Title Goes Here Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 478 https://internationalpubls.com Pseudo Symmetric Ideals in Near Subtraction Semigroups Jayarami Reddy.M1,2 a),Siva Prasad.P3, b)and Madhusudana Rao.D4C) 1Research Scholar, VFSTR Deemed to be University, Guntur,Andhra Pradesh, India 2Associate Professor, UCET, Guntur, AP, India 3Associate Professor, Dept of CSE, SOCI, VFSTR, Deemed to be University, Guntur, AP, India 4Professor of Mathematics, Government College for Women, Sambasiva Pet, Guntur, AP, India a) jayaramireddy1979@gmail.com, b) pusapatisivaprasad@gmail.com, c) dmrmaths@gmail.com Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: The current ongoing work, author introduce the theory of pseudo symmetric, PSideal, semiprime ideal, completely semi prime ideal, left pseudo commutative, right pseudo commutative, duo NSSGand PSnear subtraction SG. Keywords: Pseudo symmetric, completely semiprme, duo semigroup, near subtractionsemigroup. 1. Introduction In abstract algebra the theory of near subtraction semi groups is a rapid departing branch. The subtraction algebra was introduced by Abbott in1967 byusingthe concept of subtraction algebra, schein developed subtraction SG in the year 1992.In the year of 2007 Dheena.P was first introduced the concept of near subtraction SG. Basically, the NSSG is a fundamental notions of subtraction algebra. Jun et al., investigated about the theory of ideals in near subtraction algebra after that he characterized some elementary axioms. Later on some research scholars developed the basic properties of near subtraction SG. Later they establish a structure in NSSG and also examine different kinds of near subtraction algebra and getting the equivalent conditions for regularity. The fundamental concept of this work is to learn about PS ideals in NSSG and PSNSSG and some of the results were proved regarding NSSG. 2. Preliminaries Forfundamental conceptsmake reference to (1),(2),(3) and(4) 3. Maincontent Definition 3.1: Assume that A be an ideal of a NSSG X which is said to be a PS if x,yX, xyA =>xsyA for each sX. EXAMPLE3.2: Assume that X= {0,1,2,3,4,5} for which ‘-,’ and ‘.’ are defined by - 0 1 2 3 4 5 0 0 0 0 0 0 0 1 1 0 3 4 3 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 479 https://internationalpubls.com 2 2 5 0 2 5 4 3 3 0 3 0 3 3 4 4 0 0 4 0 4 5 5 5 0 5 5 0 . 0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 1 4 3 4 0 2 0 4 2 0 4 5 3 0 3 0 3 0 0 4 0 4 4 0 4 0 5 0 0 5 0 0 5 Then (A,-,. ) is a NSSG. Let I = {0,1,3,4} is a PS ideal of X. EXAMPLE3.3:LetX={p, q, r} for which ‘-‘ and ‘.’ are defined as follows: - p q r p p p p q q p q r r r p . p q r p p p p q p p p r p q r Then (X, -, . )is a near subtraction normal SG. The ideals of X are {p},{p, q} {p, q, r} which are PS. Theorem 3.4 :The family of non empty intersection of PS ideals of a NSSG X a PS is an ideals of X. Proof : Assume that { 𝐴𝛼}∝𝜖∆a family of PSI of X and letA=⋂ 𝐴𝛼𝛼𝜖∆ . By known theorem, A is an ideal of X. Letx,y,sX, xy∈ A, Γy∈ A ⇒xy∈⋂ 𝐴𝛼𝛼𝜖∆ ⇒ xy A for each α∈ Δ. xy A , A is a PSI of X, sX⇒xsy∈𝐴𝛼 ⇒ xsy ∈𝐴𝛼for all α ∈Δ ⇒xsy ∈⋂ 𝐴𝛼𝛼𝜖∆ ⇒xsy ∈ A. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 480 https://internationalpubls.com ∴A is a PSI of X. Theorem 3.5: Suppose that X be a NSSG and A be ideal in X. So that A is a PS iff for all a ∈ X.the setA r( a ) ={x ∈ X. : ax A } is an ideal of X. Proof :Assume that A will be a PSI in X and a∈ X Let x, y∈A r( a )⇒ ax, ay∈A. a(x – y) = ax – ay ∈A , since Ais an ideal of X and ax, ay∈A. In this manner x – y ∈ A r( a )and hence A r( a ) is a sub algebra of X. Presently x∈ A r( a )implies that ax∈ A. Presently ax∈ A, A is PS implies asx∈ A implies that sx∈A r( a ) Presently ax∈ A, s∈S, A is an ideal implies that axs∈ A implies that xs∈A r( a ) Consequently A r( a )is an ideal in X for each a∈ X. On opposite way guess that A r( a )is an ideal in X for every a∈ X . Letx ,y∈Xand xy∈A. xy∈A implies that y∈A r( a ) y ∈A r( a ), A r( a )is an ideal implies sy ∈A r( a )for every s ∈X implies that xsy∈A for each s∈X ∴ A is a PSI of X. Theorem 3.6 : Let X be a NSSG and A is an ideal in X. Then A is PS iff for all a∈X the set Al(a) = { x∈X : xa ∈ A} is an ideal of X. xy∈A implies x∈ Al(y), Al(y) is an ideal implies that xs∈ Al(y) for each s∈ X, xsy ∈A for each s ∈X. ThusA will be a PSI of X. Corollary3.7: Suppose X be a NSSG and A is an ideal in X. Then A is PS iff for each a∈ X the set A (a) ={ x∈ X : xa, ax ∈ A } is an ideal in X. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 481 https://internationalpubls.com Theorem3.8: Let A be any PSI in a NSSG X and x1, x2….,xn X. Then 1) x1x2∈ A implies x2x1∈ A. 2) x1x2….xn A if and only if …. A. Proof : 1) Suppose x1x2∈ A. Then (x2x1) 2 = x2 (x1x2)x1∈ A and hence x2x1∈ A. 2) Let x1x2….xnA. Since A is PSI of X. Then by corollary 3.6, x1∈Al(x2x3…..xn) ⇒⊆ Al(x2x3…..xn) ⇒x2x3…..xn⊆A. By property (1), we have x2x3…..xn⊆ A. Now x2∈ Al(x3x4…..xn)⇒⊆Al(x3x4…..xn). Therefore x3x4…..xn⊆ A ⇒x3x4…..xn⊆ A. Continuing this process we have x1x2….xn A⇒…..⊆ A. Conversely suppose that…. A Then x1x2….xn…. A. Thereforex1x2….xnA Corollary3.9: If A is a PSIin a NSSGX, then for eachnNanA⇒nA. Proof:By the abovecorollary 3.8 substitute a1= a2= a3….= an= a Corollary3.10 : Let A be a PSIin a NSSGX. If anA, nNthenn, n X. Theorem 3.11: Each CSPI A in aNSSGX is a PSIof X. Proof:Assume that A be a CSPI ofthe NSSG X. Let x, y X and xyA⇒(yx)2=(yx)(yx)=y(xy)x A. (yx)2A, A is CSPI implies yxA. If s X, (xsy)2= (xsy)(xsy) = xs(yx)syA. (xsy)2A, A is CSP implies that xsyA. Thus A will be a PSI of X. Note3.12 :The reverse proof of the theorem 3.11which is absurdity i.e., a PSIofaNSSG which is not to beCSP. Example 3.13: ::Let X = {p, q, r}. Define a binary operations ‘-‘ and ‘ .’ on Xas follows - p q r r p p p q q p q r r r p . p q r p p p p Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 482 https://internationalpubls.com q p p p r p q r Clearly (X, -, .) is a NSSG and{p}, {p, q}, X are the ideal of X. Here pp{p}ppp, pqp, prp{p} pq{p}ppq, pqq, prq{p} pr{p}ppr, pqr, prr{p} qp{p}qpp, qqp, qrp{p} qr{p}qpr, qqr, qrr{p} rp{p}rpp, rqp, rrp{p}. Therefore{p}is a PSI in X. Here p2=p{p}but b{p}. Therefore{p}is not a CSPI. Corollary 3.14 : Each CPI of a NSSG X is a PSI of a NSSGX. Proof : Let A is a CPI of a NSSGX. By known theorem, A is a CSPIof X. By known theorem, AisPSIof X. Theorem3.15: If A is aPSIof a NSSGX then A2 = {x/xn∈ A for every n∈ N} is a CSPI ofX. Proof : Clearly A A2 and subsequently A2 is a non-empty subset ofX. Assume thatx, y ∈A2 and s∈ X. Presently x ∈ A2x n, ym∈ A for some m, n∈ N, xn, ym∈ A ⇒x, y ∈ A, Awill be a PSI of Ximplies (x - y ) ∈ A⇒ (x - y )1∈ Aand consequently (x - y ) ∈ A2. Hence A2 is a sub algebra of X. Presently x ∈A2implies xn∈A for each n∈ N. xn∈A, s∈ X, Awill be a PSI of X ⇒(xs)n∈A, (sx)n∈A implies that xs, sx∈ A2. Accordingly A2is an ideal of X. Assume that x ∈ Xand x2∈A2. Presently x2∈A2 implies that(x2)n∈ Afor each n∈ N x2n∈A implies that x ∈A2. So A2 is a CSPI of X. Let Q to be any CSPI containing A. Let x ∈A2. Then, at that point, xn∈A for each n∈ N. by corrolary 3.7, xn∈A implies thatnAQ . Since Q is CSP nQ implies that x Q . In this way A2 is minimal CSPI of X containing A. Theorem 3..16 : If A is a PSIof a NSSG X then A2 = A4. Proof :By known theorem, A4A2. SupposexA2 then xnA for some nN. Since A is PS, xnAnAx A4. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 483 https://internationalpubls.com ∴ A2A4 and hence A2 = A4. Theorem 3.17 : If A is a PSIof a NSSG X then A4 = {x / nA forsomenN} is the minimalsemiprimeideal of X containing A . Theorem 3.18 :Let A bean ideal of a NSSG X. Then A is completely primeiff A is primeand PS. Proof :Assume A will be a CPI. By known hypothesis, A is prime. By corollary 3.14, A will be a PSI of X. On the other hand assume that A is prime and PS. Let x, y∈ X and xy∈ A . xy∈ A , A will be a PSI of X ⇒xsy ∈ A for all s ∈ X ⊆ A or ⊆ A  x ∈ X or y ∈ A . Consequently A is completely prime. Theorem.3.19: Let A be an ideal of a NSSG X. Then A is CSPiffAis semiprime and PS. Proof :Assume that A is completely semiprime. By known hypothesis, A is semiprime and furthermore by known hypothesis 3.11, A is PS. On the other hand suppose that A is semiprime and PS. Let x ∈ X and x2∈A . x2∈A , A is PS⇒⊆A ⇒⊆A ⇒x ∈A . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 484 https://internationalpubls.com In this way A is CSP. 4. Pseudo Symmetric Near Subtraction Semigroup Definition 4.1: A NSSG X is said to be PSNSSG provided every ideal in X is a PSI. Example 4.2: In example 3.3, the NSSG X is a PSNSSG. Theorem 4.3: Each left duo NSSG X is a PSSSG. Theorem 4.4 :Each right duo NSSG X is a PSNSSG. Corollary 4.5: Each duo NSSG is a PSNSSG. Proof :By theorem 4.3, and 4.4, we conclude that every duoNSSGX is a PSNSSG. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 485 https://internationalpubls.com Theorem 4.6 :Each left pseudo commutative NSSG is a PSNSSG. Proof : Let X be a left pseudo commutativeNSSG and A be any ideal of X. Let x,y∈ X, xy∈A and s ∈ X. Now xsy = sxy= s(xy) ∈A. Therefore xsy∈A for all s ∈ X. ThereforeAis aPSI. Therefore X is a PSNSSG. Theorem 4.7: Each right pseudo commutative NSSG is PSNSSG. Proof : Let X be a right pseudo commutativeNSSG and A be any ideal of X. Letx, y ∈ X, xy ∈ A and s ∈ X. Now xsy = xys = (xy)s ∈ A. Thereforexsy ∈ A for all s ∈ X. ∴A is a PSI . Hence X is a PSNSSG. Corollary 4.8: Each quasi commutative NSSG is a PSNSSG. Proof :Let X be a quasi commutative SG. By known theorem, X is a normal NSSG. By known theorem, X is a duo NSSG. By corollary 4.5, X is a PS SG. Corollary 4.9:Each generalized commutativeNSSG is a PSNSSG. Proof :Suppose thatS be a commutative NSSG. By known theorem, X is a left duo NSSG. By known theorem 4.3, X is a PSNSSG. Corollary 4.10: EachNSSG is a PSNSSG. Proof :Suppose that X be a normal SG. By known theorem, X is a duo SG. By corollary 4.5, X is a PSNSSG. Theorem 4.11: Each idempotent NSSG is a PSNSSG Proof :Suppose that X be an idempotent NSSG and A be any ideal of X. Let x, y ∈ X, xy ∈ A. Now xy ∈ A⇒(yx)= ( yx)2 = ( yx)(yx) = y(xy)x∈ A. Thereforeyx∈ A. If s ∈ X, then xsy = ( xsy)2 = ( xsy)( xsy) = xs(yx)sy ∈ A. Therefore xsy ∈ A for every s ∈ X. Hence A is a PSI. Therefore X is a PSSG. Theorem 4.12: If X is a NSSG in which every element is a midunit then X is a PS NSSG. Proof :Suppose that X be a NSSG in which every element is a midunit and A be any ideal ofX. If s ∈ X,then s is a midunit and hence xsy= xy∈A. Hence Ais aPSI. ∴ X isa PSSG. Remarks:SG-Semigroup,PS-Pseudo symmetric,PSI-Pseudo symmetric ideal NSSG-Near subtraction semigroup andPSNSSG -Pseudo Symmetric Near subtraction semigroup,CSP- completely semiprime,CPI-completely prime ideal and CSPI- completely semiprime ideal. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 486 https://internationalpubls.com REFERENCES [1] M.Jayarami Reddy,P.Siva Prasad,D.Madhusudana Rao.,Pseudo integral NSSG,Natural Volatiles &Essential Oils- 2021;Volume-8(Issue no-5):PP:5422-542. [2] M. 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