A Structure of KK-Algebras and its Properties
|
|
- Mae Franklin
- 6 years ago
- Views:
Transcription
1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 21, A Structure of KK-Algebras and its Properties S. Asawasamrit Department of Mathematics, Faculty of Applied Science, King Mongkut s University of Technology North Bangkok, Bangkok 10800, Thailand Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road, Bangkok 10400, Thailand suphawata@kmutnb.ac.th A. Sudprasert University of the Thai Chamber of Commerce, Bangkok, Thailand aisuriya sud@utcc.ac.th Abstract In this paper, we define the notions of KK-algebras, quotient KKalgebras and investigate its properties. Moreover we show the relation between ideals and congruences on KK-algebras. Mathematics Subject Classification: 06B35, 03G10, 06B99 Keywords: KK-algebra, ideal, quotient KK-algebra, congruences 1 Introduction and Preliminaries By an algebra X =(X,, 0), we mean a non-empty set X together with a binary operation and a some distinguished 0. H. Yisheng [6] studied an algebraic structure called a BCI-algebra which is an algebra (X,, 0) with a binary operation such that for all x, y, z X, satisfies the following properties: (B-1) ((x y) (x z)) (z y) =0; (B-2) x x =0; (B-3) x y = 0 and y x = 0 imply x = y, for any x, y, z X. In this paper is introduction to the general theory of KK-algebra. We will first give the notion of KK-algebra, quotient KK-algebra and investigate elementary and fundamental properties, and we will show relation between ideals and congruences.
2 1036 S. Asawasamrit and A. Sudprasert 2 KK-algebras In this section, we do define some familiar concepts as KK-algebras, both for illustration and for review of the concept. First we give a few definitions and some notation. Definition 2.1. An algebra (X;, 0) with a binary operation and a nullary operation 0. Then X is called KK-algebra if it satisfies for all x, y, z X : (KK-1) (x y) ((y z) (x z))=0; (KK-2) 0 x = x; (KK-3) x y =0and y x =0if and only if x = y. First, give example of KK-algebra. Examples 2.2. Let be defined on an abelian group G by letting x y = x 1 y, where x, y in G, with e is unity element of G. Then (G;,e) is a KK-algebra. Examples 2.3. Let X = {0, 1} and let be defined by Then (G;, 0) is a KK-algebra. * Theorem 2.3. Let (X;, 0) be a KK-algebra if and only if it satisfies the following conditions: for all x, y, z X, (1) (x y) ((y z) (x z))=0; (2) x ((x y) y) =0; (3) x x =0; (4) x y = 0 and y x = 0 if and only if x = y. Proof. Assume that (X;, 0) is a KK-algebra. From definition of KKalgebra, (1) and (4) holds. Then we see that x ((x y) y) =(0 x) ((x y) (0 y)) = 0, and x x =0 (x x) =(0 0) ((0 x) (0 x)) = 0, so (2) and (3) holds. Conversely, we need to show KK-2. By (1), (2) and (3), we see that ((0 x) x) 0 = ((0 x) x) (0 ((0 x) x)) = ((0 x) x) ((x x) ((0 x) x)) = 0. And since 0 ((0 x) x) =0. From (4), it follows that (0 x) x = 0 and x (0 x) =x ((x x) x) =0. Therefore 0 x = x, proving our theorem. Definition 2.4. Define a binary relation on KK-algebra X by letting x y if and only if y x =0.
3 A structure of KK-algebras and its properties 1037 Proposition 2.5. If (X;, 0) is a KK-algebra, then (X; ) is a partially order set. Proposition 2.6. If (X;, 0) be a KK-algebra and x 0, then x =0, for any x X. Moreover, 0 is called a minimal element in X. Proof. Let x 0, then 0 x =0. By KK-2, 0 x = x, and thus x =0. It is easy to show that the following properties are true for a KK-algebra. Theorem 2.7. Let (X;, 0) be a KK-algebra if and only if it satisfies the following conditions: for all x, y, z X, (1) ((y z) (x z)) (x y); (2) ((x y) y) x; (3) x y if and only if y x =0. Proposition 2.8. Let x, y, z be any element in a KK-algebra X. Then (1) x y implies y z x z. (2) x y implies z x z y. Proposition 2.9. Let x, y, z be any element in a KK-algebra X. Then x (y z) =y (x z). Proof. Since theorem 2.7(2), (x z) z x, and by proposition 2.8(1), we get that x (y z) ((x z) z) (y z). Putting x = y and y = x z in theorem 2.7(1), it follows that ((x z) z) (y z) y (x z). By the transitivity of gives x (y z) y (x z). And we replacing x by y and y by x, we obtain y (x z) x (y z). By the anti-symmetry of, thus x (y z) =y (x z) and finishing the proof. Corollary Let x, y, z be any element in a KK-algebra X. Then (1) y z x if and only if x z y. (2) (z x) (z y) x y. (3) x y implies x z y z. Proposition Let x, y, z be any element in a KK-algebra X. Then (1) ((x y) y) y = x y. (2) (x y) 0=(x 0) (y 0). Proof. (1) From theorem 2.3(2) and theorem 2.7(1), (((x y) y) y) (x y) x ((x y) y) =0. Thus (((x y) y) y) (x y) =0. Since (x y) (((x y) y) y) =((x y) y) ((x y) y) =0. So, by KK-3, (x y) y = x y. (2) Since (x 0) (y 0) = (x 0) (y ((x y) (x y))) = (x 0) ((x y) (y (x y))) = (x 0) ((x y) (x (y y))) = (x y) ((x 0) (x 0)) = (x y) 0. The proof is complete. In this paper we will denote N for the set of all nonnegative integers, i.e.,
4 1038 S. Asawasamrit and A. Sudprasert 0, 1, 2,..., and N for the set of all natural numbers, i.e., 1, 2, 3,..., and we will also use the following notation in brevity: y 0 x = x, y n x = y (... (y (y x))), }{{} ntimes where x, y are any elements in a KK-algebra and n N. Proposition Let x, y be any element in a KK-algebra X. Then (1) ((y x) x) n x = y n x for any n N. (2) (x n 0) 0=(x 0) n 0 for any n N. Proof. Let X be a KK-algebra and x, y X and n, m N. (1) Proceed by induction on n and defined the statement P (n), ((y x) x) n x = y n x. We see that P (0) is true, ((y x) x) 0 x = x = y 0 x. Assume that P (k) is true for some arbitrary k 0, that is ((y x) x) k x = y k x. Since ((y x) x) k+1 x =((y x) x) (((y x) x) k x) =((y x) x) (y k x) = y k (((y x) x) x) =y k (y x) =y k+1 x. This show that P (k + 1) is true and by the principle of mathematical induction, P (n) is true for each n N. (2) Since (x n 0) 0=(x (x n 1 0)) 0=(x 0) ((x n 1 0) 0) = (x 0) ((x (x n 2 0)) 0) = (x 0) ((x 0) ((x n 2 0) 0)) = (x 0) 2 ((x n 2 0) 0) =... =(x 0) n 0 Given x X if it satisfies x 0=0, that is 0 x, the element x is called a positive element of X. By definition, the zero element 0 of X is positive. Proposition Let x be any element in a KK-algebra X. Then ((x 0) 0) x is a positive element of X for every x X. Proof. Since (((x 0) 0) x) 0 = (((x 0) 0) 0)) (x 0) = (x 0) (x 0) = 0. Therefore ((x 0) 0) x is a positive element of X. 3 Ideals Definition 3.1. A non-empty subset A of a KK-algebra X is called a closed of X on condition that x y A whenever x, y A. Definition 3.2. A non-empty subset A of a KK- algebra X is called an ideal of X if it satisfies the following conditions: (I-1) 0 A (I-2) for any x, y X, x y A and x A imply y A. Examples 3.3. Let X = {0, 1, 2, 3} and let be defined by the table
5 A structure of KK-algebras and its properties 1039 * Thus, it can be easily shown that X is a KK- algebra. I = {0, 1} and J = {0, 3} are closed ideals of X. And we see that Lemma 3.4. Let A be a closed of KK-algebra X. Then A is an ideal of X if and only if x A and z y A imply z (x y) A for all x, y, z X. Proof. Let A be an ideal of X and let x A whereas z y A. Suppose that z (x y) A. By proposition 2.9, we see that x (z y) A. Since A is an ideal of X and x A, z y A, a contradiction. So z (x y) A. Conversely, assume that if x A and z y A imply z (x y) A for all x, y, z X. Since A is a closed of X, then there is x A which 0 = x x A. That is, 0 A. Now, let x y A and x A. Assume that y A. We have that 0 y = y A. It follows that 0 (x y) A. Hence x y A, contradiction. Therefore A is an ideal of X. This completes the proof. Corollary 3.5. Let A be a closed of KK-algebra X. Then A is an ideal of X if and only if x A and y A imply x y A for all x, y X. Lemma 3.6. Let A be a closed of KK-algebra X. Then A is an ideal of X if and only if x (y z) A and x z A imply y A for all x, y, z X. Proof. Let A be an ideal of X and let x (y z) A, x z A. Suppose that y A. By proposition 2.9, we have y (x z) A. Since A is an ideal of X, thus x z A, contradiction, this shows that y A. Conversely, assume that x (y z) A and x z A imply y A for all x, y, z X. Since A is a closed of X, then there is y A which 0 = y y A. Then 0 A. Let y z A, y A and suppose that z A. By KK-2, 0 (y z) A and 0 z A. By assumption, so y A, a contradiction. This proves that A is an ideal of X. Corollary 3.7. Let A be a closed of KK-algebra X. Then A is an ideal of X if and only if x y A and y A imply x A for all x, y, z X. This Lemma gives some properties of ideal of KK-algebra. Lemma 3.8. If A is an ideal of KK-algebra X and B is an ideal of A, then B is an ideal of X. Proof. Since B is an ideal of A, then 0 B. Let x, y X such that x y B and x B. It follows that that x y A and x A. By assumption, A is an ideal of X, so y A and x B. From B is an ideal of A, so y B. Therefore, B is an ideal of X.
6 1040 S. Asawasamrit and A. Sudprasert Theorem 3.9. Let {J i : i N} be a family of ideals of a KK-algebra X where J n J n+1 for all n N. Then J n is an ideal of X. Proof. Let {J i : i N} be a family of ideals of X. It can be proved easily that J n X. Since J i is an ideal of X for all i, so 0 J n. Let x y J n and x J n. It follows that x y J j for some j N and x J k for some k N. Furthermore, let J j J k. Hence x y J k and x J k. By assumption, J k is an ideal of X, it follows that y J k. Therefore, y J n, proving that J n is an ideal of X. Theorem Let {J i : i N} be a family of closed ideals of a KK-algebra X where J n J n+1 for all n N. Then J n is a closed ideal of X. Proof. Let {J i : i N} be a family of closed ideals of X. By theorem 3.9, J n is an ideal of X. We will show that J n is a closed of X. Let x, y J n. It follows that x J j for some j N and y J k for some k N. WLOG, we assume that j k, we obtain J j J k. That is, x J k and x J k. Since J k is a closed of X, we get x y J k J n. This proves that is a closed ideal of X. J n Theorem Let {I j : j J} be a family of ideals of a KK-algebra X. Then I j is an ideal of X. Proof. Let {I j : j J} be a family of ideals of X. It is obvious that I j X. Since 0 I j for all j J, it follows that 0 I j. Let x y I j and x I j. We get that x y I j and x I j for all j J, then y I j for all j J. Because I j is an ideal of X. So y I j, proving our theorem. Theorem Let {I j : j J} be a family of closed ideals of a KK-algebra X. Then I j is a closed ideal of X. Proof. Let {I j : j J} be a family of closed ideals of X. By theorem 3.11, I j is an ideal of X. We will show that I j is a closed of X. Let x, y I j. It follows that x, y I j for all j J. Since I j is a closed of X and x y I j for all j J, then x y I j. This show that I j is a closed ideal of X.
7 A structure of KK-algebras and its properties Quotient KK-Algebras In this section, we describe congruence on KK-algebras. Definition 4.1. Let I be an ideal of a KK-algebra X. Define a relation on X by x y iff x y I and y x I. Theorem 4.2. If I is an ideal of KK-algebra X, then the relation is an equivalence relation on X. Proof. Let I be an ideal of X and x, y, z X. By Theorem 2.3, x x =0 and assumption, x x I. That is, x x. Hence is reflexive. Next, suppose that x y. It follows that x y I and y x I. Then y x, so is symmetric. Finally, let x y and y z. Then x y, y x, y z, z y I and (y x) ((z y) (z x)) = 0 I. It follows that (z y) (z x) I, and since z y I, so z x I. Similarly, x z I. Thus is transitive. Therefore is an equivalence relation. Lemma 4.3. Let I be an ideal of KK-algebra X. For any x, y, u, v X, if u v and x y, then u x v y. Proof. Assume that u v and x y, for any x, y, u, v X, then u v, v u, x y, y x I and by K-1, we see that (u v) ((v x) (u x)) = 0 and (v u) ((u x) (v x)) = 0. From assumption and I is an ideal of X, these imply that (v x) (u x) I and (u x) (v x) I. This shows that v x u x. On the other hand, by corollary 2.10, we have that (y x) ((v y) (v x)) = 0 and (x y) ((v x) (v y)) = 0. From assumption and I is an ideal of X, these imply that (v y) (v x) I and (v x) (v y) I. Thus v x v y. Since is symmetric and transitive, so u x v y. Corollary 4.4. If I is an ideal of KK-algebra X, then the relation is a congruence relation on X. Proof. By theorem 4.2 and lemma 4.3. Definition 4.5. Let I be an ideal of a KK-algebra X. Given x X, the equivalence class [x] I of x is defined as the set of all element of X that are equivalent to x, that is, [x] I = {y X : x y}. We define the set X/I = {[x] I : x X} and a binary operation on X/I by [x] I [y] I =[x y] I. Theorem 4.6. If I is an ideal of KK-algebra X with X/I = {[x] I : x X} where a binary operation on a set X/I is defined by [x] I [y] I =[x y] I, then the binary operation is a mapping from X/I X/I to X/I. Proof. Let [x 1 ] I, [x 2 ] I, [y 1 ] I, [y 2 ] I X/I such that [x 1 ] I =[x 2 ] I and [y 1 ] I =
8 1042 S. Asawasamrit and A. Sudprasert [y 2 ] I. It follows that x 1 x 2 and y 1 y 2. By lemma 4.3, x 1 y 1 x 2 y 2, proving that [x 1 y 1 ] I =[x 2 y 2 ] I. Theorem 4.7. If I is an ideal of KK-algebra X, then (X/I;, [0] I ) is a KKalgebra. Moreover, the set X/I is called the quotient KK-algebra. Proof. Let [x] I, [y] I, [z] I X/I. Then ([z] I [x] I ) (([x] I [y] I ) ([z] I [y] I )) = [z x] I ([x y] I [z y] I )=[z x] I [(x y) (z y)] I =[(z x) ((x y) (z y))] I = [0] I. It is clear that [0] I [x] I =[0 x] I =[x] I. Now, let [x] I [y] I = [0] I and [y] I [x] I = [0] I. It follows that x y 0 and y x 0, that is 0 (x y), 0 (y x) I. Since I is an ideal of X and 0 I, we get that x y, y x I. Consequently, x y, proving that [x] I =[y] I. Therefore, (X/I;, [0] I ) is a KK-algebra. Example 4.8. According to example 3.3, we can get that X/I = {[0] I, [2] I }, where [0] I = [1] I = {0, 1} and [2] I = [3] I = {2, 3}. Let be defined on X/I by Then (X/I;, [0] I ) is a KK-algebra. [0] I [2] I [0] I [0] I [2] I [2] I [2] I [0] I Lemma 4.9. Let X be a KK-algebra and I,J be any sets such that I J X. Suppose that I is an ideal of X, then J is an ideal of X if and only if J/I is an ideal of X/I. Proof. Let I be an ideal of X with I J X. Suppose firstly that J is an ideal of X, then J/I = {[x] I : x J}, where [x] I = {y J : x y}, and X/I = {[x] I : x X}, where [x] I = {y X : x y}. Obviously, J/I X/I and [0] I J/I. Now, let [x] I [y] I J/I and [y] I J/I. Then [x y] I =[x] I [y] I J/I, it follows that x y J and x J. By assumption, y J. Accordingly, [y] I J/I, this shows that J/I is an ideal of X/I. On the other hand, suppose that J/I is an ideal of X/I and I is an ideal of X with I J X. Thus, 0 J. Let x y J and x J. It follows that [x y] I, [x] I J/I. Since [x y] I =[x] I [y] I, so [x] I [y] I J/I. By hypothesis, [y] I J/I implies y J, proving our lemma. Lemma Let X be a KK-algebra and I,J be any sets such that I J X. Suppose that I is a closed ideal of X. Then J is a closed ideal of X if and only if J/I is a closed ideal of X/I. Proof. Similar to that of lemma 4.9 Lemma Let I,J be ideals of a KK-algebra X with I J, then I is an ideal of X. Proof. Obvious.
9 A structure of KK-algebras and its properties 1043 Next, the basic properties of equivalence classes are considered are as the following Theorem. Theorem Let I be a closed ideal of a KK-algebra X and a, b X. Then (1) [a] I = I iff a I. (2) [a] I =[b] I or [a] I [b] I =. Proof. Let I be a closed ideal of X and a, b X. (1) It is clear due to the fact that a a for all a X and a a =0 I, so we get that a [a] I = I. Conversely, let x [a] I. Then x a, it follows that x a, a x I. By hypothesis, x I. Hence, [a] I I. To show that I [a] I, choose x I. Since I is a closed of X, we have x a, a x I. Thus, x a, this means that x [a] I and shows that I [a] I. Consequently, [a] I = I. (2) Assume that [a] I [b] I. Then there is x [a] I [b] I such that x [a] I and y [a] I. It follows that x a and x b, so a b by the symmetric and transitive properties. Thus [a] I =[b] I. Theorem If I is a closed ideal of a KK- algebra X and y I, then [y] I is closed ideal of X. Proof. Let I be a closed ideal of X and y I. It is clear that 0 [y] I. Now, suppose that a b [y] I and a [y] I. We will show that b [y] I. Then a b y and a y, it follows that y (a b) I and y a I. By assumption, a I. From proposition 2.9, a (y b) =y (a b) I, and I is a closed ideal of X and a I, therefore, y b I. By properties of X, we get that (a 0) (((a b) y) (b y)) = (a (b b)) (((a b) y) (b y)) = (b (a b)) (((a b) y) (b y)) = 0. By hypothesis, (a 0) (((a b) y) (b y)) I, and I is closed and a I, then a 0 I. Thus ((a b) y) (b y) I. From a b y and I is closed, then b y I. Hence, b y, this means a [y] I. Accordingly, [y] I is an ideal of X. Finally, let a, b [y] I. Then a y and b y, by lemma 4.3, a b y y. By theorem 2.3, it follows that a b 0. Thus a b [0] I. Now, we have 0,y I and I is closed, so 0 y I and y 0 I. That is, 0 y. Hence, [0] I =[y] I. By transitive, a b [y] I, proving our theorem. ACKNOWLEDGEMENTS. This paper is supported by Faculty of Applied Science, King Mongkut s University of Technology North Bangkok, Thailand, Grant No References [1] S. Asawasamrit and U. Leerawat, On quotient binary algebras, Scientia Magna Journal, 6(2010), No.1,
10 1044 S. Asawasamrit and A. Sudprasert [2] W.A. Dudek and X. Zhang, On proper BCC-algebras, Bull. Enst. Math. Academia Simica of Mathematics, 20(1992): [3] W.A. Dudek and X. Zhang, On ideals and congruences in BCC-algebras, Czechoslovak Math. Journal, 48(123) (1998), [4] E.H. Roh, Y.K. Seon, B.J. Young and H.S.Wook, On difference algebras, Kyungpook Math. J, 43(2003): [5] J. Soontharanon and U. Leerawat, AC-algebras, Scientia Magna Journal, 5(2009), No.4, [6] H. Yisheng, BCC-algebras, Science press, Beijing, 1st(2006), 356p. Received: November, 2011
International Journal of Algebra, Vol. 7, 2013, no. 3, HIKARI Ltd, On KUS-Algebras. and Areej T.
International Journal of Algebra, Vol. 7, 2013, no. 3, 131-144 HIKARI Ltd, www.m-hikari.com On KUS-Algebras Samy M. Mostafa a, Mokhtar A. Abdel Naby a, Fayza Abdel Halim b and Areej T. Hameed b a Department
More informationOn atoms in BCC-algebras
Discussiones Mathematicae ser. Algebra and Stochastic Methods 15 (1995), 81 85 On atoms in BCC-algebras Wies law A. DUDEK and Xiaohong ZHANG Abstract We characterize the the set of all atoms of a BCC-algebra
More informationDOI: /auom An. Şt. Univ. Ovidius Constanţa Vol. 25(1),2017, ON BI-ALGEBRAS
DOI: 10.1515/auom-2017-0014 An. Şt. Univ. Ovidius Constanţa Vol. 25(1),2017, 177 194 ON BI-ALGEBRAS Arsham Borumand Saeid, Hee Sik Kim and Akbar Rezaei Abstract In this paper, we introduce a new algebra,
More informationOn Derivations of BCC-algebras
International Journal of Algebra, Vol. 6, 2012, no. 32, 1491-1498 On Derivations of BCC-algebras N. O. Alshehri Department of Mathematics, Faculty of Sciences(Girls) King Abdulaziz University, Jeddah,
More informationScientiae Mathematicae Japonicae Online, Vol.4 (2001), a&i IDEALS ON IS ALGEBRAS Eun Hwan Roh, Seon Yu Kim and Wook Hwan Shim Abstract. In th
Scientiae Mathematicae Japonicae Online, Vol.4 (2001), 21 25 21 a&i IDEALS ON IS ALGEBRAS Eun Hwan Roh, Seon Yu Kim and Wook Hwan Shim Abstract. In this paper, we introduce the concept of an a&i-ideal
More informationON FUZZY TOPOLOGICAL BCC-ALGEBRAS 1
Discussiones Mathematicae General Algebra and Applications 20 (2000 ) 77 86 ON FUZZY TOPOLOGICAL BCC-ALGEBRAS 1 Wies law A. Dudek Institute of Mathematics Technical University Wybrzeże Wyspiańskiego 27,
More informationOn two-sided bases of ternary semigroups. 1. Introduction
Quasigroups and Related Systems 23 (2015), 319 324 On two-sided bases of ternary semigroups Boonyen Thongkam and Thawhat Changphas Abstract. We introduce the concept of two-sided bases of a ternary semigroup,
More informationON A PERIOD OF ELEMENTS OF PSEUDO-BCI-ALGEBRAS
Discussiones Mathematicae General Algebra and Applications 35 (2015) 21 31 doi:10.7151/dmgaa.1227 ON A PERIOD OF ELEMENTS OF PSEUDO-BCI-ALGEBRAS Grzegorz Dymek Institute of Mathematics and Computer Science
More informationarxiv: v1 [math.lo] 20 Oct 2007
ULTRA LI -IDEALS IN LATTICE IMPLICATION ALGEBRAS AND MTL-ALGEBRAS arxiv:0710.3887v1 [math.lo] 20 Oct 2007 Xiaohong Zhang, Ningbo, Keyun Qin, Chengdu, and Wieslaw A. Dudek, Wroclaw Abstract. A mistake concerning
More informationON FILTERS IN BE-ALGEBRAS. Biao Long Meng. Received November 30, 2009
Scientiae Mathematicae Japonicae Online, e-2010, 105 111 105 ON FILTERS IN BE-ALGEBRAS Biao Long Meng Received November 30, 2009 Abstract. In this paper we first give a procedure by which we generate a
More informationLecture 8: Equivalence Relations
Lecture 8: Equivalence Relations 1 Equivalence Relations Next interesting relation we will study is equivalence relation. Definition 1.1 (Equivalence Relation). Let A be a set and let be a relation on
More informationBulletin of the Iranian Mathematical Society Vol. 39 No.6 (2013), pp
Bulletin of the Iranian Mathematical Society Vol. 39 No.6 (2013), pp 1125-1135. COMMON FIXED POINTS OF A FINITE FAMILY OF MULTIVALUED QUASI-NONEXPANSIVE MAPPINGS IN UNIFORMLY CONVEX BANACH SPACES A. BUNYAWAT
More informationSome Results About Generalized BCH-Algebras
International Journal of Algebra, Vol. 11, 2017, no. 5, 231-246 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.712 Some Results About Generalized BCH-Algebras Muhammad Anwar Chaudhry 1
More informationFinite homogeneous and lattice ordered effect algebras
Finite homogeneous and lattice ordered effect algebras Gejza Jenča Department of Mathematics Faculty of Electrical Engineering and Information Technology Slovak Technical University Ilkovičova 3 812 19
More informationHomework #2 Solutions Due: September 5, for all n N n 3 = n2 (n + 1) 2 4
Do the following exercises from the text: Chapter (Section 3):, 1, 17(a)-(b), 3 Prove that 1 3 + 3 + + n 3 n (n + 1) for all n N Proof The proof is by induction on n For n N, let S(n) be the statement
More informationPrime and Irreducible Ideals in Subtraction Algebras
International Mathematical Forum, 3, 2008, no. 10, 457-462 Prime and Irreducible Ideals in Subtraction Algebras Young Bae Jun Department of Mathematics Education Gyeongsang National University, Chinju
More informationComplete Induction and the Well- Ordering Principle
Complete Induction and the Well- Ordering Principle Complete Induction as a Rule of Inference In mathematical proofs, complete induction (PCI) is a rule of inference of the form P (a) P (a + 1) P (b) k
More informationPure Mathematical Sciences, Vol. 1, 2012, no. 3, On CS-Algebras. Kyung Ho Kim
Pure Mathematical Sciences, Vol. 1, 2012, no. 3, 115-121 On CS-Algebras Kyung Ho Kim Department of Mathematics Korea National University of Transportation Chungju 380-702, Korea ghkim@ut.ac.kr Abstract
More informationOn Generalizations of Pseudo-Injectivity
Int. Journal of Math. Analysis, Vol. 6, 2012, no. 12, 555-562 On Generalizations of Pseudo-Injectivity S. Baupradist 1, T. Sitthiwirattham 2,3 and S. Asawasamrit 2 1 Department of Mathematics, Faculty
More informationTHE DIRECT SUM, UNION AND INTERSECTION OF POSET MATROIDS
SOOCHOW JOURNAL OF MATHEMATICS Volume 28, No. 4, pp. 347-355, October 2002 THE DIRECT SUM, UNION AND INTERSECTION OF POSET MATROIDS BY HUA MAO 1,2 AND SANYANG LIU 2 Abstract. This paper first shows how
More information1 2 3 style total. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.
1 2 3 style total Math 415 Examination 3 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. The rings
More informationON STRUCTURE AND COMMUTATIVITY OF NEAR - RINGS
Proyecciones Vol. 19, N o 2, pp. 113-124, August 2000 Universidad Católica del Norte Antofagasta - Chile ON STRUCTURE AND COMMUTATIVITY OF NEAR - RINGS H. A. S. ABUJABAL, M. A. OBAID and M. A. KHAN King
More informationCongruence Boolean Lifting Property
Congruence Boolean Lifting Property George GEORGESCU and Claudia MUREŞAN University of Bucharest Faculty of Mathematics and Computer Science Academiei 14, RO 010014, Bucharest, Romania Emails: georgescu.capreni@yahoo.com;
More informationRohit Garg Roll no Dr. Deepak Gumber
FINITE -GROUPS IN WHICH EACH CENTRAL AUTOMORPHISM FIXES THE CENTER ELEMENTWISE Thesis submitted in partial fulfillment of the requirement for the award of the degree of Masters of Science In Mathematics
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationSoft subalgebras and soft ideals of BCK/BCI-algebras related to fuzzy set theory
MATHEMATICAL COMMUNICATIONS 271 Math. Commun., Vol. 14, No. 2, pp. 271-282 (2009) Soft subalgebras and soft ideals of BCK/BCI-algebras related to fuzzy set theory Young Bae Jun 1 and Seok Zun Song 2, 1
More informationFUZZY BCK-FILTERS INDUCED BY FUZZY SETS
Scientiae Mathematicae Japonicae Online, e-2005, 99 103 99 FUZZY BCK-FILTERS INDUCED BY FUZZY SETS YOUNG BAE JUN AND SEOK ZUN SONG Received January 23, 2005 Abstract. We give the definition of fuzzy BCK-filter
More informationThe Influence of Minimal Subgroups on the Structure of Finite Groups 1
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 14, 675-683 The Influence of Minimal Subgroups on the Structure of Finite Groups 1 Honggao Zhang 1, Jianhong Huang 1,2 and Yufeng Liu 3 1. Department
More informationBG/BF 1 /B/BM-algebras are congruence permutable
Mathematica Aeterna, Vol. 5, 2015, no. 2, 351-35 BG/BF 1 /B/BM-algebras are congruence permutable Andrzej Walendziak Institute of Mathematics and Physics Siedlce University, 3 Maja 54, 08-110 Siedlce,
More informationON STRUCTURE OF KS-SEMIGROUPS
International Mathematical Forum, 1, 2006, no. 2, 67-76 ON STRUCTURE OF KS-SEMIGROUPS Kyung Ho Kim Department of Mathematics Chungju National University Chungju 380-702, Korea ghkim@chungju.ac.kr Abstract
More informationHessenberg Pairs of Linear Transformations
Hessenberg Pairs of Linear Transformations Ali Godjali November 21, 2008 arxiv:0812.0019v1 [math.ra] 28 Nov 2008 Abstract Let K denote a field and V denote a nonzero finite-dimensional vector space over
More informationORBIT-HOMOGENEITY IN PERMUTATION GROUPS
Submitted exclusively to the London Mathematical Society DOI: 10.1112/S0000000000000000 ORBIT-HOMOGENEITY IN PERMUTATION GROUPS PETER J. CAMERON and ALEXANDER W. DENT Abstract This paper introduces the
More informationNotes on Systems of Linear Congruences
MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the
More informationFINITE ABELIAN GROUPS Amin Witno
WON Series in Discrete Mathematics and Modern Algebra Volume 7 FINITE ABELIAN GROUPS Amin Witno Abstract We detail the proof of the fundamental theorem of finite abelian groups, which states that every
More informationPrime and irreducible elements of the ring of integers modulo n
Prime and irreducible elements of the ring of integers modulo n M. H. Jafari and A. R. Madadi Department of Pure Mathematics, Faculty of Mathematical Sciences University of Tabriz, Tabriz, Iran Abstract
More informationDerivations of B-algebras
JKAU: Sci, Vol No 1, pp: 71-83 (010 AD / 1431 AH); DOI: 104197 / Sci -15 Derivations of B-algebras Department of Mathematics, Faculty of Education, Science Sections, King Abdulaziz University, Jeddah,
More informationSome remarks on hyper MV -algebras
Journal of Intelligent & Fuzzy Systems 27 (2014) 2997 3005 DOI:10.3233/IFS-141258 IOS Press 2997 Some remarks on hyper MV -algebras R.A. Borzooei a, W.A. Dudek b,, A. Radfar c and O. Zahiri a a Department
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationRelations. Binary Relation. Let A and B be sets. A (binary) relation from A to B is a subset of A B. Notation. Let R A B be a relation from A to B.
Relations Binary Relation Let A and B be sets. A (binary) relation from A to B is a subset of A B. Notation Let R A B be a relation from A to B. If (a, b) R, we write a R b. 1 Binary Relation Example:
More informationClimbing an Infinite Ladder
Section 5.1 Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite
More informationDiscrete Mathematics. Spring 2017
Discrete Mathematics Spring 2017 Previous Lecture Principle of Mathematical Induction Mathematical Induction: rule of inference Mathematical Induction: Conjecturing and Proving Climbing an Infinite Ladder
More informationGROUPS WITH A MAXIMAL IRREDUNDANT 6-COVER #
Communications in Algebra, 33: 3225 3238, 2005 Copyright Taylor & Francis, Inc. ISSN: 0092-7872 print/1532-4125 online DOI: 10.1081/AB-200066157 ROUPS WITH A MAXIMAL IRREDUNDANT 6-COVER # A. Abdollahi,
More informationInterval-Valued Fuzzy KUS-Ideals in KUS-Algebras
IOSR Jornal of Mathematics (IOSR-JM) e-issn: 78-578. Volme 5, Isse 4 (Jan. - Feb. 03), PP 6-66 Samy M. Mostafa, Mokhtar.bdel Naby, Fayza bdel Halim 3, reej T. Hameed 4 and Department of Mathematics, Faclty
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationarxiv: v1 [math.ra] 1 Apr 2015
BLOCKS OF HOMOGENEOUS EFFECT ALGEBRAS GEJZA JENČA arxiv:1504.00354v1 [math.ra] 1 Apr 2015 Abstract. Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalize some
More informationAn Improved Lower Bound for. an Erdös-Szekeres-Type Problem. with Interior Points
Applied Mathematical Sciences, Vol. 6, 2012, no. 70, 3453-3459 An Improved Lower Bound for an Erdös-Szekeres-Type Problem with Interior Points Banyat Sroysang Department of Mathematics and Statistics,
More informationGroups and Symmetries
Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group
More informationOn Strongly Prime Semiring
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 30(2) (2007), 135 141 On Strongly Prime Semiring T.K. Dutta and M.L. Das Department
More informationTRANSITIVE AND ABSORBENT FILTERS OF LATTICE IMPLICATION ALGEBRAS
J. Appl. Math. & Informatics Vol. 32(2014), No. 3-4, pp. 323-330 http://dx.doi.org/10.14317/jami.2014.323 TRANSITIVE AND ABSORBENT FILTERS OF LATTICE IMPLICATION ALGEBRAS M. SAMBASIVA RAO Abstract. The
More informationMATH 215 Final. M4. For all a, b in Z, a b = b a.
MATH 215 Final We will assume the existence of a set Z, whose elements are called integers, along with a well-defined binary operation + on Z (called addition), a second well-defined binary operation on
More informationClimbing an Infinite Ladder
Section 5.1 Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite
More informationCISC-102 Fall 2017 Week 3. Principle of Mathematical Induction
Week 3 1 of 17 CISC-102 Fall 2017 Week 3 Principle of Mathematical Induction A proposition is defined as a statement that is either true or false. We will at times make a declarative statement as a proposition
More informationInvertible Matrices over Idempotent Semirings
Chamchuri Journal of Mathematics Volume 1(2009) Number 2, 55 61 http://www.math.sc.chula.ac.th/cjm Invertible Matrices over Idempotent Semirings W. Mora, A. Wasanawichit and Y. Kemprasit Received 28 Sep
More information* 8 Groups, with Appendix containing Rings and Fields.
* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that
More informationWe want to show P (n) is true for all integers
Generalized Induction Proof: Let P (n) be the proposition 1 + 2 + 2 2 + + 2 n = 2 n+1 1. We want to show P (n) is true for all integers n 0. Generalized Induction Example: Use generalized induction to
More informationMINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS
MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RŮŽIČKA Abstract. A subset X of a group (or a ring, or a field) is called generating, if the smallest subgroup
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationObstinate filters in residuated lattices
Bull. Math. Soc. Sci. Math. Roumanie Tome 55(103) No. 4, 2012, 413 422 Obstinate filters in residuated lattices by Arsham Borumand Saeid and Manijeh Pourkhatoun Abstract In this paper we introduce the
More informationSome Pre-filters in EQ-Algebras
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 12, Issue 2 (December 2017), pp. 1057-1071 Applications and Applied Mathematics: An International Journal (AAM) Some Pre-filters
More informationMinimal basis for connected Markov chain over 3 3 K contingency tables with fixed two-dimensional marginals. Satoshi AOKI and Akimichi TAKEMURA
Minimal basis for connected Markov chain over 3 3 K contingency tables with fixed two-dimensional marginals Satoshi AOKI and Akimichi TAKEMURA Graduate School of Information Science and Technology University
More informationA Note on Quasi and Bi-Ideals in Ordered Ternary Semigroups
Int. Journal of Math. Analysis, Vol. 6, 2012, no. 11, 527-532 A Note on Quasi and Bi-Ideals in Ordered Ternary Semigroups Thawhat Changphas 1 Department of Mathematics Faculty of Science Khon Kaen University
More informationA new class of generalized delta semiclosed sets using grill delta space
A new class of generalized delta semiclosed sets using grill delta space K.Priya 1 and V.Thiripurasundari 2 1 M.Phil Scholar, PG and Research Department of Mathematics, Sri S.R.N.M.College, Sattur - 626
More informationPrime Hyperideal in Multiplicative Ternary Hyperrings
International Journal of Algebra, Vol. 10, 2016, no. 5, 207-219 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.6320 Prime Hyperideal in Multiplicative Ternary Hyperrings Md. Salim Department
More informationSection 7.1 Relations and Their Properties. Definition: A binary relation R from a set A to a set B is a subset R A B.
Section 7.1 Relations and Their Properties Definition: A binary relation R from a set A to a set B is a subset R A B. Note: there are no constraints on relations as there are on functions. We have a common
More informationGeneralizations of -primary gamma-ideal of gamma-rings
Global ournal of Pure and Applied athematics ISSN 0973-1768 Volume 1, Number 1 (016), pp 435-444 Research India Publications http://wwwripublicationcom Generalizations of -primary gamma-ideal of gamma-rings
More informationDiscrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet
More informationPrimitive Ideals of Semigroup Graded Rings
Sacred Heart University DigitalCommons@SHU Mathematics Faculty Publications Mathematics Department 2004 Primitive Ideals of Semigroup Graded Rings Hema Gopalakrishnan Sacred Heart University, gopalakrishnanh@sacredheart.edu
More information. As the binomial coefficients are integers we have that. 2 n(n 1).
Math 580 Homework. 1. Divisibility. Definition 1. Let a, b be integers with a 0. Then b divides b iff there is an integer k such that b = ka. In the case we write a b. In this case we also say a is a factor
More informationDefinition 2.3. We define addition and multiplication of matrices as follows.
14 Chapter 2 Matrices In this chapter, we review matrix algebra from Linear Algebra I, consider row and column operations on matrices, and define the rank of a matrix. Along the way prove that the row
More informationZero-Sum Flows in Regular Graphs
Zero-Sum Flows in Regular Graphs S. Akbari,5, A. Daemi 2, O. Hatami, A. Javanmard 3, A. Mehrabian 4 Department of Mathematical Sciences Sharif University of Technology Tehran, Iran 2 Department of Mathematics
More informationRelations and Equivalence Relations
Relations and Equivalence Relations In this section, we shall introduce a formal definition for the notion of a relation on a set. This is something we often take for granted in elementary algebra courses,
More informationProjective Schemes with Degenerate General Hyperplane Section II
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 111-126. Projective Schemes with Degenerate General Hyperplane Section II E. Ballico N. Chiarli S. Greco
More informationCompleteness of Star-Continuity
Introduction to Kleene Algebra Lecture 5 CS786 Spring 2004 February 9, 2004 Completeness of Star-Continuity We argued in the previous lecture that the equational theory of each of the following classes
More informationA CLOSURE SYSTEM FOR ELEMENTARY SITUATIONS
Bulletin of the Section of Logic Volume 11:3/4 (1982), pp. 134 138 reedition 2009 [original edition, pp. 134 139] Bogus law Wolniewicz A CLOSURE SYSTEM FOR ELEMENTARY SITUATIONS 1. Preliminaries In [4]
More information@FMI c Kyung Moon Sa Co.
Annals of Fuzzy Mathematics and Informatics Volume 4, No. 2, October 2012), pp. 365 375 ISSN 2093 9310 http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com On soft int-groups Kenan Kaygisiz
More informationOn binary reflected Gray codes and functions
Discrete Mathematics 308 (008) 1690 1700 www.elsevier.com/locate/disc On binary reflected Gray codes and functions Martin W. Bunder, Keith P. Tognetti, Glen E. Wheeler School of Mathematics and Applied
More informationP = 1 F m(p ) = IP = P I = f(i) = QI = IQ = 1 F m(p ) = Q, so we are done.
Section 1.6: Invertible Matrices One can show (exercise) that the composition of finitely many invertible functions is invertible. As a result, we have the following: Theorem 6.1: Any admissible row operation
More informationMA441: Algebraic Structures I. Lecture 18
MA441: Algebraic Structures I Lecture 18 5 November 2003 1 Review from Lecture 17: Theorem 6.5: Aut(Z/nZ) U(n) For every positive integer n, Aut(Z/nZ) is isomorphic to U(n). The proof used the map T :
More informationHouston Journal of Mathematics. c 2007 University of Houston Volume 33, No. 1, 2007
Houston Journal of Mathematics c 2007 University of Houston Volume 33, No. 1, 2007 ROUPS WITH SPECIFIC NUMBER OF CENTRALIZERS A. ABDOLLAHI, S.M. JAFARIAN AMIRI AND A. MOHAMMADI HASSANABADI Communicated
More information1 Examples of Weak Induction
More About Mathematical Induction Mathematical induction is designed for proving that a statement holds for all nonnegative integers (or integers beyond an initial one). Here are some extra examples of
More informationSubalgebras and ideals in BCK/BCI-algebras based on Uni-hesitant fuzzy set theory
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 2, 2018, 417-430 ISSN 1307-5543 www.ejpam.com Published by New York Business Global Subalgebras and ideals in BCK/BCI-algebras based on Uni-hesitant
More informationTHE formal definition of a ternary algebraic structure was
A Study on Rough, Fuzzy and Rough Fuzzy Bi-ideals of Ternary Semigroups Sompob Saelee and Ronnason Chinram Abstract A ternary semigroup is a nonempty set together with a ternary multiplication which is
More information10. Noether Normalization and Hilbert s Nullstellensatz
10. Noether Normalization and Hilbert s Nullstellensatz 91 10. Noether Normalization and Hilbert s Nullstellensatz In the last chapter we have gained much understanding for integral and finite ring extensions.
More informationPREOPEN SETS AND RESOLVABLE SPACES
PREOPEN SETS AND RESOLVABLE SPACES Maximilian Ganster appeared in: Kyungpook Math. J. 27 (2) (1987), 135 143. Abstract This paper presents solutions to some recent questions raised by Katetov about the
More informationFields and Galois Theory. Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory.
Fields and Galois Theory Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory. This should be a reasonably logical ordering, so that a result here should
More informationNowhere zero flow. Definition: A flow on a graph G = (V, E) is a pair (D, f) such that. 1. D is an orientation of G. 2. f is a function on E.
Nowhere zero flow Definition: A flow on a graph G = (V, E) is a pair (D, f) such that 1. D is an orientation of G. 2. f is a function on E. 3. u N + D (v) f(uv) = w ND f(vw) for every (v) v V. Example:
More informationarxiv: v1 [math.ra] 25 May 2013
Quasigroups and Related Systems 20 (2012), 203 209 Congruences on completely inverse AG -groupoids Wieslaw A. Dudek and Roman S. Gigoń arxiv:1305.6858v1 [math.ra] 25 May 2013 Abstract. By a completely
More informationEquational Logic. Chapter 4
Chapter 4 Equational Logic From now on First-order Logic is considered with equality. In this chapter, I investigate properties of a set of unit equations. For a set of unit equations I write E. Full first-order
More informationCoupled -structures and its application in BCK/BCI-algebras
IJST (2013) 37A2: 133-140 Iranian Journal of Science & Technology http://www.shirazu.ac.ir/en Coupled -structures its application in BCK/BCI-algebras Young Bae Jun 1, Sun Shin Ahn 2 * D. R. Prince Williams
More informationMath Introduction to Modern Algebra
Math 343 - Introduction to Modern Algebra Notes Field Theory Basics Let R be a ring. M is called a maximal ideal of R if M is a proper ideal of R and there is no proper ideal of R that properly contains
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More informationIf F is a divisor class on the blowing up X of P 2 at n 8 general points p 1,..., p n P 2,
Proc. Amer. Math. Soc. 124, 727--733 (1996) Rational Surfaces with K 2 > 0 Brian Harbourne Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln, NE 68588-0323 email: bharbourne@unl.edu
More informationGroups with many subnormal subgroups. *
Groups with many subnormal subgroups. * Eloisa Detomi Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, via Valotti 9, 25133 Brescia, Italy. E-mail: detomi@ing.unibs.it
More informationMath.3336: Discrete Mathematics. Chapter 9 Relations
Math.3336: Discrete Mathematics Chapter 9 Relations Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall 2018
More informationBulletin of the Iranian Mathematical Society
ISSN: 117-6X (Print) ISSN: 1735-8515 (Online) Bulletin of the Iranian Mathematical Society Vol. 4 (14), No. 6, pp. 1491 154. Title: The locating chromatic number of the join of graphs Author(s): A. Behtoei
More informationHowever another possibility is
19. Special Domains Let R be an integral domain. Recall that an element a 0, of R is said to be prime, if the corresponding principal ideal p is prime and a is not a unit. Definition 19.1. Let a and b
More informationSpectrally arbitrary star sign patterns
Linear Algebra and its Applications 400 (2005) 99 119 wwwelseviercom/locate/laa Spectrally arbitrary star sign patterns G MacGillivray, RM Tifenbach, P van den Driessche Department of Mathematics and Statistics,
More informationLAYERED NETWORKS, THE DISCRETE LAPLACIAN, AND A CONTINUED FRACTION IDENTITY
LAYERE NETWORKS, THE ISCRETE LAPLACIAN, AN A CONTINUE FRACTION IENTITY OWEN. BIESEL, AVI V. INGERMAN, JAMES A. MORROW, AN WILLIAM T. SHORE ABSTRACT. We find explicit formulas for the conductivities of
More informationSection 2: Classes of Sets
Section 2: Classes of Sets Notation: If A, B are subsets of X, then A \ B denotes the set difference, A \ B = {x A : x B}. A B denotes the symmetric difference. A B = (A \ B) (B \ A) = (A B) \ (A B). Remarks
More information