On Pure PO-Ternary Γ-Ideals in Ordered Ternary Γ-Semirings

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1 OSR Joural of Mathematcs (OSR-JM) e-ssn: , p-ssn: X. Volume 11, ssue 5 Ver. V (Sep. - Oct. 215), PP O Pure PO-Terary Γ-deals Ordered Terary Γ-Semrgs V. Syam Julus Rajedra 1, Dr. D. Madhusudhaa Rao 2 ad M. Saja Lavaya 3 1 Lecturer, Departmet of Mathematcs, A.C. College, Guture, A.P. da. 2 Head, Departmet of Mathematcs, V. S. R & N.V.R. College, Teal, A.P. da. 3 Lecturer, Departmet of Mathematcs, A.C. College, Gutur, A.P. da. Abstract: ths paper, we troduce the cocepts of pure po-terary Γ-deal, weakly pure po-terary Γ-deal ad purely prme po-terary Γ-deal a ordered terary Γ-semrg. We obta some characterzatos of pure po-terary Γ-deals ad prove that the set of all purely prme po-terary Γ-deals s topologzed. eywords: terary Γ-semrg; ordered terary Γ-semrg; weakly regular; pure po-teraryγ-deal; weakly pure po-terary Γ-deal; purely prme po-terary Γ-deal; topology.. troducto [1], Ahsa ad Takahash troduced the otos of pure deal ad purely prme deal a semgroup. Recetly, Bashr ad Shabr [2] defed the cocepts of pure deal, weakly pure deal ad purely prme deal a terary semgroup wthout order. The authors gave some characterzatos of pure deals ad showed that the set of all purely prme deals of a terary semgroup s topologzed. ths paper, we troduce the cocepts of purepo-terary Γ-deal, weakly pure po-terary Γ-deal ad purely prme po-terary Γ-deal a ordered terary Γ-semrg. We characterze pure po-terary Γ-deals ad prove that the set of all purely prme po-terary Γ-deals of a ordered terary Γ-semrg s topologzed. Note that the results o terary Γ-semrg wthout order become the specal cases.. Prelmares Defto 2.1: Let T ad Γ be two addtve commutatve semgroups. T s sad to be a Terary Γ-semrg f there exst a mappg from T Γ T Γ T to T whch maps ( x,, x,, x ) x x x satsfyg the codtos : ) [[aαbβc]γdδe] = [aα[bβcγd]δe] = [aαbβ[cγdδe]] )[(a + b)αcβd] = [aαcβd] + [bαcβd] ) [a (b + c)βd] = [aαbβd] + [aαcβd] v) [aαbβ(c + d)] = [aαbβc] + [aαbβd] for all a, b, c, d T ad α, β, γ, δ Γ. Obvously, every terary semrg T s a terary Γ-semrg. Let T be a terary semrg ad Γ be a commutatve terary semgroup. Defe a mappg T Γ T Γ T T by aαbβc = abc for all a, b, c T ad α, β Γ. The T s a terary Γ-semrg. Note 2.2 :Let (T,Γ, +, [ ]) be a terary Γ-semrg. For oempty subsets A 1,A 2 ad A 3 of T, let [AΓBΓC]= a b c : a A, b B, c C,,.For x T, let [xγa 1 ΓA 2 ] = [{x}γa 1 ΓA 2 ]. The other cases ca be defed aalogously. Note2.3 : Let T be a terary semrg. f A, B are two subsets of T, we shall deote the set A + B = a b : a A, b B ad 2A = { a + a : a A}. Defto 2.4 :A terary Γ-semrg T s called a ordered terary Γ-semrg f there s a partal order o T such that x y mples that () a + c b + c ad c + a c + b () [aαcβd] [bαcβd], [cαaβd] [cαbβd] ad [cαdβa] [cαdβb]for all a, b, c, d T ad α, β, γ, δ Γ. Note2.5 : For the coveece we wrte x x x stead of x x x PO-Terary Γ-deals : Defto 3.1: Let T be PO-terary Γ-semrg. A oempty subset S s sad to be a PO-terary Γsubsemrg of T f () S s a addtve subsemgroup of T, () aαbβc S for all a, b, c S, α, β Γ. DO: 1.979/ Page

2 O Pure PO-Terary Γ-deals Ordered Terary Γ-Semrgs () T T, s S, t s t S. Example 3.2 :LetT= M 2 ( Z ) ad Γ = M 2 ( Z ) defe the orderg as a b. The T be the PO-terary Γ- semrg of the set of all 2x2 square matrces over Z, the set of all o-postve tegers. Let a S / a Z. The S s a PO-terary Γ-subsemrg of T. Notato 3.3 :LetT be PO-terary Γ-semrg ad S be a oempty subset of T. f H s a oempty subset of S, we deote {s S :s h for some h H} by (H] S. Notato 3.4: Let T be PO-terary Γ-semrg ad S be a oempty subset of T. f H s a oempty subset of S, we deote {s S :h s for some h H} by [H) S. Note 3.5 : (H] T ad [H) T are smply deoted by (H] ad [H) respectvely. Note 3.6: A oempty subset S of a po-terary Γ-semrg T s apo-terary Γ-subsemrg of T ff (1)S + S S (2) SΓSΓS S, (2) (S] S. Theorem 3.7 : Let S be po-terary Γ-semrg ad A S, B S. The () A (A], () ((A]] = (A], () (A]Γ(B]Γ(C] (AΓBΓC] ad (v) A B A (B], (v) A B (A] (B], (v) (A B] = (A] (B], (v) (A B] = (A] (B]. Defto 3.8 : A oempty subset A of a PO-terary Γ-semrg T s sad to be left PO-terary Γ-deal of T f (1) a, b A mples a + b A. (2) b, c T, a A, α, β Γ mples bαcβa A. (3) t T, a A, t a t A. Note 3.9 : A oempty subset A of a PO-terary Γ-semrg T s a left PO-terary Γ-deal of T f ad oly f A s addtve subsemgroup of T, TΓTΓA A ad (A] A. Note 3.1:Let T be a PO-terary Γ-semrg. The the set (TΓTΓa] = {t T / t x y a for some x, y T,, 1 ad N}. Example 3.11 : the PO-terary Γ-semrg Z, Z s a left PO-terary Γ-deal for ay N. Theorem 3.12: Let Tbe a PO-terary Γ-semrg. The (TΓTΓa] s a left PO-terary Γ-deal of T for all a T. Defto 3.13: A left PO-terary Γ-deal A of a PO-terary Γ-semrg T s sad to be the prcpal left POterary Γ-deal geerated by a f A s a left PO-terary Γ-deal geerated by a for some a T. t s deoted by L (a) or <a> l. Theorem 3.14 : f T s a PO-terary Γ-semrg ad a T the L(a) = (A] where A= r t a a : r, t T,, a d z 1 ad z s the set of all postve teger wth zero. DO: 1.979/ Page, ad deotes a fte sum Proof :Gve that A = r t a a : r, t T,, a d z. Let a, b A. 1 a, b A. The a = r t a a ad b = r t a a for r j j j j, t, r j, t j T,,,, ad j j z. Now a + b = r t a a + r t a a a + b s a fte sum. j j j j Therefore a + b A ad hece A s a addtve subsemgroup of T. For t 1,t 2 T ad a A. The t 1 αt 2 βa = t 1 αt 2 ( r t a a ) = r t ( t t a ) ( t t a ) A

3 O Pure PO-Terary Γ-deals Ordered Terary Γ-Semrgs Therefore t 1 αt 2 βa A ad hece A s a left terary Γ-deal of T. By theorem 3.18,we have (A] s a left ordered terary Γ-deal of T cotag a. Thus L(a) (A].O the other had, L(a) s also aleft ordered Γ-deal of T cotag a, so we have A L(a). Thus (A] L(a) sce (A] s a left ordered terary deal of T geerated by A. Therefore L(a) = (A], as requred. Defto 3.15 : A oempty subset of a PO-terary Γ-semrg T s sad to be a lateral PO-terary deal of T f (1) a, b A mples a + b A. (2) b, c T, α, β Γ, a A mples bαaβc A. (3) t T, a A, t a t A. Note 3.16: A oempty subset of A of a PO-terary semrg T s a lateral PO-terary Γ-deal of T f ad oly f A s addtve subsemgroup of T, TΓAΓT A ad (A] A. Theorem 3.18: Let T be a PO-terary Γ-semrg. The (TΓaΓT] s a lateral PO-terary Γ-deal of T for all a T. Theorem 3.18: Let T be a PO-terary Γ-semrg. The (TΓTΓaΓTΓT] s a lateral PO-terary Γ-deal of T for all a T. Defto 3.19 : A lateral PO-terary Γ-deal A of a PO-terary Γ-semrg T s sad to be the prcpal lateral PO-terary Γ-deal geerated by a f A s a lateral PO-terary Γ-deal geerated by a for some a T. t s deoted by M (a) (or) <a> m. Theorem 3.2 : f T s a PO-terary Γ-semrg ad a T the M(a) = (A], where A = r a t u v a p q a : r, t, u v p q T,,,,,, a d z 1 j1 j j j j j j j j j j j j j j j j, ad deotes a fte sum ad z s the set of all postve teger wth zero. Defto 3.21 : A oempty subset A of a PO-terary Γ-semrg T s a rght PO-terary Γ-dealof T f (1) a, b A mples a + b A. (2) b, c T, α, β Γ, a A mples aαbβc A. (3) t T, a A, t a t A. Note 3.22 : A oempty subset A of a PO-terary Γ-semrg T s a rghtpo-terary Γ-deal of T f ad oly f A s addtve subsemgroup of T, AΓTΓT A ad (A] A. Defto 3.23 : A rght PO-terary Γ-deal A of a PO-terary Γ-semrg T s sad to be a prcpal rght POterary Γ-deal geerated by a f A s a rght PO-terary Γ-deal geerated by a for some a T. t s deoted by R (a) (or) <a> r. Theorem 3.24 : f T s a po-terary Γ-semrg ad a T the R(a) = (A], where A = a r t a : r, t T,, a d z 1 z s the set of all postve teger wth zero., deotes a fte sum ad Defto 3.25 : A oempty subset A of a PO-terary Γ-semrg T s a two sded PO-terary Γ-dealof T f (1) a, b A mples a + b A (2) b, c T,α, β Γ,a A mples bαcβa A, aαbβc A. (3) t T, a A, t a t A. Note 3.26: A oempty subset A of a PO-terary Γ-semrg T s a two sded PO-terary Γ-deal of T f ad oly f t s both a left PO-terary Γ-deal ad a rght PO-terary Γ-deal of T. DO: 1.979/ Page

4 O Pure PO-Terary Γ-deals Ordered Terary Γ-Semrgs Defto 3.27 : A two sded PO-terary Γ-deal A of a PO-terary Γ-semrg T s sad to be the prcpal two sded PO-terary Γ-deal provded A s a two sded PO-terary Γ-deal geerated by a for somea T. t s deoted by T (a) (or) <a> t. Theorem 3.28 : f T s a PO-terary Γ-semrg ad a T the T(a) = (A], where r s a : a t u j j j j l m a p q a k k k k k k k k A = j1 k 1 ad deotes a 1 r, s, t, u, l m, p, q T,,,,,,,, a d Z j j k k k k j j k k k k fte sum ad z s the set of all postve teger wth zero. Defto 3.29 : A oempty subset A of a PO-terary Γ-semrg T s sad to be PO-terary Γ-dealof T f (1) a, b A mples a + b A (2) b, c T,α, β Γ,a A mples bαcβa A, bαaβc A, aαbβc A. (3) t T, a A, t a t A. Note 3.3 : A oempty subset A of a PO-terary Γ-semrg T s apo-terary Γ-deal of T f ad oly f t s left PO-terary Γ-deal, lateral PO-terary Γ-deal ad rght PO-terary Γ-deal of T. Defto 3.31 :A elemet a of a PO-terary Γ-semrg. T s sad to be regular f there exst x, y T such that a aαxβaγyδa for all α, β, γ, δ Γ. V. Pure po-teraryγ-deals ordered terary Γ-semrg ths secto we defe pure po-terary Γ-deals ordered terary Γ-semrg. Defto 4.1: Let T be a ordered terary Γ-semrg. A two-sded po-terary Γ-deal A of T s called a left (respectvely, rght) pure two-sded po-terary Γ-deal f for each x A there exst y,z A, α, β Γ where Δsuch that x y z x (respectvely, x x y z 1 1 ). Apo-terary Γ- deal A of Ts called left (respectvely, rght) pure po-terary Γ-deal f for each x A there exst y,z A, α, β Γ where Δsuch that x y z x (respectvely x x y z 1 ). Smlarly, we defe oe-sded left ad rght pure poterary Γ-deals. 1 Theorem 4.2: Let T be a ordered terary Γ-semrg ad A a two-sded po-teraryγ-deal of T. The A s rght pure po-terary two-sded Γ-deal f ad oly f B\A = ([BΓAΓA]] for all rght po-terary Γ-deals B of T. Proof:Assume that A s rght pure two-sded po-terary Γ-deal. Let B be a rght po-terary Γ-deal of T. We have [BΓAΓA] [BΓTΓT] B. The ([BΓAΓA]] (B] = B. Sce [BΓAΓA] [TΓTΓA] A, so ([BΓAΓA]] (A] = A.Hece ([BΓAΓA]] B A.To prove the reverse cluso, let x B A. By assumpto, there exst y,z A, α, β Γ where Δsuch that x x y z 1. Sce 1 DO: 1.979/ Page x y z [BΓAΓA], we obta x ([BΓAΓA]]. Thus B A ([BΓAΓA]]. Coversely, suppose that B A = ([BΓAΓA]] for all rght po-terary Γ-deals B of T. Let x A. Sce ({x} [xγtγt]] s a rght po-terary Γ-deal of T ad [TΓTΓA] A, we have ({x} [xγtγt]] A = ([({x} [xγtγt]]γaγa]] ([xγaγa] [[xγtγt]γaγa]] ([xγaγa]]. Sce x ({x} [xγtγt]] A, x ([xγaγa]]. Hece A s a rght pure two-sded po-teraryγ-deal of T. Defto 4.3: A ordered terary Γ-semrg T s sad to be rght weakly regular f for ay x T, x ([[xγtγt]γ[xγtγt]γ[xγtγt]]]. Note that every regular ordered terary Γ-semrg s rght weakly regular. Theorem 4.4: Let T be a ordered terary Γ-semrg. The followg are equvalet. () T s rght weakly regular. () ([AΓAΓA]] = A for all rght po-terary Γ-deals A of T. () B A = ([BΓAΓA]] for all rght po-terary Γ-deals B ad all two-sded po-terary Γ-deals A of T. (v) B A = ([BΓAΓA]] for all rght po-teraryγ-deals B ad all po-terary Γ-deals A of T.

5 O Pure PO-Terary Γ-deals Ordered Terary Γ-Semrgs Proof:() (). Assume that T s rght weakly regular. Let A be a rght po-terary Γ-deal of T. Sce [AΓAΓA] [AΓTΓT] A, we have ([AΓAΓA]] A. Let x A. By assumpto, x ([[xγtγt]γ[xγtγt]γ[xγtγt]]] ([AΓAΓA]]. The A ([AΓAΓA]], whece ([AΓAΓA]] = A. () (). Assume that ([AΓAΓA]] = A for all rght po-terary Γ-deals A of T. Let x T. Sce ({x} [xγtγt]] s a rght po-terary Γ-deal of T, we have ({x} [xγtγt]] = ([({x} [xγtγt]]γ({x} [xγtγt]]γ({x} [xγtγt]]] ({[xγxγx]} [[xγxγx]tγt] [[xγxγt]γtγx] [[xγxγt]γtγ[xγtγt]] [[xγtγt]γxγx] [[xγtγt]γxγ[xγtγt]] [[xγtγt]γ[xγtγt]γx] [[xγtγt]γ[xγtγt]γ[xγtγt]]]. Sce x ({x} [xγtγt]], we obta (by calculatos) x ([[xγtγt]γ[xγtγt]γ[xγtγt]]]. Hece T s rght weakly regular. () (). Assume that T s rght weakly regular. Let B ad A be a rght po-terary Γ-deal ad a two-sded poterary Γ-deal of T, respectvely. Sce [BΓAΓA] [BΓTΓT] B, ([BΓAΓA]] B. Smlarly, ([BΓAΓA]] A.The ([BΓAΓA]] B A. Let x B A. We have ([[xγtγt]γ[xγtγt]γ[xγtγt]]] ([BΓAΓA]]. By assumpto, we get x ([[xγtγt]γ[xγtγt]γ[xγtγt]]], hece x ([BΓAΓA]]. Thus B A ([BΓAΓA]], whece B A = ([BΓAΓA]]. That () (v) s clear. (v) (). Assume that B A = ([BΓAΓA]] for all rght po-terary Γ-deals B ad all po-terary Γ-deals A of T. To prove that T s rght weakly regular, let x T. We have ({x} [xγtγt]] ad ({x} [xγtγt] [TΓTΓx] [TΓxΓT] [TΓ[TΓxΓT]ΓT]] are rghtpo-terary Γ- deal ad deal of T, respectvely. The ({x} [xγtγt]] ({x} [xγtγt] [TΓTΓx] [TΓxΓT] [TΓ[TΓxΓT]ΓT]] =(({x} [xγtγt]]γ({x} [xγtγt] [TΓTΓx] [TΓxΓT] [TΓ[TΓxΓT]ΓT]]Γ ({x} [xγtγt] [TΓTΓx] [TΓxΓT] [TΓ[TΓxΓT]ΓT]]] ({[xγxγx]} [[xγxγx]γtγt] [xγ[xγtγt]x] [xγ[xγtγx]γt] [xγ[xγtγt]γ[xγtγt]] [xγ[tγtγx]γx] [[xγtγt]γ[xγxγt]γt] [[xγtγt]γ[xγtγt]γx] [[xγtγt]γ[xγtγx]γt] [[xγtγt]γ[xγtγt]γ[xγtγt]] [[xγtγx]γtγx] [[xγtγx]γ[tγxγt]γt] [[xγtγx]γ[tγtγx]γt]] ([[xγtγt]γ[xγtγt]γ[xγtγt]]]. Thus x ([[xγtγt]γ[xγtγt]γ[xγtγt]]]. Hece T s rght weakly regular ordered terary Γ-semrg. Theorem 4.5.Let T be a ordered terary Γ-semrg. The followg areequvalet. () T s rght weakly regular. () Every two-sded po-teraryγ-deal A of T s rght pure. () Every po-teraryγ-deal A of T s rght pure. Proof:Ths follows from Theorem 4.2, ad Theorem 4.4. Defto4.6: A elemet a of a po-terary Γ-semgroup T s sad to be zero of T provded aαbβc = bαaβc = bαcβa = a ad a t b, c T, α, β Γ. Theorem 4.7.Let T be a ordered terary Γ-semrg wth zero. () {}s a rght pure po-teraryγ-deal of T. () Uo of ay rght pure two-sded po-teraryγ-deals (respectvely, po-terary Γ-deal) of T s a rght pure two-sded po-teraryγ-deal (respectvely, po-terary Γ-deals) of T. () Fte tersecto of rght pure two-sded deals (respectvely, deal) of T sa rght pure two-sded poteraryγ-deal (respectvely, po-terary Γ-deals) of T. Proof:() Ths s obvous. () Let A, be rght pure two-sded po-terary Γ-deals of T. We have A s a two-sded po-terary Γ-deal of T. Let x A. The x A j for some j. Sce A j s rght pure two-sded po-terary Γ-deal, there exst y, z A j, α, β Γ such that x [xαyβz]. Sce y, z A j A, we have A s rght pure. () Let A 1, A 2, A be rght pure two-sded po-terary Γ-deals of T. DO: 1.979/ Page

6 O Pure PO-Terary Γ-deals Ordered Terary Γ-Semrgs The A s a two-sded po-terary Γ-deal of T. 1 Let x A. For k {1,2,.., }, there exst y k, z k A k,α, β Γsuch that x [xαy k βz k ]. 1 We have x [[xαy βz ] [y 2 αz 2 βy 1 ]γz 1 ].Sce [[y αz βy -1 ] [y 2 γz 2 δy 1 ]εz 1 ] rght pure two-sded po-terary Γ-deal of T. DO: 1.979/ Page 1 A, we have Theorem 4.8.Let T be a ordered terary Γ-semrg wth zero ad A s a two-sded po-terary Γ-deal of T. The A cotas the largest rght pure two-sded po-terary Γ-deal of T, deoted by S(A). S(A) s called the pure part of A. Proof: Sce {}s a rght pure two-sded po-terary deal of T cotaed A, t follows thatthe uo of all rght pure two-sded po-terary Γ-deals of T cotaed A exsts, ad hecet s the largest rght pure twosded po-terary Γ-deal of T cotaed A. Smlarly, we have the followg. Theorem 4.9.Let T be a ordered terary Γ-semrg wth zero ad A s a po-terary Γ-deal of T. The A cotas the largest rght pure po-terary Γ-deal of T. Theorem 4.1.Let T be a ordered terary Γ-semrg wth zero. Let A, B ad A, be two-sded po-terary Γ-deals of T. () S(A B) = S(A) S(B). () S ( A ) S ( ) A. Proof:() Sce S(A) A ad S(B) B, we have S(A) S(B) A B. HeceS(A) S(B) S(A B). Sce S(A B) A B A, we get S(A B) S(A). Smlarly, S(A B) S(B). The S(A B) S(A) S(B), whece S(A B) = S(A) S(B). () Sce S(A ) A for all, we have S ( A ) A. The ( ) S A S ( ) A. Defto4.11: A rght pure two-sded po-terary Γ-deal A of a ordered terary Γ-semrg T s sad to be purely maxmal f for ay proper rght pure two-sded po-terary Γ-deal B of T, A B mples A = B. Defto 4.12: A proper rght pure two-sded po-terary Γ-deal A of a ordered terary Γ-semrg T s sad to be purely prme f for ay rght pure two-sded po-terary Γ-deals B 1,B 2 of T, B 1 B 2 A mples B 1 A or B 2 A. Theorem 4.13:Every purely maxmal two-sded po-terary Γ-deal of a ordered terary Γ-semrg T s purely prme. Proof: Let A be a purely maxmal two-sded po-terary Γ-deal of T. Let B ad C be rght pure two-sded poterary Γ-deals of T such that B C A ad B A. Sce B A s a rght pure two-sded po-terary Γ-deal such that A B A, so T = B A. We have C = C T = C (B A) = (C B) (C A) A. The A s purely prme. Theorem 4.14:The pure part of ay maxmal two-sded po-terary Γ-deal of a ordered terary Γsemrg T wth zero s purely prme. Proof: Let A be a maxmal two-sde po-terary Γ-deal of T. To show thats(a) s purely prme, let B,C be rght pure two-sde po-terary Γ-deals of T such that B C S(A). f B A, the B S(A). Suppose that B A. We have B A s a two-sde po-terary Γ-deal of T. By maxmalty of A, T = B A, ad hece C A. Thus C S(A). Theorem 4.15: Let T be a ordered terary Γ-semrg ad Aa rght pure two-sded po-terary Γ-deal of T. f x T \A, the there exsts a purely prme two-sded po-terary Γ-deal B of T such that A B ad x B. Proof: Let P = {B B s a rght pure two-sded po-terary Γ-deal of T, A B ad x B}. Sce A P, P. Uder the usual cluso, P s a partally ordered set. Let B k,k be a totally ordered subset of P. By Theorem 4.7, B s a rght pure two-sded po-terary Γ-deal of T. Sce A k B ad x k B, we have k B P. k k By Zor s Lemma, P has a maxmal elemet, say M. The M s a rght pure two-sded po-terary Γ- deal, A M ad x M. We shall show that M s purely prme. Let A 1 ad A 2 be rght pure two-sded po-terary Γ-deals of T such that A 1 M ad A 2 M. Sce A 1, A 2 ad M are rght pure two-sded po-terary Γ-deals of T, we obta A 1 M ad A 2 M are rght pure two-sded po-terary Γ-deals of T such that M A 1 M ad k k 1 k A s a

7 O Pure PO-Terary Γ-deals Ordered Terary Γ-Semrgs M A 2 M. Thus x A M (k = 1,2). Sce x M, x A 1 A 2.Hece A 1 A 2 M. Ths shows that M s purely prme. Theorem 4.16:Ay proper rght pure two-sded po-terary Γ-deal A of a ordered terary Γ-semrg T s the tersecto of all the purely prme two-sded po-terary Γ-deals of T cotag A. Proof:By Theorem 4.15, there exsts purely prme po-terary Γ-deals cotag A. Let {B : }be the set of all purely prme two-sded po-terary Γ-deals of T cotag A. We have A B. To show that Bk k A B j ad x B j. Hece A. Let x A. By Theorem 4.15, there exsts purely prme po-terary Γ-deal B j such that x. B V. Weakly pure deals ordered terary PO-sem-rgs ths secto, we troduce the cocept of weakly pure po-terary Γ-deal ordered terary Γ- semrg. Defto 5.1.Let T be a ordered terary Γ-semrg. A two-sded po-terary Γ-deal A of T s called left (respectvely, rght) weakly pure f A B = ([AΓAΓB]](respectvely,A B = ([BΓAΓA]]) for all two-sded poterary Γ-deals B of S. a ordered terary Γ-semrg, every left (rght) pure two-sded po-terary Γ-deals s left(rght) weakly pure. Theorem 5.2:Let Tbe a ordered terary semgroup wth zero. f A ad B are two-sdedpo-terary Γ- deals of T, the BΓA -1 = {t T x, y A, α, β Γ,[xαyβt] B} A -1 ΓB= {t T x, y A, α, β Γ,[tαxβy] B} are two-sded po-terary Γ-deals of T. Proof: We shall show that BΓA 1 s a two-sded po-terary Γ-deal of T. That A 1 ΓB s a twosded po-terary Γ- deal of T ca be proved smlarly. Clearly, BΓA 1.Let u, v T, α, β Γ ad t BΓA 1. To show that [uαvβt] BΓA 1, letx, y A. Sce [yγuαv] A for γ, δ Γ, we have [xεy[uδvβt]] = [xε[yγuδv]βt] B. Thus [uαvβt] BΓA 1. Let x BΓA 1 ad y T be such that y x. Let z, w A. Sce [zαwβy] [ zαwγx] ad [zαwγx] B, we have [zαwβy] B. Hece y BΓA 1. Therefore, BΓA 1 s a two-sded po-terary Γ-deal of T. Theorem 5.3:Let T be a ordered terary Γ-semrg ad Aa two-sded po-terary Γ-deal of T. The A s left (rght) weakly pure two-sded po-terary Γ-deal f ad oly f (BΓA 1 ) A = A B ((BΓA 1 ) A = A B) for all po-terary Γ-deals B of T. Proof:Suppose that A s left weakly pure two-sded po-terary Γ-deal. Let B be a po-terary Γ-deal of T. By Theorem 5.2, BΓA 1 s a two-sde po-terary Γ-deal of T, ad thus A BΓA 1 = ([AΓAΓ(BΓA 1 )]]. Sce [AΓAΓ(BΓA 1 )] [AΓTΓT] A, we have ([AΓAΓ(BΓA 1 ])] (A] = A. Let t ([AΓAΓ(BΓA 1 )]] be such that t [xαyβz] for some x,y A,α, β Γ,z BΓA 1. By Defto of BΓA 1, [xαyβz] B. Thus t B. Ths proves that A BΓA 1 A B. For the reverse cluso, let a A B. Sce [xαyβa] B for ay x,y A,α, β Γ, we have a BΓA 1. We get a BΓA 1 A, ad the A B BΓA 1 A. Coversely, assume that (BΓA 1 ) A = A B for all po-terary Γ-deal B of T. To showthat A s left weakly pure two-sded po-terary Γ-deal, let C be ay po-terary Γ-deal of T. To show that A C = ([AΓAΓC]]. By assumpto, A C = CΓA 1 A. Sce [AΓAΓC] [AΓTΓT] A, ([AΓAΓC]] A.Let t ([AΓAΓC]] such that t [xαyβz] for some x,y A,α, β Γ,z C ad let a,b A. Sce [a[bβxγy]δz] = aαbβ[xγyδz] C, we obta [xγyδz] CΓA 1, ad so t CΓA 1. The ([AΓAΓC]] CΓA 1.Ths proves that ([AΓAΓC]] A C. For the reverse cluso, we have C ([AΓAΓC]]ΓA 1 because c C,a,b A, α, β Γ mples [aαbβc] [AΓAΓC] ([AΓAΓC]]. The A C ([AΓAΓC]]ΓA 1 A = A ([AΓAΓC]] ([AΓAΓC]]. Theorem 5.4: Let T be a ordered terary Γ-semrg. The followg are equvalet. () Every two-sded po-terary Γ-deal s left weakly pure two-sded po-terary Γ-deal. () For every two-sded po-terary Γ-deal A of T, [AΓAΓA] = A..e. each two-sded po-terary Γ-deal s dempotet. () Every two-sded po-terary Γ-deal s rght weakly pure two-sded po-terary Γ-deal. Proof : () () Suppose that each two-sded po-terary Γ-deal of T s left weakly pure. Let A be the two sded po-terary Γ-deal of T, the for each two- sded po-terary Γ-deal B of T we have A B = AΓAΓB. partcular A = A A = AΓAΓA. Therefore each two-sded po-terary Γ-deal of T s dempotet. DO: 1.979/ Page

8 O Pure PO-Terary Γ-deals Ordered Terary Γ-Semrgs () () Suppose that each two-sded po-terary Γ-deal of T s dempotet. Let A be a two-sded po-terary Γ-deal of T, the for ay two-sded po-terary Γ-deal B of T we always have AΓAΓB = A B. O the other had, A B = (A B)Γ(A B)Γ(A B) AΓAΓB. Hece we have A B = AΓAΓB. Thus A s left weakly pure. () () Smlarly as () () () () Suppose that each two-sded po-terary Γ-deal of T s rght weakly pure two-sded po-terary Γ- deal. Let A be ay two-sded po-terary Γ-deal of T. The A s rght weakly pure. Therefore for each twosded po-terary Γ-deal B of T, we have A B = BΓAΓA. partcular A A = AΓAΓA. Thus each twosded po-terary Γ-deal of T s dempotet. 6. Pure spectrum of a ordered terary Γ-semrg Notato 6.1 :Let T be a ordered terary Γ-semrg wth zero such that [TΓTΓT] = T.The set of all rght pure poterary Γ-deals of T ad the set of all proper purely prme po-terary Γ-deals of T wll be deoted by P(T) ad P (T), respectvely. For A P(T), let A = {J P (T) A J} ad τ(t) = { A A P(T)}. Theorem 6.2: τ(t) forms a topology o P (T). Proof: Sce {} s a rght pure po-terary Γ-deal of T ad {} =, we have τ(t). Sce T s a rght pure poterary Γ-deal of T such that T = P (T), we get P (T) τ(t). Let { A { J P ( T ) : A J } = A1 A A1 A 2 2 α Λ} τ(t). We have = { J P ( T ) : A A J for some α Λ} = A, therefore let J. Whece τ (T). Let A Sce J s purely prme, A 1 J or A 2 J. A cotradcto. The J the reverse cluso, let J topology o P (T).. Cosequetly,, τ(t). We shall show that. We have J P (T), A 1 J ad A 2 J. Suppose that A 1 A 2 J., hece. Sce A 1 A 2 J, A 1 J ad A 2 J. Ths mples that J A1 A A1 A 2 2 V., whch mples Cocluso. For, thus τ(t). Therefore τ(t) forms a ths paper maly we start the study of pure po-terary Γ-deals, weakly pure po-terary Γ-deals ad purely prme po-terary Γ-deals po-terary Γ-semrgs. We characterze po-terary Γ-semrgs by the propertes of pure ad weakly pure po-terary Γ-deals. Ackowledgmets The frst ad thrd authors express ther warmest thaks to the Uversty Grats Commsso(UGC), da, for dog ths research uder Faculty Developmet Programme. The authors would lke to thak the experts who have cotrbuted towards preparato ad developmet of the paper ad the referees, Chef Edtor for the valuable suggestos ad correctos for mprovemet of ths paper. Refereces [1]. Ahsa. J, Takahash. M, Pure spectrum of a mood wth zero, obe J. Mathematcs 6 (1989) [2]. Bashr. S, Shabr. M, Pure deals terary semgroups, Quasgroups ad Related Systems 17 (29) [3]. Chagphas. T, B-deals ordered terary semgroups, Appled Mathematcal Sceces 6 (212) [4]. Chagphas. T, B-deals terary semgroups, Appled Mathematcal Sceces 6 (212) [5]. Chram. R, Saelee. S, Fuzzy deals ad fuzzy flters of ordered terary semgroups, Joural of Mathematcs Research 2 (21) [6]. Dxt. V. N, Dewa. S, A ote o quas ad b-deals terary semgroups, terat. J. Math. Math. Sc. 18 (3) (1995) [7]. ampa., O ordered deal extesos of ordered terary semgroups, Lobachevsk J. Math. 31 (1) (21) [8]. Los. J, O the extedg of models, Fudam. Math. 42 (1995) [9]. Satago. M. L, Sr Bala. S, Terary semgroups, Semgroup Forum 81 (21) [1]. Soso F. M, deal theory terary semgroups, Math. Japa 1 (1965) [11]. Syam julus rajedra. V, Dr. Madhusudhaa rao. D, Saja lavaya. M-A Study o Completely Regular po-terary Γ-Semrg, Asa Academc Research Joural of Multdscplary, Volume 2, ssue 4, September 215, PP: DO: 1.979/ Page

9 O Pure PO-Terary Γ-deals Ordered Terary Γ-Semrgs Authors s Bref Bography: 2 Dr. D. MadhusudhaaRao: He completed hs M.Sc. from Osmaa Uversty, Hyderabad, Telagaa, da. M. Phl. from M.. Uversty, Madura, Taml Nadu, da. Ph. D. from Acharya Nagarjua Uversty, Adhra Pradesh, da. He joed as Lecturer Mathematcs, the departmet of Mathematcs, VSR & NVR College, Teal, A. P. da the year 1997, after that he promoted as Head, Departmet of Mathematcs, VSR & NVR College, Teal. He helped more tha 5 Ph. D s. At preset he s gudg 7 Ph. D. Scholars ad 3 M. Phl., Scholars the departmet of Mathematcs, AcharyaNagarjua Uversty, Nagarjua Nagar, Gutur, A. P. A major part of hs research work has bee devoted to the use of semgroups, Gamma semgroups, duo gamma semgroups, partally ordered gamma semgroups ad terary semgroups, Gamma semrgs ad terary semrgs, Near rgs ect. He actg as peer revew member to (1) Brtsh Joural of Mathematcs & Computer Scece, (2) teratoal Joural of Mathematcs ad Computer Applcatos Research, (3) Joural of Advaces Mathematcs ad Edtoral Board Member of (4) teratoal Joural of New Techology ad Research. He s lfe member of (1) Adhra Pradesh Socety for Mathematcal Sceces, (2) Heath Awareess Research sttuto Techology Assocato, (3) Asa Coucl of Scece Edtors, Membershp No: , (4) Coucl for ovatve Research for Joural of Advaces Mathematcs. He publshed more tha 62 research papers dfferet teratoal Jourals to hs credt the last four academc years. 1 Mr. V. Syam Julus Rajedra: He completed hs M.Sc. from Madras Chrsta College, uder the jursdcto of Uversty of Madras, Chea,Tamladu. After that he dd hs M.Phl. from M.. Uversty, Madura, Tamladu, da. He joed as lecturer Mathematcs, the departmet of Mathematcs, A. C. College, Gutur, Adhra Pradesh, da the year At preset he s pursug Ph.D. uder gudace of Dr. D. Mathusudhaa Rao, Head, Departmet of Mathematcs, VSR & NVR College, Teal, Gutur(Dt), A.P. da Acharya Nagarjua Uversty. Hs area of terests are terary semrgs, ordered terary semrgs, semrgs ad topology. He publshed more tha 2 research papers dfferet teratoal Jourals to hs credt. Presetly he s workg o Partally Ordered Terary Γ-Semrgs. 3 Mrs. M. Saja Lavaya : She completed her M.Sc. from Hdu College, Gutur, uder the jursdcto of Acharya Nagarjua Uversty, Gutur, Adhra Pradesh, da. She joed as lecturer Mathematcs, the departmet of Mathematcs, A. C. College, Gutur, Adhra Pradesh, da the year At preset she pursug Ph.D. uder gudace of Dr. D. Mathusudhaa Rao, Head, Departmet of Mathematcs, VSR & NVR College, Teal, Gutur(Dt), A.P. da Acharya Nagarjua Uversty. Her area of terests are terary semrgs, ordered terary semrgs, semrgs ad topology. He publshed more tha 2 research papers dfferet teratoal Jourals to hs credt. Presetly she s workg o Terary Γ-Semrgs. DO: 1.979/ Page