: Let A1,A2 be A*- algebras and B1,B2 be Boolean algebras then       I. A1 ≅ A2 iff B(A1) ≅ B(A2). II. B1 ≅ B2 iff A(B1) ≅ A(B2).   MAIN RESULTS  Theorem: For every non-empty set C, there exists a free A*-algebra on C.  Proof: Let F = {θ / θ: C→3 = {0, 1, 2}, θ(c) = 0 for all most all c∈C } Define ∨, ∧, (-)~,(-)pi, *, 0,1, 2 as follows: Let θ, ξ ∈ F. For all c∈C, define (θ ∨ ξ)(c) = (θ)(c) ∨ (ξ)(c) (θ ∧ ξ)(c) = (θ)(c) ∧ (ξ)(c) (θ * ξ)(c) = (θ)(c) * (ξ)(c) θ~(c) = (θ(c))~ 0(c) = 0 1(c) = 1 2(c) = 2 Then F is an A*-algebra. Now define f : C→ F as follows: For any c∈C, f(c) : C → 3 by f(c)(d) = 1 if d = c                                                                  = 0 if d ≠ c  Claim: (F, f ) is a free A*-algebra on C. Let A be any A*-algebra and g : C → A  be a mapping. Define h :  F  → A as follows: Let θ ∈ F  ( ) ( ) ( )c Ch c g cθ θ∈= ∨  Where θ(c).g(c) = (θ(c).g(c))pi * (θ(c).g(c))# (θ(c).g(c))pi = (θ(c) ∧g(c))pi ∨ (θ(c)# ∧g(c)#) (θ(c).g(c))# = (θ(c)pi ∧g(c) #) ∨ (θ(c)# ∧g(c) pi) Clearly h is an A*- homomorphism ( ( )) ( ( ))( ). ( )t Ch f c f c t g t∈= ∨ = g(c) Therefore hog = g. Uniqueness of h: Let θ ∈ F, t∈ C θ (t) = θ (t).1  ( ).[ ( )]( )[ ( ). ( )]( )c Cc Cc f c tc f c tθθ∈∈= ∨= ∨  ( ) ( )c CTherefore c f cθ θ∈= ∨  Suppose : →h F A is another A*- homomorphism such that =ho f g   ( ) ( ( ) ( ))( ) ( ( ))( ) ( )( )c Cc Cc Ch h c f cc h f cc g chθ θθθθ∈∈∈= ∨= ∨= ∨= Therefore =h h   Remerk: The free A*-algebra F constructed as in the above theorem is called the free A*-algebra on C.   Definition: Suppose A is an A*-algebra and A .⊆C Then A is said to be freely generated by C if   (i) <C> = A      (ii) Every mapping f : C → Al, Al is another A*-algebra, can be extending uniquely to a homomorphism h : A→ Al .  Theorem: If A is an A*-algebra and B = B(A), A.⊆C  then A is freely generated by C if and only if B is freely generated by Cpi, where Cpi = {api / a ∈ C}.  Proof: Suppose A is an A*-algebra, B = B(A) and A.⊆C  .Assume that A is freely generated by C. We have to prove that B is freely generated by Cpi = {api / a ∈ C}. Let g : C → B´  is a mapping where B´  is another Boolean algebra. Define f : C → A(B´ ) by f(a) = (g(a)pi, g(a)pi~). Since C generates A, f can be extended to a unique homomorphism             h :A → A(B ´  ). hpi : B →B(A(B ´  )) is also a unique homomorphism of Boolean Rao and Rao   12algebras. Since ( )( )B ´  ´≅A B B , let I : B(A(B´  )) is an isomorphism such that  Then I hpi : B → B´  is a homomorphism and I hpi is a unique extension of g. Therefore Cpi generates B freely. Conversely assume that Cpi generates B freely. We have to prove that A is freely generated by C. Suppose f : C → A ´  is another mapping where A ´  is another  A*-algebra. Define fpi : Cpi  → B(A ´  ) by fpi(a) = (f(b)) pi Since Cpi generates B freely ∃  a unique homomorphism  g : B → B(A´  )  Such that fpi(a) = g(a) a Cpi∀ ∈ . Define h : A → A ´  by  Then h is a homomorphism. Since g is unique, his also unique. We now show that h = f on C. Let a C ∈ ⇒ ∈a Cpi pi                ~ ~~( ) ( ( ) ( ) ) ( ) ( ( ) )[ ( ) (( ( ) ]( )h a g a g a f a f af a f af api pi pi pi pi pipi pi pi= ∗ = ∗= ∗= Therefore h is a unique extension of f. Therefore C generates A freely.   Definition: Let X be a non-empty set. A class  F∗= {(A1, A2) / A1, A2 ⊆X, 1 2A A }∩ = φ  is called an A*-field of subsets of X if  (i) (X, φ ) ∈ F ∗  (ii) (A1, A2) ∈ F ∗ ⇒  (A2, A1) ∈ F ∗  (iii) (A1, A2), (B1, B2) ∈ F ∗ ⇒  (A1B1 , A1B2 + A2B1 + A2B2) (iv) (A1, A2), (B1, B2) ∈ F ∗ ⇒  (A1, A1C B2 ) ∈ F ∗        Juxtaposition and addition stand for intersection and union of sets      From the above definition immediately we have the following theorem.   Theorem: Let X be a non-empty set. A class F ∗  = {(A1, A2) / A1, A2 ⊆ X,  A1IA2 = φ}. Then F∗ is an A*-algebra with the following operations: (i) 1 = (X, φ ), 0 = ( , )Xφ , 2 = ( , )φ φ  (ii) (A1, A2) pi = (A2, A1C) (iii) (A1, A2)C = (A2, A1) (iv) (A1, A2) ∗  (B1, B2) = (A1, A1C B2 ) (v) (A1, A2) ∧ (B1, B2) = (A1B1 , A1B2 + A2B1 + A2B2) (vi) (A1, A2) ∨ B1, B2) = (A1B1 + A1B2 + A2B1 , A2B2) Proof: It is routine to verify the axioms in 1.6. Theorem: Let F∗  be an A*- field of subsets of a non-empty set X and  Then F is a field of subsets of X and B(F∗ ) ≅  F.  Proof: Suppose F∗ is an A*- field of subsets of X. We have to prove that is a field of subsets of X.  Consider   = (AB, ABC + AC(B + BC) )∈ F∗  = (AB, ABC + AC)∈ F∗(AB, (A + AC).(BC + AC) )∈ F∗  =(AB, AC+BC )∈ F∗  = (AB, (AB)C )∈ F∗  = AB∈ F Let A ∈ F. Then (A, AC)∈ F∗⇒ (AC, A) ∈ F∗ ⇒  ⇒   Therefore F is a field of subsets of X and clearly F ≅  B(F∗ ) Theorem: Let F is a field of subsets of a non-empty set X. Then A(F)  is an A*- field of subsets of X.  Proof: It is routine to verify the axioms in 2.5. We now prove the following theorem.  Theorem: Every A*- algebra A is isomorphic to an A*- field of subsets F∗ of a Stone space.  Proff: Let A be an A*- algebra. Let B = B(A). Then there exists a Stone space H such that B ≅  F, where F is the field of clopen subsets of H. By the known result A(B) ≅  A(F).    But A ≅  A(B). Thus A ≅  A(F).  ACKNOWLEDGEMENT       Bringing out a special issue in Journal of Experimental Sciences is a great honor to the eminent mathematician Prof.N.Ch.Pattabhi RamaCharyulu. We feel that we are very happy because our article is published in this Special issue.  REFERENCES  [1] Koteswara Rao, P “A*-algebras and If –Then –Else Structures”, Ph.D Thesis, Nagarjuna University (1994), Andhra Pradesh, India. [2] Sikorski, R “Boolean Algebras”, Acaddemic Press, New York, 1968. [3] Siosen, F. M “ Free algebraic characterizations of primal and independent Algebras, Proc. Amer. Math. Soc, Vol. 41 (1937), pp. 375-481. [4] Stone, M. H “Applications of the theory Boolean rings to the General topology”, Trans. Amer. Math. Soc, Vol.41 (1937), pp. 375-481.

Free A*-algebras

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Free A*-algebras

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