In this paper we will define topological BP-algebras and find some properties of these algebras. Also we show how to connect certain topologies with ideals of BP-algebras. Furthermore, we derived certain standard properties using topological concepts and we give a characterization of topological BP-algebras in terms of neighborhoods

N, Kandaraj

V, Abisha

English

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Ratio Mathematica                                                                                               Topology on BP-Algebras   V. Abisha* N. Kandaraj†    Abstract  In this paper we will define topological BP-algebras and find some properties of these algebras. Also we show how to connect certain topologies with ideals of BP-algebras. Furthermore, we derived certain standard properties using topological concepts and we give a characterization of topological BP-algebras in terms of neighborhoods.   Keywords: BP-algebras, Topology, Topological BP-algebras Hausdroff.  2010 AMS subject classification: 54A05, 46H05, 22A26, 06F35, 03G25.                                                         *Research Scholar, PG and Researc h Department of Mathematics, Saiva Bhanu Kshatriya College,     Aruppukottai-626 101,Affiliated to Madurai Kamaraj University, Madurai-625 021, Tamil Nadu, India. abisha1997@gmail.com. †Associate Professor,PG and Research Department of Mathematics, Saiva Bhanu Kshatriya College, Aruppukottai-626 101, Affiliated to Madurai Kamaraj University, Madurai-625 021,Tamil Nadu, India. n.kandarajsbkc1998@gmail.com. †Received       on October 2023. Accepted on November 30, 2023. Published on December 30, 2023.DOI:10.23755/rm.v49i0.1433. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper ispublished under the CC-BY licence agreement.Volume 49, 2023116V. Abisha and N. Kandaraj   1. Introduction  The two classes of abstract algebras namely BCK-algebras and BCI-algebras are introduced by Imai Y and Iseki K [5].It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. Hu Q.P and Li X [4] introduced a wide class of abstract algebras: BCH-algebras. Also it is known that the class of BCI-algebras is a proper subclass of the class of BCH-algebras. Ahn S.S and Han J.S [1] introduced the concepts of BP-algebras and they discussed some relations with several algebras. Moreover they have discussed quadratic BP-algebras and shown that the quadratic BP-algebras is equivalent to several quadratic algebras. Alo R and Deeba E [3] attempted to study the topological concepts of the BCK-structure. Theories of topological groups, topological rings and topological modules are well known and still investigated by many Mathematicians. Topological universal algebraic structures have been studied by some authors.  Motivated by this, in this paper,  we study the connection between BP-Algebras and topology. We will define topological BP-algebras (TBP-algebras) and find some properties of TBP-algebras. Also we show how to connect certain topologies with ideals of BP-algebras. Furthermore, we derive certain standard properties using topological concepts and we give a characterization of topological BP-algebras in terms of neighborhoods.  Finally, we show that a TBP-algebra B is Hausdroff if and only if {0} is closed in B.  2. Preliminaries Definition 2.1 [1]. Let X be a set with a binary operation  and a constant 0.  Then (X, , 0) is called a BP-algebra if it satisfies the following axioms. 1. x  x = 0 2. x  (x  y) = y 3. (x  z)  (y  z) = x  y for any x, y, z . Example 2.2 [1]. Let X = {0, a, b, c} be a set with following table   Table 1 Then (X, , 0) is a BP-algebra.  117Topology on BP-Algebras   Theorem 2.3 [1]. If (X, , 0) is a BP-algebra, then the following results hold: for any x, y  1. x  0 = x. 2. x  y = 0 implies y  x = 0. 3. 0  x = 0  y implies x = y. 4. 0  x = y implies 0  y = x. 5. 0  x = x implies x  y = y  x    Proposition 2.4 [1]. If (X, , 0) is a BP-algebra with (x  y)  z = x  (z  y)  for any x, y, z , then 0  x = x for any x . Theorem 2.5 [1]. If (X, , 0) is a BP-algebra with x  y = 0 and y  x = 0, then  x = y.  Definition 2.6 [10]. Let S be a non-empty subset of a BP-algebra X, then S is  called BP-subalgebra of X if x  y  S for all x, y  S. Definition 2.7 [10]. Let (X, , 0) be a BP-algebra and I be a non-empty subset  of X. Then I is called an ideal of X, if it satisfies the following conditions. 1. 0  I. 2. x  y  I and y  I  x  I. Definition 2.8 [9]. Let X be a non-empty set. A Collection  of subsets of X is  called a Topology if it satisfies the following conditions. 1.  and X is in . 2. Arbitrary union of the members of any finite sub collection of   is in . 3. Intersection of the members of any finite sub collection of   is in . The tuple (X, ) is called a topological space. Definition 2.9 [9]. Let (X, ) is a Topological space, we say that a subset U of  X is an open set of X if U belongs to the collection  . Definition 2.10 [9]. The complement of an open set is called a closed set.  Definition 2.11 [9]. A subset A of X is said to be neighborhood of  x  A if  there exist an open set U such that x  U  A. Definition 2.12[9]. Let (X, ) be a topological space. (X, ) is called a Ti-space  i  {0, 1, 2} if it satisfies the following axioms. 118V. Abisha and N. Kandaraj   T0: For every x, y  X and x  y, there exist at least one open neighbourhood U  that contains one point of the pair (either x or y) excluding the other point.  T1: For each x, y  X and x  y, there exist open neighbourhoods U1 and U2  such that U1 contains x but not y and U2 contains y but not x.  T2 (Hausdroff):  For each x, y  X and x  y, both have disjoint open  neighbourhoods U, V such that x  U and y  V Definition 2.13 [9]. Let (X, ) be a topological space and A  X.  Define A = { U  A / U }. Then (A, A) is a topological space and the  topology A is called the subspace topology on A. Definition 2.14 [9]. Let (X, ) be a topological space and A  X. Let a  A is  called an interior point of A, if there exist an open set U of ‘a’ such that U  A. Remark 2.15 ([9]). 1. Restriction function of a continuous function is a continuous.   2. Every T2-space is a T1-space. Furthermore, every T1-space is a T0-space.   3. Topological BP-algebras  Definition 3.1: Let B be a BP-algebra and X, Y be any two non-empty subsets  of B. We define a subset X  of B as follows. X  = {a  b / a , Y}. Example 3.2: Let B= ({0, p, q, r}, ,  0 ) be a BP-algebra with the binary  operation  defined by the Cayley table.   Table 2 Let X= {0. p} and Y= {0, q} be two subsets of X,    X  = {0  0, 0  q,  p  0,  p  q}  = {0, p, p, r}  = {0, p, r} B. 119Topology on BP-Algebras   Definition 3.3: Let B be a BP-algebra. Let T be the collection of subsets of B, T  is said to be a topology on B if the following conditions are satisfied. 1. B , T   2. Arbitrary union of members of T is in T. 3. Finite intersection of members of T is in T. Definition 3.4: Let (B, , 0) be a BP-algebra and T be a topology on B.  Then  (B, , T) is called a topological BP-algebra if the operation  is continuous. Equivalently, for a, b B if there exist an open set U of a b then there exist two  open sets X and Y of a and b respectively such that X  U. Example 3.5: Let B= ({0, p, q, r}, ,  0 ) be a BP-algebra with the binary  operation  defined by as on example 3.2. Define a topology T= {B, , {0}, {0, p, q}} Let 0,  p  B and U= {0, p, q} Now a  b = p 0 = p U. Then there exist an open set X= {0, p, q}, Y= {0} if p and 0 respectively such  that X  = {0  0, p  0, q  0}      = {0, p, q} U. Therefore every open set U for a  b then there exist an open sets X and Y of a  and b respectively such that X  U for all a, b  B. Hence (B, , T) is a Topological BP-algebra and it is denoted by TBP-algebra. Example 3.6: Let B= ({0, p, q, r}, , 0) be a BP-algebra with the binary  operation  defined in Example 3.2. Define a topology T= {B, , {q}, {r}} Let 0, q  B, q 0 = q = {q} = U. Let X={q} and Y={B}, then X  = { q  0, q  p, q  q, q  r} = {q, r, 0, p} U. Hence (B, , T) is not a Topological BP-algebra. Remark 3.7: 1. The member of T is known as open sets in B. 2. An open set of an element a  B is a member of T containing B.  3. The complement of C  T that is B\C is called a closed set in B. Example 3.8: Let B= ({0, p, q, r}, ,  0) be a BP-algebra with the binary  operation  defined in Example 3.5. Define a topology T= {B, , {0, p, q }, {r}}. 120V. Abisha and N. Kandaraj   Here {0, p, q} is open in B. Now {r} is closed in B.  An open set of p  B. is {0, p, q}. Theorem 3.9: Let (B, , T) be a TBP-algebra. If {0} is an open set in TBP-algebra, then T is a discrete topology in TBP-algebra. Proof. Let (B, , T) be a TBP-algebra. For every a  B. a a = 0.  Since {0} is an open in TBP-algebra, there exist two neighborhoods X and Y of  a such that X {0}  X {0}               ---------------------(1) Let U = X  Y. Clearly U U = {0} by (1) Since a  b = 0 and b  a = 0  a =b, which implies U = {a}. Therefore T is discrete.  4. Properties of topological BP-algebras Definition 4.1: Let (B, , T) be a TBP-algebra. Then B is said to be T0-space  BP-algebra if it satisfies the following axiom: For every a, b  B and a  b, there exist at least one open neighborhood U that contains one point of the pair (either a or b) excluding the other point. Definition 4.2: Let (B, , T) be a TBP-algebra. Then B is said to be T1-space  BP-algebra if it satisfies the axiom: For each a, b  B and a  b, there exist an open neighborhoods X1 and  X2 such that X1 contains a but not b and X2 contains b but not a.  Definition 4.3: Let (B, , T) be a TBP-algebra. Then B is said to be T2-space  (Hausdroff) BP-algebra if it satisfies the axiom:For each a, b  B and a  b, both have disjoint open neighborhoods X, Y such that  a  X and b  Y. Example 4.4: Consider the TBP-algebra (B = {0, p, q}, , T) with the following  Cayley table.          Table 3  Consider the  topology T = { , B, {0, p }, {q}, {q, 0}, {0}}. Then (B, , T) is called a T0-space BP-algebra. 121Topology on BP-Algebras   Example 4.5: From the TBP-algebra given in Example 3.13. Consider the topology Ton B is given by T={ , B, {0, p }, {q}, {q, 0}, {0}, {p},  {q, p}}. Here T satisfies the T1 and T2 axioms. Thus (B, , T) is both T1-space BP-algebra and T2-space BP-algebra. Theorem 4.6: Let B be a TBP-algebra and {0} is closed set on B if and only if  B is Hausdroff. Proof. Let (B, , T) be a TBP-algebra. Let a, b  B. Assume that a and b are different in B, then a  b 0 or b  a 0 We can assume that a  b 0, then there exist some open sets X and Y of a and b  respectively such that X Y B \ {0}, since {0} is closed and B is a TBP-algebra. Suppose, let a  X  Y. Then a  X and a  Y implies a  a = 0  X  Y. This contradicts to the fact that X Y B \ {0}  X  Y= . Therefore B is Hausdroff. Conversely, Assume that B is Hausdroff Claim: {0} is closed in TBP-algebra We want to prove B \ {0}, there exist two neighborhoods X and Y of a and 0  respectively such that X  Y=  . Since B is Hausdroff, then X  B \ {0}  B \ {0} is an open so that {0} is closed in TBP-algebra. Theorem 4.7: Let B be a TBP-algebra and {0} be closed, then 0 = {0}  where 0 is the neighborhood system of 0. Proof. Let B be a TBP-algebra Since {0} is closed. By the above theorem 3.15, B is Hausdroff. Given an element a  B \ {0}, 0 has a neighborhood X such that a  X and  a 0.Hence 0 = {0}. Definition 4.8: Let (B, , T) be a TBP-algebra and I be a non-empty subset of  B. Then I is called an ideal of B if it satisfies the following conditions. 1. 0  I. 2. If a  b  I and b  I a  I. Example 4.9: Let (B = {0, p, q, r}, , T ) be a TBP-algebra with the binary operation  defined by the example 3.2. I = {0} is an ideal. Theorem 4.10: Let B be a TBP-algebra. If D is an open set in B which is also a  BP-ideal of B, then it is closed set in B. 122V. Abisha and N. Kandaraj   Proof. Let D be a BP-ideal which is an open set in B and let a  B \ D. Then  there exist a neighborhood X of a such that X X D. Since a a = 0  D and D is open. Claim: X  B \ D.  If X  B \ D, then there exists b  X  D. Note that c  b  X X D foe all c  X Since b  D and D is a BP-ideal, it follows that c  D. Which show that X D, a contradiction.Hence D is closed. Definition 4.11: Let (B, , T) be a TBP-algebra. Let A  B, x  A is called an  interior point of A, if there exist a neighborhood X of x contained in A. Example 4.12: Let (B= {0, p, q}, , T) be a TBP-algebra with the binary  operation  defined by the example 4.4. Define a topology T = {B, , {p, 0 }, {q} }. Let A = {p, q}. Here, r is an interior point of A. Theorem 4.13: Let A be a BP-ideal of a TBP-algebra B. If 0 is an interior point  of A, then A is an open. Proof. Let a  A. Since a  a = 0  A and 0 is an interior point of A, there exist a neighborhood X  of 0 which is contained in A. Then there exist neighborhoods G and H of a such that G  H  X  A On the other hand for every b  G, b  a  G  H  A. Since A is a BP-ideal and a  A, it follows that b  A so that a  G  A Hence A is open. Definition 4.14: Let B and C be a TBP-algebra. A map f: B  C is continuous.  If for every a  B and any open set Y of f (a) there exist an open set X  containing a such that f(X)  Y. Example 4.15: Consider a TBP-algebra (B = {0, p, q}, , T) as an example 3.13.  Define a topology T = {B, , {0, p }}. Define a function f: B  B by f (a) = a for all a  B. Clearly, f is continuous.   Notation 4.16: Let B be TBP-algebra and a, b  B, then {b  A / b a mod I} 123Topology on BP-Algebras    = {b  A / a  b, b a  I}. Theorem 4.17: Let B be a TBP-algebra. If I is an (closed) open ideal, then for  each a  B, then {b  B / b a mod I} is closed (open). Proof. Define a map Ga: B  B such that Ga(b) = a  b. Since B is TBP-algebra, then Ga is continuous. Define a map Ha: B  B such that Ha(b) = b  a. Since B is a TBP-algebra, then Ha is continuous. Ga-1(I)  Ha-1(I) = {b  B / / a  b, b a  I}.  Inverse image of an (closed) open set is an closed (open)  Ga-1(I)  Ha-1(I) is an closed (open) Therefore {b  B / b a mod I} is an closed (open). Theorem 4.18: Suppose that {0} is closed (open), then {a} is closed (open) for  all x  B. Proof. Define a map g: B B  B such that g (x, y) = x  y. Since B is a TBP-algebra ‘ ’ is continuous. which implies g is continuous. Define a map h: B B B  B B such that h(x, y, z) = (x  y, y  z) which implies h is continuous.              (Since g is continuous and Remark 2.15) Since {0} is closed, then {(0, 0)} B B is closed. Let x  B and let f: B  B B such that f(y) = h(x, y, x) = (x  y, y  x) Here f is the restriction of h to {x}     (since h to {x}  B  {x)) and hence f is continuous                    (since h is continuous and Remark 2.15) f-1(0, 0) = {y / x  y = 0 and y  x = 0} = {x}. Thus {x} is also closed. Similarly, for the other case where {0} is an open. Theorem 4.19: B is T1-space BP-algebra if and only if B is T2-space BP- algebra. Proof. Let B be a T1-space BP-algebra. Let a, b  A and a  b, there exist an open neighborhoods X1 and X2 such that  X1 contains a but not b and X2 contains b but not a.  {a} is closed, for all a  B  {0} is closed. By theorem 3.15, we have B is a TBP-algebra. Conversely, Let B be a T2-space BP-algebra. 124V. Abisha and N. Kandaraj   Let a, b  B and a  b both have disjoint open neighborhoods X and Y such that  a  X and b  Y and a  Y and b  X.  B is T1-space BP-algebra. Theorem 4.20: Let (B, , T) be a TBP-algebra. If S is a subset of B such that  foreach a, b  S, a  b  S, then (S, , TS) is a TBP-algebra, where TS is the  subspace topology from B. Proof. Let a, b  S  a  b  S Let U be an open set of a  b in B.  U  S is an open set for a  b in S. Since B is a TBP-algebra, there exist an open sets X and Y of a and b  respectively such that  X  Y  U   There exist an open sets X  S and Y  S of a and b in S such that  (X  S) (Y  S)  U  S   (S, , TS) is a TBP-algebra.  4  Conclusion and Discussion  In this paper, we have introduced the topological concepts on BP-algebra, which is induced by the binary operation * of the BP-algebra and provided some important properties. We have studied the connection of the topologies of the BP-algebra with the ideals. In addition, we have shown that if 0 is an interior point of a BP-ideal of a TBP-algebra B, then the BP-ideal of a TBP-algebra B is open.   Finally, studying a topological properties on other algebras (BF/BH/TM-algebras) is an interesting open problem.  Acknowledgements  The authors would like highly grateful to referees for their valuable suggestions   and comments which improved the paper.  References   [1] S.S. Ahn and J.S. Han: On BP-algebras, Hacettepe Journal of Mathematics and Statistics,Volume 42 (5), (2013), 551-557. [2] S.S. Ahn and S.H. Kwon: Topological properties in BCC-algebras, Commun.  Korean Math.Soc. 23 (2008), No.2, 169-178. [3] R.A. Alo and E.Y. Deeba: Topologies of BCK-algebras, Math. Japonica 31 (No.6) (1986), 841-853. 125Topology on BP-Algebras   [4] Q.P. Hu and X. Li: On BCH-algebras, Math. Seminar Notes11 (1983), 313-320. [5] Y. Imai and K. Iseki: On axiom system of propositional calculi XIV, Proc. Japan Academy 42 (1966), 19-22. [6] K. Iseki and S. Tanaka: An introduction to theory of BCK-algebras, Math. Japo. 23 (1978),1-26. [7] Y.B. Jun, X.L. Xin and D.S. Lee: On topological BCI-algebras, Information Sciences,Vol.116, Issues 2-4, 1999, 253-261 [8] D.S. Lee and D.N. Ryu: Notes on topological BCK-algebras, Scientiae Mathematicae, Vol.1No.2 (1998), 231-235 [9] James R. Munkers: Topology, Second Edition, Published by Person Education Inc., Published as Person Prentice Hall (2012). [10] J. Neggers and H.S. Kim: On d-algebra, Math. Solvaca, Vol.49 (1999), 19-26. 126

Topology on BP-Algebras

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Topology on BP-Algebras

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