Andrzej Walendziak, Magdalena Wojciechowska-Rysiawa BIPARTITE PSEUDO-BL ALGEBRAS
|
|
- Ronald Wilson
- 5 years ago
- Views:
Transcription
1 DEMONSTRATIO MATHEMATICA Vol. XLIII No Andrzej Walendziak, Magdalena Wojciechowska-Rysiawa BIPARTITE PSEUDO-BL ALGEBRAS Abstract. The class of bipartite pseudo-bl algebras (denoted by BP) and the class of strongly bipartite pseudo-bl algebras (denoted by BP 0 ) are investigated. We prove that the class BP 0 is a variety and show that BP is closed under subalgebras and arbitrary direct products but it is not a variety. We also study connections between bipartite pseudo-bl algebras and other classes of pseudo-bl algebras. 1. Introduction BL algebras were introduced by Hájek [9] in MV algebras introduced by Chang [1] are contained in the class of BL algebras. Georgescu and Iorgulescu [6] introduced pseudo-mv algebras as a noncommutative generalization of MV algebras. In 2000, in a natural way, there were introduced pseudo-bl algebras as a generalization of BL algebras and MV algebras. A pseudo-bl algebra is a pseudo-mv algebra if and only if the pseudo-double Negation condition (pdn, for short) is satisfied, that is, (x ) = (x ) = x for all x. Main properties of pseudo-bl algebras were studied in [2] and [3]. Pseudo-BL algebras correspond to a pseudo-basic fuzzy logic (see [10] and [11]). Bipartite MV algebras were defined and studied by Di Nola, Liguori and Sessa in [4]. Dymek [5] investigated bipartite pseudo-mv algebras. Georgescu and Leuştean [8] introduced the class BP of pseudo-bl algebras bipartite by some ultafilter and the subclass BP 0 of pseudo-bl algebras bipartite by all ultrafilters. In this paper we give some characterizations of bipartite and strongly bipartite pseudo-bl algebras. We prove that the class BP 0 is a variety and show that BP is closed under subalgebras and arbitrary direct products but it is not a variety. We also study connections between bipartite pseudo-bl algebras and other classes of pseudo-bl algebras Mathematics Subject Classification: 03G25, 06F05. Key words and phrases: pseudo-bl algebra, filter, ultrafilter, (strongly) bipartite pseudo-bl algebra.
2 488 A. Walendziak, M. Wojciechowska-Rysiawa 2. Preliminaries Definition 2.1. ([2]) Let (A,,,,,, 0, 1) be an algebra of type (2, 2, 2, 2, 2, 0, 0). The algebra A is called a pseudo-bl algebra if it satisfies the following axioms, for any x, y, z A : (C1) (A,,, 0, 1) is a bounded lattice, (C2) (A,, 1) is a monoid, (C3) x y z x y z y x z, (C4) x y = (x y) x = x (x y), (C5) (x y) (y x) = (x y) (y x) = 1. Throughout this paper A will denote a pseudo-bl algebra. For any x A and n = 0, 1,..., we put x 0 = 1 and x n+1 = x n x. Proposition 2.2. ([2])The following properties hold in A for all x, y A : (a) x y x y = 1 x y = 1, (b) x y x and x y y. Let us define x = x 0 and x = x 0 for all x A. Proposition 2.3. ([2]) The following properties hold in A for all x, y A: (a) x (x ) and x (x ), (b) x x = x x = 0, (c) x y implies y x and y x. Definition 2.4. A nonempty set F is called a filter of A if the following conditions hold: (F1) If x, y F, then x y F, (F2) if x F, y A, x y then y F. The filter F is called proper if F A. The set of all filters of A is denoted by Fil(A). For every subset X A, the smallest filter of A which contains X, that is the intersection of all filters F X, is said to be the filter generated by X and will be denoted by [X). Proposition 2.5. ([2]) If X A, then [X) = {y A : x 1 x n y for some n 1 and x 1,...,x n X}. Definition 2.6. Let F be a proper filter of A. (a) F is called prime iff for all x, y A, x y F implies x F or y F. (b) F is called maximal (or ultrafilter) iff whenever H is a filter such that F H A, then either H = F or H = A.
3 Bipartite pseudo-bl algebras 489 We denote by Max(A) the set of ultrafilters of A. Definition 2.7. A filter H of A is called normal if for every x, y A x y H x y H. Proposition 2.8. ([2]) Any ultrafilter of A is a prime filter of A. Proposition 2.9. ([2]) Any proper filter of A can be extended to an ultrafilter. and Following [8], for any F A, we define two sets F and F as follows: By Remark 1.13 of [8] we have F = {x A : x f for some f F } F = {x A : x f for some f F }. F = {x A : x F } and F = {x A : x F }. Lemma If F is a proper filter of A, then: (a) F F =, (b) F F =, (c) F A F, (d) F A F. Proof. (a) Suppose that x F F. Then x F and x f for some f F. Since F is a filter, from definition it follows that f F and f F. Using Proposition 2.3 (b) we have 0 = f f F. This contradicts the fact that F is proper. (b) Similar to (a). (c) Let x F. Then x F. Suppose that x F. Applying Proposition 2.3 (b) we obtain x x = 0 F. This is a contradiction, because F is proper. (d) Similar to (c). Proposition Let F be a proper filter of A. Then the following conditions are equivalent: (a) A = F F = F F, (b) F = F = A F, (c) x A (x F or (x F and x F)), (d) x A (x x, x x F) and F is prime. Proof. (a) (b). Follows easily from Lemma 2.10 (a) and (b). (b) (c). Let x A F. Therefore x F = F. Hence x F and x F. (c) (d). See Proposition 5.1 of [8].
4 490 A. Walendziak, M. Wojciechowska-Rysiawa (c) (a). Obvious. Proposition If F is a proper filter of A and one of the equivalent conditions of Proposition 2.11 holds, then F is an ultrafilter. Proof. Suppose that x / F and let U = [F {x}). We show that A = U. It suffices to prove that 0 U. Let x F and hence x F. Therefore x U. Consequently, 0 = x x U. Let h : A B be a homomorphism of pseudo-bl algebras. The set Ker(h) = {x A : h(x) = 1} is called the kernel of h. Proposition ([8]) Let h : A B be a homomorphism of pseudo-bl algebras. Then: (a) Ker(h) is a normal filter of A, (b) A/Ker(h) = B. Proposition ([8]) If H is a normal filter of A, then there is a bijection between the ultrafilters of A containing H and the ultrafilters of A/H. 3. Bipartite pseudo-bl algebras Definition 3.1. ([8]) A is called bipartite if A = F F = F F for some ultrafilter F. Define the class BP as follows: A BP A is bipartite. Let us denote by sup(a) the set {x x : x A} {x x : x A}. Proposition 3.2. ([8]) sup(a) = {x A : x x or x x }. Proposition 3.3. Let sup(a) be a proper filter. Then A BP. Proof. Suppose that sup(a) is a proper filter. By Proposition 2.9 there exists an ultrafilter F of A such that sup(a) F. From Proposition 2.8 we conclude that F is prime. Applying Propositions 2.11 and 2.12 we deduce that A BP. Proposition 3.4. A BP [sup(a)) A. Proof. : Assume that A BP and [sup(a)) = A. By Proposition 2.11, there exists an ultrafilter F of A such that x x, x x F for all x A. Then sup(a) F. Consequently, A = [sup(a)) F and hence A = F, a contradiction. : Suppose that [sup(a)) A. By Proposition 2.9, [sup(a)) can be extended to an ultrafilter F. From Proposition 2.11 we have A = F F = F F. Thus A BP. Proposition 3.5. If F = A {0} is an ultrafilter of A, then A is bipartite.
5 Bipartite pseudo-bl algebras 491 Proof. Let x A. Then x F or x = 0. If x = 0, then x = x = 1 F. By Proposition 2.11, A = F F = F F, and hence A is bipartite. Proposition 3.6. Any subalgebra of a bipartite pseudo-bl algebra is bipartite. Proof. Let A BP and suppose that B is a subalgebra of A. Let F be a proper filter of A satisfying the condition (d) of Proposition Then U = F B is a prime filter of B and supb U. By Propositions 2.11 and 2.12, U is an ultrafilter of B and B = U U = U U. Hence B is a bipartite pseudo-bl algebra. Proposition 3.7. Let A and A t (t T) be pseudo-bl algebras and A = t T A t. If A t0 is bipartite for some t 0 T, then A is bipartite. Proof. Let U t0 be a prime filter of A t0 such that sup(a t0 ) U t0. Let U = U t0 s =t 0 A s. It is obvious that U is a prime filter of A. For every x = (a t ) t T A, x x = (a t a t ) t T U and x x = (a t a t ) t T U. Therefore, A is bipartite. Corollary 3.8. Let A t (t T) be bipartite pseudo-bl algebras. Then A = t T A t is a bipartite pseudo-bl algebra. Proposition 3.9. A homomorphic image of a bipartite pseudo-bl algebra is not bipartite in general. Proof. Let A = A 1 A 2, where A 1 BP and A 2 / BP. We consider the projection map π 2 : A A 2. Obviously π 2 is a homomorphism from A onto A 2. From Proposition 3.7 we see that A is bipartite but, by assumption, A 2 is not bipartite. Corollary The class BP is not a variety. 4. Strongly bipartite pseudo-bl algebras We define the class BP 0 of pseudo-bl algebras as follows: A BP 0 iff A = F F = F F for any ultrafilter F of A. Algebras from the class BP 0 are called strongly bipartite. Of course, BP 0 BP. Proposition 4.1. The following conditions are equivalent: (a) (b) A is strongly bipartite, F Max(A) x A [x / F n N((x n ) F and (x n ) F)]. Proof. (a) (b). Let A BP 0 and let F be an ultrafilter. Suppose that x A F. By Proposition 2.11, x F and x F. Applying Propositions
6 492 A. Walendziak, M. Wojciechowska-Rysiawa 2.2 (b) and 2.3 (c) we have x (x n ) and x (x n ) for all n N. Then (x n ) F and (x n ) F. (b) (a). Let the condition (b) be satisfied and F be an ultrafilter of A. Suppose that x / F. Then (x n ) F and (x n ) F for n N. In particular, x F and x F. Thus the condition (c) of Proposition 2.11 holds. Consequently, A = F F = F F. Therefore, A is strongly bipartite. Proposition 4.2. ([8]) The following conditions are equivalent: (a) (b) A is strongly bipartite, sup(a) M(A), where M(A) = {F : F is an ultrafilter of A}. In [3], there were defined two sets: and U(A) := {x A : (x n ) x for all n N} V (A) := {x A : (x n ) x for all n N}. Proposition 4.3. ([3]) M(A) U(A) V (A). Proposition 4.4. U(A) V (A) sup(a). Proof. Let x U(A). Then (x n ) x for all n N. In particular, x x. By Proposition 3.2, x sup(a). Thus U(A) sup(a). Similarly, V (A) sup(a). From Propositions 4.3 and 4.4 we obtain Corrolary 4.5. M(A) sup(a). Theorem 4.6. The following are equivalent: (a) A BP 0, (b) sup(a) = U(A) = V (A) = M(A), (c) F Max(A) sup(a) F. Proof. (a) (b). We have U(A) sup(a) (by Proposition 4.4) M(A) (by Proposition 4.2) U(A) (by Proposition 4.3). Therefore sup(a) = U(A) = M(A). Similarly, sup(a) = V (A) = M(A). (b) (c). Obvious. (c) (a). Let (c) hold. Then sup(a) M(A) and hence, by Proposition 4.2, A is strongly bipartite. Proposition 4.7. Any subalgebra of strongly bipartite pseudo-bl algebra is strongly bipartite.
7 Bipartite pseudo-bl algebras 493 Proof. Let A be strongly bipartite and B be a subalgebra of A. Let F be an ultrafilter of B and F be the filter generated by F in A. Then F = {y A : y x for some x F } by Proposition 2.5. Suppose that 0 F. Hence 0 F. This contradicts the fact that F is proper. Then F is proper too. By Proposition 2.9, there is an ultrafilter U of A such that U F. It is easy to see that U B Fil(B) and U B F. Since F Max(B), it follows that U B = F. We obtain sup(b) sup(a) U, because A is strongly bipartite. As B is a subalgebra we have sup(b) B. Consequently, sup(b) U B = F. By Theorem 4.6, B is strongly bipartite. Proposition 4.8. The class BP 0 is closed under direct products. Proof. Let A = t T A t, and A t be bipartite for t T. Let F Max(A). Then there is t 0 T such that F = F t0 s =t 0 A s, where F t0 Max(A t0 ). Let x = (a t ) t T A. It is easily seen that x x = (a t a t ) t T F and x x = (a t a t ) t T F. Thus sup(a) F for each F Max(A), and therefore A BP 0 by Theorem 4.6. Proposition 4.9. Let A BP 0 and h : A B be a surjective homomorphism. Then B BP 0. Proof. Write H = Ker(h). By Proposition 2.13, H is a normal filter and B = A/H. From Proposition 2.14 it follows that every ultrafilter of A/H has a form F/H, where F is an ultrafilter of A containing H. We have sup(a/h) = {a/h (a/h) : a A} {a/h (a/h) : a A} = {a a /H : a A} {a a /H : a A} F/H, because sup(a) F. Consequently, B is strongly bipartite. Propositions yield Theorem The class BP 0 is a variety. Let B(A) denote the set of all complemented elements in the distributive lattice L(A) = (A,,, 0, 1) of a pseudo-bl algebra A. Proposition ([3]) The following are equivalent: (a) x B(A), (b) x x = 1, (c) x x = 1.
8 494 A. Walendziak, M. Wojciechowska-Rysiawa Write M n (A) = {F : F is a normal ultrafilter of A}. Recall that A is called semisimple iff M n (A) = {1}. Proposition Let A be a semisimple pseudo-bl algebra. Then A is strongly bipartite if and only if A = B(A). Proof. Let A BP 0. Then sup(a) M(A). It is easily seen that M(A) M n (A). Since A is semisimple, M n (A) = {1}. Consequently, sup(a) = {1}. Hence x x = 1 for all x A and by Proposition 4.11, B(A) = A. Assume now that x x = 1 for all x A. Therefore sup(a) = {1}. Hence sup(a) F for all F Max(A). From Theorem 4.6 it follows that A is strongly bipartite. A pseudo-bl algebra A is called good if it satisfies the following condition: (a ) = (a ) for all a A. We say that A is local if it has a unique ultrafilter. The order of a A, in symbols ord(a), is the smallest natural number n such that a n = 0. If no such n exists, then ord(a) =. A good pseudo-bl algebra A is called perfect if it is local and for any a A, Following [8], we define two sets: ord(a) < ord(a ) =. D(A) = {a A : ord(a) = } and D(A) = {a A : ord(a) < }. It is obvious that D(A) D(A) = and D(A) D(A) = A. Proposition ([8]) The following conditions are equivalent: (a) A is local; (b) D(A) is the unique ultrafilter of A. Proposition ([8]) Let A be a local good pseudo-bl algebra. The following are equivalent: (a) A is perfect, (b) D(A) = D(A) = D(A). Proposition Every perfect pseudo-bl algebra is strongly bipartite. Proof. Let A be perfect. Then it is local, and so, by Proposition 4.13, D(A) is the unique ultrafilter of A. We have A = D(A) D(A) and from Proposition 4.14 it follows that D(A) = D(A) = D(A). Consequently, A is strongly bipartite. Example ([13]) Let a, b, c, d R, where R is the set of all real numbers. We put by definition (a, b) (c, d) a < c or (a = c and b d).
9 Bipartite pseudo-bl algebras 495 For any x, y R, we define operations and as follows: x y = min{x, y} and x y = max{x, y}. The meet and the join are defined on R R component-wise. Let {( ) } 1 A = 2, b : b 0 {(a, b) : 1 < a < 1, b R} {(1, b) : b 0}. 2 For any (a, b), (c, d) A, we put: ( ) 1 (a, b) (c, d) = 2, 0 (ac, bc + d), ( ) [( 1 c (a, b) (c, d) = 2, 0 a, d b ) ] (1, 0), a ( ) [( ) ] 1 c (a, b) (c, d) = 2, 0 ad bc, (1, 0). a a Then (A,,,,,, ( 1 2, 0), (1, 0)) is a pseudo-bl algebra. Let (a, b) A. We have ( ) ( ) [( ) ] (a, b) = (a, b) 2, 0 = 2, 0 2a, b (1, 0) a and (a, b) = (a, b) It is easy to see that ( ) ( ) [( , 0 = 2, 0 2a, b ) ] (1, 0). 2a ((a, b) ) = (a, b) = ((a, b) ). Then A satisfies condition (pdn) and hence A is a good pseudo-bl algebra. (Moreover, A is a pseudo-mv algebra.) Let F = {(1, b) : b 0}. In [13], we proved that F is the unique ultrafilter of A. Consequently, A is local. Since F is normal (see [13]), we have M n (A) = {F } {(1, 0)}, and therefore A is not semisimple. Now we show that condition (c) of Proposition 2.11 is not satisfied. Indeed, let x = ( 3 4, 1). Then x / F and x = ( ) [( ) ] ( ) , 0 3, 4 (1, 0) = 3 3, 4 / F. 3 Therefore, A is not bipartite and obviously it is not strongly bipartite. Define {( ) } 1 B = 2, b R 2 : b 0 {(1, b) R 2 : b 0}. It is easy to see that (B,,,,,, ( 1 2, 0), (1, 0)) is a subalgebra of A. The subset F is also the unique ultrafilter of B and hence B is local. Now we
10 496 A. Walendziak, M. Wojciechowska-Rysiawa prove that B is perfect. By Proposition 4.13, D(B) = F. Let x = (a, b) B. We have ord(x) < x B F x F ord(x ) =. Thus B is perfect. From Proposition 4.15 it follows that B is strongly bipartite. Since A is not strongly bipartite, we see that A is not perfect by Proposition Acknowledgments. The authors are highly grateful to referee for her/his remarks and suggestions for improving the paper. References [1] C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), [2] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: Part I, Multiple- Valued Logic 8 (2002), [3] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: Part II, Multiple- Valued Logic 8 (2002), [4] A. Di Nola, F. Liguori, S. Sessa, Using maximal ideals in the classification of MV algebras, Portugal. Math. 50 (1993), [5] G. Dymek, Bipartite pseudo-mv algebras, Discuss. Math., General Algebra and Applications 26 (2006), [6] G. Georgescu, A. Iorgulescu, Pseudo-MV algebras: a noncommutative extension of MV algebras, The Proceedings of the Fourth International Symposium on Economic Informatics, Bucharest, Romania, May 1999, [7] G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: a noncommutative extension of BL algebras, Abstracts of the Fifth International Conference FSTA 2000, Slovakia 2000, [8] G. Georgescu, L. Leuştean, Some classes of pseudo-bl algebras, J. Austral. Math. Soc. 73 (2002), [9] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Amsterdam, [10] P. Hájek, Fuzzy logics with noncommutative conjuctions, J. Logic Comput. 13 (2003), [11] P. Hájek, Observations on non-commutative fuzzy logic, Soft Computing 8 (2003), [12] J. Rachůnek, A non-commutative generalizations of MV algebras, Math. Slovaca 52 (2002), [13] A. Walendziak, M. Wojciechowska, Semisimple and semilocal pseudo-bl algebras, Demonstratio Math. 42 (2009), Andrzej Walendziak WARSAW SCHOOL OF INFORMATION TECHNOLOGY Newelska 6 PL WARSZAWA, POLAND walent@interia.pl Magdalena Wojciechowska-Rysiawa UNIVERSITY OF PODLASIE 3Maja54 PL SIEDLCE, POLAND magdawojciechowska6@wp.pl Received March 23, 2009; revised version August 5, 2009.
ON FUZZY IDEALS OF PSEUDO MV -ALGEBRAS
Discussiones Mathematicae General Algebra and Applications 28 (2008 ) 63 75 ON FUZZY IDEALS OF PSEUDO MV -ALGEBRAS Grzegorz Dymek Institute of Mathematics and Physics University of Podlasie 3 Maja 54,
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 informationFuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras
Fuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras Jiří Rachůnek 1 Dana Šalounová2 1 Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova
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 informationBM-ALGEBRAS AND RELATED TOPICS. 1. Introduction
ao DOI: 10.2478/s12175-014-0259-x Math. Slovaca 64 (2014), No. 5, 1075 1082 BM-ALGEBRAS AND RELATED TOPICS Andrzej Walendziak (Communicated by Jiří Rachůnek ) ABSTRACT. Some connections between BM-algebras
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 informationPseudo MV-Algebras Induced by Functions
International Mathematical Forum, 4, 009, no., 89-99 Pseudo MV-Algebras Induced by Functions Yong Chan Kim Department of Mathematics, Kangnung National University, Gangneung, Gangwondo 10-70, Korea yck@kangnung.ac.kr
More informationThe logic of perfect MV-algebras
The logic of perfect MV-algebras L. P. Belluce Department of Mathematics, British Columbia University, Vancouver, B.C. Canada. belluce@math.ubc.ca A. Di Nola DMI University of Salerno, Salerno, Italy adinola@unisa.it
More informationMathematica Slovaca. Ján Jakubík On the α-completeness of pseudo MV-algebras. Terms of use: Persistent URL:
Mathematica Slovaca Ján Jakubík On the α-completeness of pseudo MV-algebras Mathematica Slovaca, Vol. 52 (2002), No. 5, 511--516 Persistent URL: http://dml.cz/dmlcz/130365 Terms of use: Mathematical Institute
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 informationThe Blok-Ferreirim theorem for normal GBL-algebras and its application
The Blok-Ferreirim theorem for normal GBL-algebras and its application Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics
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 informationANNIHILATOR IDEALS IN ALMOST SEMILATTICE
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 7(2017), 339-352 DOI: 10.7251/BIMVI1702339R Former BULLETIN
More informationMV-algebras and fuzzy topologies: Stone duality extended
MV-algebras and fuzzy topologies: Stone duality extended Dipartimento di Matematica Università di Salerno, Italy Algebra and Coalgebra meet Proof Theory Universität Bern April 27 29, 2011 Outline 1 MV-algebras
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 informationComputers and Mathematics with Applications
Computers and Mathematics with Applications 58 (2009) 248 256 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Some
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 informationRINGS IN POST ALGEBRAS. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXVI, 2(2007), pp. 263 272 263 RINGS IN POST ALGEBRAS S. RUDEANU Abstract. Serfati [7] defined a ring structure on every Post algebra. In this paper we determine all the
More informationOn the filter theory of residuated lattices
On the filter theory of residuated lattices Jiří Rachůnek and Dana Šalounová Palacký University in Olomouc VŠB Technical University of Ostrava Czech Republic Orange, August 5, 2013 J. Rachůnek, D. Šalounová
More informationITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N ( ) 631
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 38 2017 (631 642) 631 n-fold (POSITIVE) IMPLICATIVE FILTERS OF HOOPS Chengfang Luo Xiaolong Xin School of Mathematics Northwest University Xi an 710127
More informationz -FILTERS AND RELATED IDEALS IN C(X) Communicated by B. Davvaz
Algebraic Structures and Their Applications Vol. 2 No. 2 ( 2015 ), pp 57-66. z -FILTERS AND RELATED IDEALS IN C(X) R. MOHAMADIAN Communicated by B. Davvaz Abstract. In this article we introduce the concept
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 informationThe variety generated by perfect BL-algebras: an algebraic approach in a fuzzy logic setting
Annals of Mathematics and Artificial Intelligence 35: 197 214, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands. The variety generated by perfect BL-algebras: an algebraic approach in
More informationThe prime spectrum of MV-algebras based on a joint work with A. Di Nola and P. Belluce
The prime spectrum of MV-algebras based on a joint work with A. Di Nola and P. Belluce Luca Spada Department of Mathematics and Computer Science University of Salerno www.logica.dmi.unisa.it/lucaspada
More informationContents. Introduction
Contents Introduction iii Chapter 1. Residuated lattices 1 1. Definitions and preliminaries 1 2. Boolean center of a residuated lattice 10 3. The lattice of deductive systems of a residuated lattice 14
More informationStrong Tensor Non-commutative Residuated Lattices
Strong Tensor Non-commutative Residuated Lattices Hongxing Liu Abstract In this paper, we study the properties of tensor operators on non-commutative residuated lattices. We give some equivalent conditions
More informationFleas and fuzzy logic a survey
Fleas and fuzzy logic a survey Petr Hájek Institute of Computer Science AS CR Prague hajek@cs.cas.cz Dedicated to Professor Gert H. Müller on the occasion of his 80 th birthday Keywords: mathematical fuzzy
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 informationMonadic GMV -algebras
Department of Algebra and Geometry Faculty of Sciences Palacký University of Olomouc Czech Republic TANCL 07, Oxford 2007 monadic structures = algebras with quantifiers = algebraic models for one-variable
More informationOn the lattice of congruence filters of a residuated lattice
Annals of University of Craiova, Math. Comp. Sci. Ser. Volume 33, 2006, Pages 174 188 ISSN: 1223-6934 On the lattice of congruence filters of a residuated lattice Raluca Creţan and Antoaneta Jeflea Abstract.
More informationINTRODUCING MV-ALGEBRAS. Daniele Mundici
INTRODUCING MV-ALGEBRAS Daniele Mundici Contents Chapter 1. Chang subdirect representation 5 1. MV-algebras 5 2. Homomorphisms and ideals 8 3. Chang subdirect representation theorem 11 4. MV-equations
More informationDirect Product of BF-Algebras
International Journal of Algebra, Vol. 10, 2016, no. 3, 125-132 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.614 Direct Product of BF-Algebras Randy C. Teves and Joemar C. Endam Department
More informationProbabilistic averaging in bounded commutative residuated l-monoids
Discrete Mathematics 306 (2006) 1317 1326 www.elsevier.com/locate/disc Probabilistic averaging in bounded commutative residuated l-monoids Anatolij Dvurečenskij a, Jiří Rachůnek b a Mathematical Institute,
More informationARCHIVUM MATHEMATICUM (BRNO) Tomus 48 (2012), M. Sambasiva Rao
ARCHIVUM MATHEMATICUM (BRNO) Tomus 48 (2012), 97 105 δ-ideals IN PSEUDO-COMPLEMENTED DISTRIBUTIVE LATTICES M. Sambasiva Rao Abstract. The concept of δ-ideals is introduced in a pseudo-complemented distributive
More informationClasses of Commutative Clean Rings
Classes of Commutative Clean Rings Wolf Iberkleid and Warren Wm. McGovern September 3, 2009 Abstract Let A be a commutative ring with identity and I an ideal of A. A is said to be I-clean if for every
More informationMODAL OPERATORS ON COMMUTATIVE RESIDUATED LATTICES. 1. Introduction
ao DOI: 10.2478/s12175-010-0055-1 Math. Slovaca 61 (2011), No. 1, 1 14 MODAL OPERATORS ON COMMUTATIVE RESIDUATED LATTICES M. Kondo (Communicated by Jiří Rachůnek ) ABSTRACT. We prove some fundamental properties
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 informationBoolean Algebras, Boolean Rings and Stone s Representation Theorem
Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to
More informationWEAK EFFECT ALGEBRAS
WEAK EFFECT ALGEBRAS THOMAS VETTERLEIN Abstract. Weak effect algebras are based on a commutative, associative and cancellative partial addition; they are moreover endowed with a partial order which is
More informationSoft set theoretical approach to residuated lattices. 1. Introduction. Young Bae Jun and Xiaohong Zhang
Quasigroups and Related Systems 24 2016, 231 246 Soft set theoretical approach to residuated lattices Young Bae Jun and Xiaohong Zhang Abstract. Molodtsov's soft set theory is applied to residuated lattices.
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 informationRestricted versions of the Tukey-Teichmüller Theorem that are equivalent to the Boolean Prime Ideal Theorem
Restricted versions of the Tukey-Teichmüller Theorem that are equivalent to the Boolean Prime Ideal Theorem R.E. Hodel Dedicated to W.W. Comfort on the occasion of his seventieth birthday. Abstract We
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationOn Homomorphism and Algebra of Functions on BE-algebras
On Homomorphism and Algebra of Functions on BE-algebras Kulajit Pathak 1, Biman Ch. Chetia 2 1. Assistant Professor, Department of Mathematics, B.H. College, Howly, Assam, India, 781316. 2. Principal,
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More informationFinite groups determined by an inequality of the orders of their elements
Publ. Math. Debrecen 80/3-4 (2012), 457 463 DOI: 10.5486/PMD.2012.5168 Finite groups determined by an inequality of the orders of their elements By MARIUS TĂRNĂUCEANU (Iaşi) Abstract. In this note we introduce
More informationThe Space of Maximal Ideals in an Almost Distributive Lattice
International Mathematical Forum, Vol. 6, 2011, no. 28, 1387-1396 The Space of Maximal Ideals in an Almost Distributive Lattice Y. S. Pawar Department of Mathematics Solapur University Solapur-413255,
More informationDUAL BCK-ALGEBRA AND MV-ALGEBRA. Kyung Ho Kim and Yong Ho Yon. Received March 23, 2007
Scientiae Mathematicae Japonicae Online, e-2007, 393 399 393 DUAL BCK-ALGEBRA AND MV-ALGEBRA Kyung Ho Kim and Yong Ho Yon Received March 23, 2007 Abstract. The aim of this paper is to study the properties
More informationFuzzy Dot Subalgebras and Fuzzy Dot Ideals of B-algebras
Journal of Uncertain Systems Vol.8, No.1, pp.22-30, 2014 Online at: www.jus.org.uk Fuzzy Dot Subalgebras and Fuzzy Dot Ideals of B-algebras Tapan Senapati a,, Monoranjan Bhowmik b, Madhumangal Pal c a
More informationON k-subspaces OF L-VECTOR-SPACES. George M. Bergman
ON k-subspaces OF L-VECTOR-SPACES George M. Bergman Department of Mathematics University of California, Berkeley CA 94720-3840, USA gbergman@math.berkeley.edu ABSTRACT. Let k L be division rings, with
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 informationOUTER MEASURES ON A COMMUTATIVE RING INDUCED BY MEASURES ON ITS SPECTRUM. Dariusz Dudzik, Marcin Skrzyński. 1. Preliminaries and introduction
Annales Mathematicae Silesianae 31 (2017), 63 70 DOI: 10.1515/amsil-2016-0020 OUTER MEASURES ON A COMMUTATIVE RING INDUCED BY MEASURES ON ITS SPECTRUM Dariusz Dudzik, Marcin Skrzyński Abstract. On a commutative
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 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 information8. Distributive Lattices. Every dog must have his day.
8. Distributive Lattices Every dog must have his day. In this chapter and the next we will look at the two most important lattice varieties: distributive and modular lattices. Let us set the context for
More informationCHAPTEER - TWO SUBGROUPS. ( Z, + ) is subgroup of ( R, + ). 1) Find all subgroups of the group ( Z 8, + 8 ).
CHAPTEER - TWO SUBGROUPS Definition 2-1. Let (G, ) be a group and H G be a nonempty subset of G. The pair ( H, ) is said to be a SUBGROUP of (G, ) if ( H, ) is group. Example. ( Z, + ) is subgroup of (
More informationD. S. Passman. University of Wisconsin-Madison
ULTRAPRODUCTS AND THE GROUP RING SEMIPRIMITIVITY PROBLEM D. S. Passman University of Wisconsin-Madison Abstract. This expository paper is a slightly expanded version of the final talk I gave at the group
More informationAN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC
Iranian Journal of Fuzzy Systems Vol. 9, No. 6, (2012) pp. 31-41 31 AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC E. ESLAMI Abstract. In this paper we extend the notion of degrees of membership
More informationA New Characterization of Boolean Rings with Identity
Irish Math. Soc. Bulletin Number 76, Winter 2015, 55 60 ISSN 0791-5578 A New Characterization of Boolean Rings with Identity PETER DANCHEV Abstract. We define the class of nil-regular rings and show that
More informationTROPICAL SCHEME THEORY
TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),
More informationEQ-algebras: primary concepts and properties
UNIVERSITY OF OSTRAVA Institute for Research and Applications of Fuzzy Modeling EQ-algebras: primary concepts and properties Vilém Novák Research report No. 101 Submitted/to appear: Int. Joint, Czech Republic-Japan
More informationChapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples
Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter
More informationStandard Ideals in BCL + Algebras
Journal of Mathematics Research; Vol. 8, No. 2; April 2016 SSN 1916-9795 E-SSN 1916-9809 Published by Canadian Center of Science and Education Standard deals in BCL + Algebras Yonghong Liu School of Automation,
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 informationThe Square of Opposition in Orthomodular Logic
The Square of Opposition in Orthomodular Logic H. Freytes, C. de Ronde and G. Domenech Abstract. In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative,
More informationGENERALIZED DIFFERENCE POSETS AND ORTHOALGEBRAS. 0. Introduction
Acta Math. Univ. Comenianae Vol. LXV, 2(1996), pp. 247 279 247 GENERALIZED DIFFERENCE POSETS AND ORTHOALGEBRAS J. HEDLÍKOVÁ and S. PULMANNOVÁ Abstract. A difference on a poset (P, ) is a partial binary
More informationLogic via Algebra. Sam Chong Tay. A Senior Exercise in Mathematics Kenyon College November 29, 2012
Logic via Algebra Sam Chong Tay A Senior Exercise in Mathematics Kenyon College November 29, 2012 Abstract The purpose of this paper is to gain insight to mathematical logic through an algebraic perspective.
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 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 informationL fuzzy ideals in Γ semiring. M. Murali Krishna Rao, B. Vekateswarlu
Annals of Fuzzy Mathematics and Informatics Volume 10, No. 1, (July 2015), pp. 1 16 ISSN: 2093 9310 (print version) ISSN: 2287 6235 (electronic version) http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com
More informationarxiv: v1 [math-ph] 23 Jul 2010
EXTENSIONS OF WITNESS MAPPINGS GEJZA JENČA arxiv:1007.4081v1 [math-ph] 23 Jul 2010 Abstract. We deal with the problem of coexistence in interval effect algebras using the notion of a witness mapping. Suppose
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 informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationSome Characterizations of 0-Distributive Semilattices
BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY http:/math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 37(4) (2014), 1103 1110 Some Characterizations of 0-Distributive Semilattices 1 H. S.
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato June 23, 2015 This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato September 10, 2015 ABSTRACT. This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a
More informationJoseph Muscat Universal Algebras. 1 March 2013
Joseph Muscat 2015 1 Universal Algebras 1 Operations joseph.muscat@um.edu.mt 1 March 2013 A universal algebra is a set X with some operations : X n X and relations 1 X m. For example, there may be specific
More informationNeighborhood spaces and convergence
Volume 35, 2010 Pages 165 175 http://topology.auburn.edu/tp/ Neighborhood spaces and convergence by Tom Richmond and Josef Šlapal Electronically published on July 14, 2009 Topology Proceedings Web: http://topology.auburn.edu/tp/
More informationIntroduction to Neutrosophic BCI/BCK-Algebras
Introduction to Neutrosophic BCI/BCK-Algebras A.A.A. Agboola 1 and B. Davvaz 2 1 Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria aaaola2003@yahoo.com 2 Department of Mathematics,
More informationA CLASS OF INFINITE CONVEX GEOMETRIES
A CLASS OF INFINITE CONVEX GEOMETRIES KIRA ADARICHEVA AND J. B. NATION Abstract. Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly
More informationLATTICE BASIS AND ENTROPY
LATTICE BASIS AND ENTROPY Vinod Kumar.P.B 1 and K.Babu Joseph 2 Dept. of Mathematics Rajagiri School of Engineering & Technology Rajagiri Valley.P.O, Cochin 682039 Kerala, India. ABSTRACT: We introduce
More informationInternational 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 informationUniversal Algebra for Logics
Universal Algebra for Logics Joanna GRYGIEL University of Czestochowa Poland j.grygiel@ajd.czest.pl 2005 These notes form Lecture Notes of a short course which I will give at 1st School on Universal Logic
More informationGeneralized Fuzzy Ideals of BCI-Algebras
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 32(2) (2009), 119 130 Generalized Fuzzy Ideals of BCI-Algebras 1 Jianming Zhan and
More informationStat 451: Solutions to Assignment #1
Stat 451: Solutions to Assignment #1 2.1) By definition, 2 Ω is the set of all subsets of Ω. Therefore, to show that 2 Ω is a σ-algebra we must show that the conditions of the definition σ-algebra are
More informationMV -ALGEBRAS ARE CATEGORICALLY EQUIVALENT TO A CLASS OF DRl 1(i) -SEMIGROUPS. (Received August 27, 1997)
123 (1998) MATHEMATICA BOHEMICA No. 4, 437 441 MV -ALGEBRAS ARE CATEGORICALLY EQUIVALENT TO A CLASS OF DRl 1(i) -SEMIGROUPS Jiří Rachůnek, Olomouc (Received August 27, 1997) Abstract. In the paper it is
More informationOn z -ideals in C(X) F. A z a r p a n a h, O. A. S. K a r a m z a d e h and A. R e z a i A l i a b a d (Ahvaz)
F U N D A M E N T A MATHEMATICAE 160 (1999) On z -ideals in C(X) by F. A z a r p a n a h, O. A. S. K a r a m z a d e h and A. R e z a i A l i a b a d (Ahvaz) Abstract. An ideal I in a commutative ring
More informationRESIDUATION SUBREDUCTS OF POCRIGS
Bulletin of the Section of Logic Volume 39:1/2 (2010), pp. 11 16 Jānis Cīrulis RESIDUATION SUBREDUCTS OF POCRIGS Abstract A pocrig (A,,, 1) is a partially ordered commutative residuated integral groupoid.
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 informationCorrect classes of modules
Algebra and Discrete Mathematics Number?. (????). pp. 1 13 c Journal Algebra and Discrete Mathematics RESEARCH ARTICLE Correct classes of modules Robert Wisbauer Abstract. For a ring R, call a class C
More information@FMI c Kyung Moon Sa Co.
Annals of Fuzzy Mathematics and Informatics Volume 1, No. 2, April 2011, pp. 163-169 ISSN 2093 9310 http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com Semicompactness in L-fuzzy topological
More informationMathematica Slovaca. Jan Kühr Prime ideals and polars in DRl-monoids and BL-algebras. Terms of use: Persistent URL:
Mathematica Slovaca Jan Kühr Prime ideals and polars in DRl-monoids and BL-algebras Mathematica Slovaca, Vol. 53 (2003), No. 3, 233--246 Persistent URL: http://dml.cz/dmlcz/136885 Terms of use: Mathematical
More informationChapter 0. Introduction: Prerequisites and Preliminaries
Chapter 0. Sections 0.1 to 0.5 1 Chapter 0. Introduction: Prerequisites and Preliminaries Note. The content of Sections 0.1 through 0.6 should be very familiar to you. However, in order to keep these notes
More informationEquational Logic. Chapter Syntax Terms and Term Algebras
Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from
More informationPseudo-BCK algebras as partial algebras
Pseudo-BCK algebras as partial algebras Thomas Vetterlein Institute for Medical Expert and Knowledge-Based Systems Medical University of Vienna Spitalgasse 23, 1090 Wien, Austria Thomas.Vetterlein@meduniwien.ac.at
More informationAN AXIOMATIC FORMATION THAT IS NOT A VARIETY
AN AXIOMATIC FORMATION THAT IS NOT A VARIETY KEITH A. KEARNES Abstract. We show that any variety of groups that contains a finite nonsolvable group contains an axiomatic formation that is not a subvariety.
More informationContents. Index... 15
Contents Filter Bases and Nets................................................................................ 5 Filter Bases and Ultrafilters: A Brief Overview.........................................................
More informationON POSITIVE, LINEAR AND QUADRATIC BOOLEAN FUNCTIONS
Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 177 187. ON POSITIVE, LINEAR AND QUADRATIC BOOLEAN FUNCTIONS SERGIU RUDEANU Abstract. In [1], [2] it was proved that a function f : {0,
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 informationVARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES
Bull. Austral. Math. Soc. 78 (2008), 487 495 doi:10.1017/s0004972708000877 VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES CAROLYN E. MCPHAIL and SIDNEY A. MORRIS (Received 3 March 2008) Abstract
More information