ABSTRACT: This paper is a study on algebraic structure of Pre A* - algebra and defines the relation R on an. and prove that the relations

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1 Interntionl Journl of Engineering Science nd Innovtive Technology (IJESIT) PRE A*-ALGEBRA AND CONGRUENCE RELATION DrYPRAROOPA 1 KMoses 2 AShou Reddy 3 1DrMrsYPRAROOPA, Associte Professor, Deprtment of Mthemtics,S&H,Andhr Loyol Institute of Engineering& Technology, Vijywd -8APIndi 2 KMoses, Senior Lecturer in Mthemtics, ACCollege, Guntur, AP 3AShou Reddy,Senior Lecturer in Mthemtics,Andhr Loyol College,VijywdAP ABSTRACT: This pper is study on lgebric structure of Pre A* - lgebr nd defines the reltion R on n lgebr A is sid to be congruence reltion if R is n equivlence reltion nd it preserves the opertions of R (ie, R is closed under the opertions in A {(, ) / nd prove tht the reltions y A A y} {(, y) A A / y}, A re congruence reltions on Pre A*-lgebr KEYWORDS: Congruence reltion, Equivlence reltion, Pre A*-lgebr, I INTRODUCTION In drft pper [4], The Equtionl theory of Disjoint Alterntives, round 1989, EGMnes introduced the concept of Ad (Algebr of disjoint lterntives) (A,, V, (-) I, (-), 0, 1, 2) (Where,V re binry opertions on A, (-) I, (-) re unry opertions nd 0,1,2 re distinguished elements on A)which is however differ from the definition of the Ad of his lter pper[5] Ads nd the equtionl theory of if-then-else in 1993 While the Ad of the erlier drft seems to be bsed on etending the If-Then-Else concept more on the bsis of Boolen lgebrs nd the lter concept is bsed on C-lgebrs (A,,V,(-) ~ ) )(where,v re binry opertions on A,(-) ~ is unry opertion ) introduced by Fernndo Guzmn nd Crig C Squir[2] In 1994, PKoteswr Ro[3] first introduced the concept of A*-lgebr (A,, V, *, (-) ~, (-), 0, 1, 2) )( where,v, * re binry opertions on A, (-) ~,(-) re unry opertions nd 0,1,2 re distinguished elements on A) not only studied the equivlence with Ad, C-lgebr, Ad s connection with 3-Ring, Stone type representtion but lso introduced the concept of A*-clone, the If-Then-Else structure over A*-lgebr nd Idel of A*-lgebr In 2000, JVenkteswr Ro[6] introduced the concept Pre A*-lgebr (A,,, (-) ~ )(where,v re binry opertions on A,(-) ~ is unry opertion on A nlogous to C-lgebr s reduct of A*- lgebr, studied their sub direct representtions, obtined the results tht 2= {0, 1} nd 3= {0, 1, 2} re the sub directly irreducible Pre-A*- lgebrs nd every Pre-A*-lgebr cn be imbedded in 3 X ( where 3 X is the set of ll mppings from nonempty set X into 3= {0, 1, 2}) PrroopY[13] introduced the specific concepts on Pre A*-lgebr nd of the ppers[8],[9],studied Pre A*-lgebr s semi lttice, lttice in Pre A*-lgebr II PRILIMINARIES DEFINITION: An lgebr (A,,, (-) ~ ) stisfying () ( ~ ) ~ =, A =, A y = y,,y A ( y) ~ = ~ y ~,,y A (e) (y z) = ( y) z,,y,z A (f) (y z) = ( y) ( z),,y,z A 307

2 EXAMPLE ISSN: Interntionl Journl of Engineering Science nd Innovtive Technology (IJESIT) (g) y = ( ~ y),,y A is clled Pre A* - lgebr 3 = { 0,1,2 } with opertions, (-) ~ defined below is Pre A* - lgebr Note The elements 0,1,2 in the bove emple stisfy the following lws: () 2 ~ = 2 1 = for ll 3 0 =, 3 2 = 2 =2, 3 Emple : 2={0,1} with opertions,, (-) ~ defined below is Pre A*-lgebr NOTE :(2,,, (-) ~ ) is Boolen lgebr So every Boolen lgebr is Pre A* -lgebr III CONGRUENCE RELATION ON PRE A* - ALGEBRA [13] Definition: A reltion on Pre A * -lgebr equivlence reltion (ii) is closed under,, ~ Lemm[13]: Let ( A,,,( ) ) be Pre A*-lgebr nd let A {(, y) A A / y} is congruence reltion A,,, ~ is clled congruence reltion if (i) is n Then the reltion Theorem: Let A be Pre A* lgebr with 1 nd,then following re equivlent () 308

3 Interntionl Journl of Engineering Science nd Innovtive Technology (IJESIT) (e) is congruence reltion on A is refleive on A is symmetric on A is refleive reltion on Pre A * -lgebr with 1 nd B(A) Note: If hence congruence reltion Lemm: Let A,,, ~ be Pre A*-lgebr nd let {(, y) A A / y} is () congruence reltion b b A then is symmetric nd trnsitive, Then the reltion we will write y to indicte ( y, ) Lemm: Let A be Pre A*-lgebr nd b, BA ( ), then Proof: By the lemm 12 we hve b b b b (, y) Let us ssume tht b, then ( b) ( b) y Now ( ( b)) (( b) ) (( b) y) ( ( b)) y y (Since b, BA ( ) ) (, y) This implies tht (, y), similrly we cn prove tht b Therefore ( y, ) b This implies tht b b 309

4 Interntionl Journl of Engineering Science nd Innovtive Technology (IJESIT) Hence s required Theorem Let A be Pre A*-lgebr, then A/ { / A} = A*-Algebr, whose opertions re defined s is Pre A*-lgebr, is clled quotient Pre (i) b b (ii) b (iii) ( ) b Definition: Let A be Pre A*- lgebr nd A Define {( p, q) A A / ( p, q) p} is not n equivlence reltion For emple, in Pre A*-lgebr A, {( p, q) A A / ( p, q) p} 2 2 {( p, q) A A/ (2 p) (2 q) p} {(2, q) A A/ q A} is not refleive, but, we hve the following theorem Theorem: Let A be Pre A* lgebr with 1 nd,then following re equivlent () is congruence reltion on A (e) is refleive on A is symmetric on A Congruence reltion A,,, ~ Lemm: Let be Pre A*-lgebr nd let {(, ) / } reltion y A A y is A Then the () congruence reltion 310

5 Interntionl Journl of Engineering Science nd Innovtive Technology (IJESIT) b b We will write y to indicte ( y, ) Lemm: Let A be Pre A*-lgebr nd, b, then b b Lemm: Let A be Pre A*-lgebr, we let A {(, ) / A} then () A if nd only if, A A denote the trivil congruence on A: AA if nd only if, A ( b, ), b then b If then A Theorem Let A be Pre A*-lgebr, then A / { / } = A is Pre A*-lgebr, is clled quotient Pre A*- Algebr, whose opertions re defined s (i) b b (ii) b b (iii) ( ) IV CONCLUSION This pper studies the lgebric structure of Pre A* - lgebr nd defines the reltion R on n lgebr A is sid to be congruence reltion if R is n equivlence reltion nd it preserves the opertions of R (ie, R is closed under the opertions in A nd prove tht the reltions {(, y) A A / y} {(, y) A A / y}, A re congruence reltions on Pre A*-lgebr REFERENCES [1] Birkoff G: Lttice theory, Americn Mthemticl Society, Colloquium Publictions, Vol25, New York, 1948 [2] Fernndo Guzmn nd Crig CSquir: The Algebr of Conditionl logic, Algebr Universlis 27(1990), [3] Koteswr RoP, A*-Algebr, n If-Then-Else Structures (thesis) 1994, Ngrjun University, AP, Indi [4] Mnes EG: The Equtionl Theory of Disjoint Alterntives, Personl Communiction to ProfNVSubrhmnym (1989) [5] Mnes EG: Ad nd the Equtionl Theory of If-Then-Else, Algebr Universlis 30(1993),

6 Interntionl Journl of Engineering Science nd Innovtive Technology (IJESIT) [6] Venkteswr RoJ, On A*-Algebrs (Thesis) 2000, Ngrjun University, AP, Indi [7] Venkteswr RoJ, PrroopY Boolen lgebrs nd Pre A*-Algebrs, Act Cienci Indic (Mthemtics), (ISSN: ), (2006) 32: pp 71-76) [8] Venkteswr RoJ nd PrroopY, Pre A*-Algebr s semi lttice, Asin Journl of Algebr, (2011)Volume 4, Number 1, [9] Venkteswr RoJ nd Prroop Y, Lttice in Pre A*-Algebr, Asin Journl of Algebr, ISSN X Volume 4, Number 1, 1-11, 2010 [10] Venkteswr RoJ nd Prroop Y, Pre A*-Algebrs nd Rings, Interntionl Journl of Computtionl Science nd Mthemtics ISSN Volume 3, Number 2 (2009), pp , 2011 [11] Venkteswr RoJ nd Prroop Y Homomorphisms, Idels nd Congruence Reltions on Pre A*-Algebr, Globl Journl of Mthemticl Sciences: Theory nd Prcticl ISSN No Volume 3, Number 2 (2011), pp , My, 2011 [12] VenkteswrRoJ nd Prroop Y Logic circuits nd Gtes in Pre A*- Algebr, Asin Journl of Applied Sciences,4(1): 89-96, 2011 [13] PrroopY, On Pre A*-Algebrs (Thesis) 2012, Ngrjun University, AP, Indi [14] Prroop Y, PRE A*-ALGEBRA AND HOMOMORPHISM, Interntionl Journl of Engineering Technology, Mngement nd Applied Sciences, November 2014, Volume 2 Issue 6, ISSN