ON THREE-VALUED MOISIL ALGEBRAS. Manuel ARAD and Luiz MONTEIRO

Size: px
Start display at page:

Download "ON THREE-VALUED MOISIL ALGEBRAS. Manuel ARAD and Luiz MONTEIRO"

Transcription

1 ON THREE-VALUED MOISIL ALGEBRAS Manuel ARAD and Luiz MONTEIRO In this paper we investigate the properties of the family I(A) of all intervals of the form [x, x] with x x, in a De Morgan algebra A, and we obtain a necessary and sufficient condition for a complete De Morgan algebra to be a Kleene algebra in terms of I(A) (Lemma 11). We prove that I(A) is a complete Boolean algebra if A is a complete Moisi! algebra (Lemma 12). Let A be a distributive lattice which is bounded, that is, has a largest element 1 and a smallest element O. We use x V y for the join, and x Ay fo r the meet of two elements x, y in A. I f is a unary operation defined on A satisfying x = x and (x y ) = x A - A is called a De Morgan algebra. The set o f all complemented elements of A is noted by B (A). A Kleene algebra is a De Morgan algebra satisfying: x A - algebra is a system (L, 1,, A, V, V) such that (L, 1,, A, V) is a De Morgan algebra and V is a unary operation (called possibility operation) defined on A with the properties: xv Vx = 1, x A - - The necessity operation A is defined by Ax = x. L is called a centered three-valued Lukasiewicz algebra, o r a three-valued Post algebra, if it has a center, that is, an element c of L such that c c. The center of L (if it exists) is unique [6], [7], [8], [12], [15]. An axis of a three-valued Lukasiewicz algebra L is an element e of L with the properties: Ae = 0 and V x A x v Ve for all x of L. Again, if the axis of L exists, it is unique. Following A. Monteiro, L is called a t h r e e of three valued Moisil algebras we refer to [12], [13], [15]. Let E be a three-valued Moisil algebra, and e E E the axis of E. In [12] L. Monteiro proved that: (1) x ( A x ye) A Vx, for all x ee, and (2) x = (Ax - Let us introduce the following set:

2 408 M A N U E L ABAD and LUIZ MONTEIRO S = { t E E : x = (Ax t ) A Vx, for all x EE}. From (I) and (2) it is clear that e and e belong to S. On the other hand, we know that Ae = 0 and this is equivalent to say that e e, thus the interval [e, e LEMMA 1. S = [e, e Proof. If y [ e, e], e y e, then we have x = (Ax e ) A Ve (A x V y ) A V x ( A x (Ax y ) A Vx for ail x ee, whence y If s ES, then x = (A x Vs) A Vx for all x EE. Then e s, for e = (A e Vs ) A V e =- ( 0 Vs ) A V e = s A Ve s, a n d s (A e Vs ) A V e = (A e Vs ) A 1 = A e V s It is well known that S = [e, e] is a distributive lattice, the elements e and e being the least and greastest elements o f S respectively [4]. I t is not difficult to see that (S, e,, A, V) is a Kleene algebra. Furthemore, S is a Boolean algebra. For, if s ES, then e ES and t = s A s ES, then (i) x = (Ax ( s A x EE. Since (ii) At = A(s A s) =- As A A s 0, from (i) and (ii) from definition and uniqueness of the axis, we have (iii) s A whence (iv) s s = e. So S is a Boolean algebra. Now we define the following mapping f: E > [e, el, f(x) = ( e A x) v e, for each x LEMMA 2. The mapping f has the following properties. Fl) f (l) = e ; F2) f(0) = e; F3) f(x A y) = fix) A f(y); F4) f( x) = f(x); F5) f(x y ) = f(x) v f(y); F6) f(s) = s if and only if S [ e, e From Fi to F4, it follows that f is an M-homomorphism (homomorphism between Morgan algebras). From F6, it follows that f is an M-epimorphism. L. Monteiro [12] proved that if E is an algebra with axis e, then E is isomorphic to the direct product of a Boolean algebra B and a centered three-valued Lukasiewicz algebra C. More precisely, B i s t h e principal ideal generated b y Ve, I( ve) = {x EE: x y e }, and C is I(Ve)=-

3 ON THREE-VALUED MOISIL ALGEBRAS LEMMA 3. I( Ve) and S are isomorphic Boolean algebras. Proof. The correspondence h(x) = x A Ye sets up an M-epimorphism from E onto l( ye). I t is an easy verification that Ker f = Ker h, which completes the proof. LEMMA 4. A t h r e e bra if S = L : x = (Ax Vs) A Vx, for all x cl }, is a nonvoid set. Proof. It is clear that S is closed under A and V. I f s es, then x ( x Vs) A V x, for all x G L, hence x = (Y x A s) V A x for all x e L a n d then s G S. F ro m th is, s A s S f o r s ES. B u t A(s A s) 0 and x = (Ax ( s A s)) A Vx f or all x EL, therefore e = s A s is the axis of algebra L. From lemma I, S = [e, e We remark that L has a center c if and only if S = tel. For if L has a center c, we know that c is also an axis [12] and then S [ c. cl [c, c] = {e an axis of L and moreover c e S = Icl, thus c = c and so c is the center of L. We now deal with the dual concept of ideal. Recall that a filter is a nonvoid subset F of a lattice L with the properties: a ef, x E L, a imply x EF, and a EF, b e F imply a Ab CF. Giv en an element a in any lattice L, the set F(a) = E L : a LÇ,. x i s evidently a filter; it is called a principal filter of L. LEMMA 5. Let E be a three-valued Moisil algebra, e its axis. Then the ordered sets F(e), I( e) and B(E) are isomorphic. Proof. Define H: F(e) B ( E ) by H(x) = Ax, and G: F(e) ) R e) by G(x) = x. DEFINITION 6. Let (A, I,, A, V) be any De Morgan algebra, x Yo elements of A such that x be Boolean if the system (A, y

4 410 M A N U E L ABAD and LUTZ MONTEIRO EXAMPLES. - x 0 a 1 d Fig. I - x 0 a 1 d d In the algebra of fig. 1, [a, d] is Boolean, and [0, 1] is not Boolean. In the algebra of fig. 2, [a, e] with its natural ordering is a Boolean algebra, but it is not a Boolean interval because c = c *d. The only Boolean intervals are [c, cl and [d, dl. REMARK. I f S [ x x A -x = x SgS, and conversely, if [x

5 ON THREE-VALUED MOISIL ALGEBRAS belong to [x every Boolean interval can be written as [ x However, an interval of this kind need not be Boolean as we can see for [0, o f the examples above. It should pointed out that a De Morgan algebra need not have Boolean intervals, as the following example shows. EXAMPLE. Let T { 0, c, 1}, with 0 < c < 1, and 0 = 1, c = c, 1 = O. T is a Kleene algebra, and if N is the set of all positive integers, it is readily verified that A = TN is a Kleene algebra with pointwise operations. Let K and K2 the set of all f e TN such that there exists a nonvoid finite subset F of N with f(x) = c for all x ef, and f(x) { 0, 1} if x F. Clearly K a proper Kleene subalgebra of A. Suppose [f, I] is a Boolean interval of K. Since f f, then f(x) BO, cl for all x en, so there exists a finite subset F of N such that f(x) c for all x e F and f(y) = 0 if y ÊF. Let i if x EF f i b A b = b w h i c h contradicts that [f, f] is Boolean. LEMMA 7 I f A is a Kleene algebra and there exists a Boolean interval in A, it is unique. Proof. Let S Since A is a Kleene algebra: (1) e = e A e v f = f. Then e v f f Vf = f, hence f e vf f, so e V f 0 S (e Vt) A (e vt) = f. By (I), t - - e then e f V e e v e e, hence e V f S and e = f. Then S Denote the set of intervals of a De Morgan algebra A of the form [x, xl with x x by I(A). I(A) 4-0 seeing that A = [0, I ] ei(a). Notice that I (A) = t x A - x, x - x l, x c A l. LEMMA 8. Any minimal element o f the ordered set (I(A), g ), whenever it exists, is a Boolean interval.

6 412 M A N U E L ABAD and LUIZ MONTEIRO Proof. Let S = [xo, x01 be minimal element of I(A). Clearly xo t A t x S t A t = xo for every t ES- Thus Sis Boolean. LEMMA 9. I f A is a complete De Morgan algebra, then the ordered set I(A) has the Zom's property for descending chains. Proof. Let C = {S Since A is complete, there exists the element x = x that x x x x [ t x that x COROLLARY 10. Any complete De Morgan algebra has Boolean intervals. LEMMA 11. A complete De Morgan algebra A is a Kleene algebra if and only if I(A) has least element. Proof. By the above lemmas, if A is a Kleene algebra, it has a unique Boolean interval, which is the least element in I(A). For the converse let P = [b, b] be the least element in I(A). Then, Pg [x A x, x V x] for all x EA. Since b EP, then x A x b for all x EA and b y V y fo r y EA. Therefore x A x y V y fo r x, y EA, that is, A is a Kleene algebra. (Note that in the converse, the completeness of A has not been used). Let A be Kleene algebra. Define the following binary operations on I(A): [x, x] v [y, y] ---- [x Ay, x V y Tx, xl A [y, y] = [x Vy, x A From Kleene condition, it follows at once that [x Ay, x y ] and [x v y, x A y] belong to I(A), if x x and y 5.. y.

7 ON THREE-VALUED MOBIL ALGEBRAS Moreover, it is easy to check that (I(A), A, A, V) is a distributive lattice with greatest element A. In fact, [x, x] A [y, y] [ x, x] if and only if [x, x] g [y, y] for [x, x] and [y, y] E I(A). Observe that if A is a complete three-valued Moisil algebra, then I(A) has least element [e, e Indeed, we know that [x, x] L e, e] if and only if x e l and x [ e, e [e, e element of I(A) is Boolean, and since [e, el is the unique Boolean interval of A, thus [e, e] is the unique minimal interval of A. Then [e, e LEMMA 12. I f A is a complete Mo isil algebra, then I (A ) is a complete Boolean algebra. Proof. To prove that for all [x, x] I ( A ) there exists [y, y] el(a) such that [x, x] A [y, --y] = [e, e] and [x, x] v [y, y] = [0, I] it is equivalent to prove that for all x e there exists y e such that x Vy = e and x Ay = O. Let us put y = A x Ae. Then ( i) x A(A x A e) 0 because x A A x O. (ii) x V(A x Ae) = e. I n d e e d, A ( x V (A x Ae)) = Ax V (A x A Ae) = A x = 0 = A e a n d 7 ( x v (A x Ac)) Vx V (A x A ye) ( V x V A x) A (Vx V Ve) = (Vx v A x) A Ve = I A Ve = Ve, and by determination principle (see [12] and [141) (ii) holds. The completeness of 1(A) follows from n [xi, -x i] =, Y Observe that the mapping H: F( e) I ( A ) defined by H( y) = [y, y], for all element y E R e), is an order isomorphism. Indeed, H is clearly a bijection and y y ' if and only if H( y) Thus we can state COROLLARY 13. I f A is a three-valued Post algebra such that I(A) is a complete Boolean algebra, then A is complete. Proof. By the above remark, if 1(A) is a complete Boolean algebra,

8 414 M A N U E L ABAD and LUTZ MONTEIRO then F( e) is a complete Boolean algebra. But in a three-valued Post algebra, e = e, thus F( e) = F(e), which is isomorphic (Lemma 5) to the set B(A) of all Boolean elements of A. Then B(A) is complete, and this is sufficient (see t12]) to assure that A is complete. Universidad Nacional del Comahue M a n u e l ABAD Universidad Nacional del Sur Av. Alem Bahia Blanca Argentina Luiz MO NTEI RO REFERENCES Bathes R. and Dwinger P., Distributive Lattices, University of Missouri Press, Columbia, Missouri 65201, (1974). [2] Bialynicki-Birula A., Remarks on quasi-boolean algebras. Bull. Acad. PoIon. Sci. Cl. III 5(1957), [3] Bialynicki-Birula A. and Rasiowa H., On the representation o f quasi-boolean algebras. Bull. Acad. Polon. Sci. Cl. III 5(1957) [4] Birkhoff G., Lattice Theory, Amer. Math. Soc. Colloq. Pub., 25, 3' dence (1967). [51 Kalman J.. Lattices with involution, Trans. Amer. Math. Soc. 87 (1958), [6] Moisil Gr. C., Recherches sur les logiques non-chrysippiennes, Ann. Sci. Univ. Jassy. 26 (1940), [71 Moisi! Gr.C., Sur les anneaux de caractéristique 2 ou 3 et leurs applications de l'ecole Polytechnique de Bucarest 12 (1941), [8] Moi sil Gr.C., Sur les ideaux des algèbres Lukasiewicziennes trivalentes, Analele Universitatti Cl. Parhon. Seria Acta Logica. 3 (1960), [91 Montciro A., Matrices de Morgan caractéristiques pour h classique, An. Acad. Brasil. Ci. 32 (1960), 1-7. [10] Monteiro A., Algebras de Ltikasiewicz trivalentes. Lectures given at the Univ. Nac. del Sur, Bahia Blanca (1963). [111 Monteiro A., Sur la définition des algèbres de Lukasiewicz trivalentes, Bull. Math. Soc. Sci Math. Phy. R.P. Roum_ 7 (55)(1963), Monteiro L., Algebras de Lukasiewicz trivalentes montidicas, Notas de LOgica Matematica N" 32, Univ. Nac. del Sur, Bahia Blanca (1974). 1 1 et Analyse, 79 (1977), N" 9-10 (1%8), [15] Varlet J.C., Considerations sur les algèbres de Lukasiewicz trivalentes, Bull. Soc. Roy. Sc. Liege. N" 9-10 (1%9)

INJECTIVE DE MORGAN AND KLEENE ALGEBRAS1

INJECTIVE DE MORGAN AND KLEENE ALGEBRAS1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 2, February 1975 INJECTIVE DE MORGAN AND KLEENE ALGEBRAS1 ROBERTO CIGNOLI ABSTRACT. The aim of this paper is to characterize the injective

More information

defined on A satisfying axioms L 1), L 2), and L 3). Then (A, 1, A, In 1 we introduce an alternative definition o Lukasiewicz L 3) V(xA y) VxA Vy.

defined on A satisfying axioms L 1), L 2), and L 3). Then (A, 1, A, In 1 we introduce an alternative definition o Lukasiewicz L 3) V(xA y) VxA Vy. 676 [Vol. 41, 147. Boolean Elements in Lukasiewicz Algebras. II By Roberto CIGNOLI and Antonio MONTEIR0 Instituto de Matem,tica Universidad Nacional del Sur, Bahia Blanca, Argentina (Comm. by Kinjir6 Kuo,.J.A.,

More information

A note on 1-l3-algebras

A note on 1-l3-algebras Revista Colombiana de Matematicas Volumen 31 (1997), paginas 1 5-196 A note on 1-l3-algebras SERGIO A. CELANI Universidad Nacional del Centro, Tandil, ARGENTINA ABSTRACT. In this note we shall generalize

More information

arxiv: v2 [math.lo] 12 May 2013

arxiv: v2 [math.lo] 12 May 2013 arxiv:1304.6776v2 [math.lo] 12 May 2013 Several characterizations of the 4 valued modal algebras Aldo V. Figallo and Paolo Landini Abstract A. Monteiro, in 1978, defined the algebras he named tetravalent

More information

Mathematica Bohemica

Mathematica Bohemica Mathematica Bohemica A. V. Figallo; Inés Pascual; Alicia Ziliani Notes on monadic n-valued Łukasiewicz algebras Mathematica Bohemica, Vol. 129 (2004), No. 3, 255 271 Persistent URL: http://dml.cz/dmlcz/134149

More information

RINGS IN POST ALGEBRAS. 1. Introduction

RINGS 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 information

Computation of De Morgan and Quasi-De Morgan Functions

Computation of De Morgan and Quasi-De Morgan Functions Computation of De Morgan and Quasi-De Morgan Functions Yu.M. Movsisyan Yerevan State University Yerevan, Armenia e-mail: yurimovsisyan@yahoo.com V.A. Aslanyan Yerevan State University Yerevan, Armenia

More information

Implicational classes ofde Morgan Boolean algebras

Implicational classes ofde Morgan Boolean algebras Discrete Mathematics 232 (2001) 59 66 www.elsevier.com/locate/disc Implicational classes ofde Morgan Boolean algebras Alexej P. Pynko Department of Digital Automata Theory, V.M. Glushkov Institute of Cybernetics,

More information

ARCHIVUM MATHEMATICUM (BRNO) Tomus 48 (2012), M. Sambasiva Rao

ARCHIVUM 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 information

Tense Operators on m Symmetric Algebras

Tense Operators on m Symmetric Algebras International Mathematical Forum, Vol. 6, 2011, no. 41, 2007-2014 Tense Operators on m Symmetric Algebras Aldo V. Figallo Universidad Nacional de San Juan. Instituto de Ciencias Básicas Avda. I. de la

More information

A strongly rigid binary relation

A strongly rigid binary relation A strongly rigid binary relation Anne Fearnley 8 November 1994 Abstract A binary relation ρ on a set U is strongly rigid if every universal algebra on U such that ρ is a subuniverse of its square is trivial.

More information

THEOREM FOR DISTRIBUTIVE LATTICES

THEOREM FOR DISTRIBUTIVE LATTICES Re ista de la Un 6n Matem&tica Argentina Vo umen 25, 1971. A HAH~-BANACH THEOREM FOR DISTRIBUTIVE LATTICES by Roberto Cignoli The familiar liahn-banach theorem for real linear spaces ([5],p 9) is still

More information

ANNIHILATOR IDEALS IN ALMOST SEMILATTICE

ANNIHILATOR 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 information

Finite pseudocomplemented lattices: The spectra and the Glivenko congruence

Finite pseudocomplemented lattices: The spectra and the Glivenko congruence Finite pseudocomplemented lattices: The spectra and the Glivenko congruence T. Katriňák and J. Guričan Abstract. Recently, Grätzer, Gunderson and Quackenbush have characterized the spectra of finite pseudocomplemented

More information

Given a lattice L we will note the set of atoms of L by At (L), and with CoAt (L) the set of co-atoms of L.

Given a lattice L we will note the set of atoms of L by At (L), and with CoAt (L) the set of co-atoms of L. ACTAS DEL VIII CONGRESO DR. ANTONIO A. R. MONTEIRO (2005), Páginas 25 32 SOME REMARKS ON OCKHAM CONGRUENCES LEONARDO CABRER AND SERGIO CELANI ABSTRACT. In this work we shall describe the lattice of congruences

More information

Lukasiewicz Implication Prealgebras

Lukasiewicz Implication Prealgebras Annals of the University of Craiova, Mathematics and Computer Science Series Volume 44(1), 2017, Pages 115 125 ISSN: 1223-6934 Lukasiewicz Implication Prealgebras Aldo V. Figallo and Gustavo Pelaitay Abstract.

More information

Congruence Coherent Symmetric Extended de Morgan Algebras

Congruence Coherent Symmetric Extended de Morgan Algebras T.S. Blyth Jie Fang Congruence Coherent Symmetric Extended de Morgan Algebras Abstract. An algebra A is said to be congruence coherent if every subalgebra of A that contains a class of some congruence

More information

Disjointness conditions in free products of. distributive lattices: An application of Ramsay's theorem. Harry Lakser< 1)

Disjointness conditions in free products of. distributive lattices: An application of Ramsay's theorem. Harry Lakser< 1) Proc. Univ. of Houston Lattice Theory Conf..Houston 1973 Disjointness conditions in free products of distributive lattices: An application of Ramsay's theorem. Harry Lakser< 1) 1. Introduction. Let L be

More information

13-V ALGEBRAS. Aldo V. FIGALLO. VoL XVII (]983), pag Rev-Uta Colomb~Yl.a de Matema.u.c..M

13-V ALGEBRAS. Aldo V. FIGALLO. VoL XVII (]983), pag Rev-Uta Colomb~Yl.a de Matema.u.c..M Rev-Uta Colomb~Yl.a de Matema.u.c..M VoL XVII (]983), pag-6. 105-116 13-V ALGEBRAS by Aldo V. FIGALLO 1. In t roducc f on. In this note we present an algebraic study of a fragment of the three-valued propositional

More information

Bivalent Semantics for De Morgan Logic (The Uselessness of Four-valuedness)

Bivalent Semantics for De Morgan Logic (The Uselessness of Four-valuedness) Bivalent Semantics for De Morgan Logic (The Uselessness of Four-valuedness) Jean-Yves Béziau Dedicated to Newton da Costa for his 79th birthday abstract. In this paper we present a bivalent semantics for

More information

FREE SYMMETRIC BOOLEAN ALGEBRAS. Manuel Abad and Luiz Monteiro

FREE SYMMETRIC BOOLEAN ALGEBRAS. Manuel Abad and Luiz Monteiro Revista de la Unión Matemática Argentina Volumen 27, 1976. FREE SYMMETRIC BOOLEAN ALGEBRAS * Manuel Abad and Luiz Monteiro Our purpose is to give a construction of the symmetric Boolean algebras with a

More information

DUALITY FOR BOOLEAN EQUALITIES

DUALITY FOR BOOLEAN EQUALITIES DUALITY FOR BOOLEAN EQUALITIES LEON LEBLANC Introduction. Boolean equalities occur naturally in Logic and especially in the theory of polyadic algebras. For instance, they have been used in [2] under the

More information

MATRIX LUKASIEWICZ ALGEBRAS

MATRIX LUKASIEWICZ ALGEBRAS Bulletin of the Section of Logic Volume 3:3/4 (1974), pp. 9 14 reedition 2012 [original edition, pp. 9 16] Wojciech Suchoń MATRIX LUKASIEWICZ ALGEBRAS This paper was presented at the seminar of the Department

More information

arxiv: v3 [math.rt] 19 Feb 2011

arxiv: v3 [math.rt] 19 Feb 2011 REPRESENTATION OF NELSON ALGEBRAS BY ROUGH SETS DETERMINED BY QUASIORDERS JOUNI JÄRVINEN AND SÁNDOR RADELECZKI arxiv:1007.1199v3 [math.rt] 19 Feb 2011 Abstract. In this paper, we show that every quasiorder

More information

Q-LINEAR FUNCTIONS, FUNCTIONS WITH DENSE GRAPH, AND EVERYWHERE SURJECTIVITY

Q-LINEAR FUNCTIONS, FUNCTIONS WITH DENSE GRAPH, AND EVERYWHERE SURJECTIVITY MATH. SCAND. 102 (2008), 156 160 Q-LINEAR FUNCTIONS, FUNCTIONS WITH DENSE GRAPH, AND EVERYWHERE SURJECTIVITY F. J. GARCÍA-PACHECO, F. RAMBLA-BARRENO, J. B. SEOANE-SEPÚLVEDA Abstract Let L, S and D denote,

More information

Averaging Operators on the Unit Interval

Averaging Operators on the Unit Interval Averaging Operators on the Unit Interval Mai Gehrke Carol Walker Elbert Walker New Mexico State University Las Cruces, New Mexico Abstract In working with negations and t-norms, it is not uncommon to call

More information

Houston Journal of Mathematics. c 2004 University of Houston Volume 30, No. 4, 2004

Houston Journal of Mathematics. c 2004 University of Houston Volume 30, No. 4, 2004 Houston Journal of Mathematics c 2004 University of Houston Volume 30, No. 4, 2004 MACNEILLE COMPLETIONS OF HEYTING ALGEBRAS JOHN HARDING AND GURAM BEZHANISHVILI Communicated by Klaus Kaiser Abstract.

More information

Uni\JHRsitutis-ZcientioRurn

Uni\JHRsitutis-ZcientioRurn Separatllm. -f~nn~l &)- Uni\JHRsitutis-ZcientioRurn, Budap6SHn 1lSis- -de- Rolando-Hotuos-nominatmt SECTIO MATHE~lATICA TOMUS I. G. GRATZER and E. T. SCHMIDT TWO NOTES ON LATTICE-CONGRUENCES TWO NOTES

More information

DISTRIBUTIVE, STANDARD AND NEUTRAL ELEMENTS IN TRELLISES. 1. Introduction

DISTRIBUTIVE, STANDARD AND NEUTRAL ELEMENTS IN TRELLISES. 1. Introduction Acta Math. Univ. Comenianae Vol. LXXVII, 2 (2008), pp. 167 174 167 DISTRIBUTIVE, STANDARD AND NEUTRAL ELEMENTS IN TRELLISES SHASHIREKHA B. RAI Abstract. In this paper, the concepts of distributive, standard

More information

Related structures with involution

Related structures with involution Related structures with involution R. Pöschel 1 and S. Radeleczki 2 1 Institut für Algebra, TU Dresden, 01062 Dresden, Germany e-mail: Reinhard.Poeschel@tu-dresden.de 2 Institute of Mathematics, University

More information

8. Distributive Lattices. Every dog must have his day.

8. 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 information

LATTICE AND BOOLEAN ALGEBRA

LATTICE AND BOOLEAN ALGEBRA 2 LATTICE AND BOOLEAN ALGEBRA This chapter presents, lattice and Boolean algebra, which are basis of switching theory. Also presented are some algebraic systems such as groups, rings, and fields. 2.1 ALGEBRA

More information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2014 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element in the set

More information

ORTHOPOSETS WITH QUANTIFIERS

ORTHOPOSETS WITH QUANTIFIERS Bulletin of the Section of Logic Volume 41:1/2 (2012), pp. 1 12 Jānis Cīrulis ORTHOPOSETS WITH QUANTIFIERS Abstract A quantifier on an orthoposet is a closure operator whose range is closed under orthocomplementation

More information

GENERALIZED DIFFERENCE POSETS AND ORTHOALGEBRAS. 0. Introduction

GENERALIZED 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 information

Properties of Boolean Algebras

Properties of Boolean Algebras Phillip James Swansea University December 15, 2008 Plan For Today Boolean Algebras and Order..... Brief Re-cap Order A Boolean algebra is a set A together with the distinguished elements 0 and 1, the binary

More information

FREE NON-DISTRIBUTIVE MORGAN-STONE ALGEBRAS

FREE NON-DISTRIBUTIVE MORGAN-STONE ALGEBRAS NEW ZEALAND JOURNAL OF MATHEMATICS Volume 25 (1996), 85-94 FREE NON-DISTRIBUTIVE MORGAN-STONE ALGEBRAS D a n i e l S e v c o v i c (Received July 1994) Abstract. In this paper we investigate free non-distributive

More information

Tense Operators on Basic Algebras

Tense Operators on Basic Algebras Int J Theor Phys (2011) 50:3737 3749 DOI 10.1007/s10773-011-0748-4 Tense Operators on Basic Algebras M. Botur I. Chajda R. Halaš M. Kolařík Received: 10 November 2010 / Accepted: 2 March 2011 / Published

More information

Characterizing the Equational Theory

Characterizing the Equational Theory Introduction to Kleene Algebra Lecture 4 CS786 Spring 2004 February 2, 2004 Characterizing the Equational Theory Most of the early work on Kleene algebra was directed toward characterizing the equational

More information

ON GENERATING DISTRIBUTIVE SUBLATTICES OF ORTHOMODULAR LATTICES

ON GENERATING DISTRIBUTIVE SUBLATTICES OF ORTHOMODULAR LATTICES PKOCkLDINGS OF Tlik AM1 KlCAN MATIiL'MATlCAL SOCIETY Vulume 67. Numher 1. November 1977 ON GENERATING DISTRIBUTIVE SUBLATTICES OF ORTHOMODULAR LATTICES RICHARD J. GREECHIE ABSTRACT.A Foulis-Holland set

More information

A logic for rough sets

A logic for rough sets A logic for rough sets Ivo Düntsch School of Information and Software Engineering, University of Ulster, Newtownabbey, BT 37 0QB, N.Ireland I.Duentsch@ulst.ac.uk Abstract The collection of all subsets

More information

Polynomials as Generators of Minimal Clones

Polynomials as Generators of Minimal Clones Polynomials as Generators of Minimal Clones Hajime Machida Michael Pinser Abstract A minimal clone is an atom of the lattice of clones. A minimal function is a function which generates a minimal clone.

More information

Periodic lattice-ordered pregroups are distributive

Periodic lattice-ordered pregroups are distributive Periodic lattice-ordered pregroups are distributive Nikolaos Galatos and Peter Jipsen Abstract It is proved that any lattice-ordered pregroup that satisfies an identity of the form x lll = x rrr (for the

More information

TORSION CLASSES AND PURE SUBGROUPS

TORSION CLASSES AND PURE SUBGROUPS PACIFIC JOURNAL OF MATHEMATICS Vol. 33 ; No. 1, 1970 TORSION CLASSES AND PURE SUBGROUPS B. J. GARDNER In this note we obtain a classification of the classes J?~ of abelian groups satisfying the following

More information

Logic Synthesis and Verification

Logic Synthesis and Verification Logic Synthesis and Verification Boolean Algebra Jie-Hong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 2014 1 2 Boolean Algebra Reading F. M. Brown. Boolean Reasoning:

More information

TIGHT RESIDUATED MAPPINGS. by Erik A. Schreiner. 1. Introduction In this note we examine the connection

TIGHT RESIDUATED MAPPINGS. by Erik A. Schreiner. 1. Introduction In this note we examine the connection Proc. Univ. of Houston Lattice Theory Conf..Houston 1973 TIGHT RESIDUATED MAPPINGS by Erik A. Schreiner 1. Introduction In this note we examine the connection between certain residuated mappings on a complete

More information

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica Ivan Chajda; Radomír Halaš; Josef Zedník Filters and annihilators in implication algebras Acta Universitatis Palackianae

More information

10. Finite Lattices and their Congruence Lattices. If memories are all I sing I d rather drive a truck. Ricky Nelson

10. Finite Lattices and their Congruence Lattices. If memories are all I sing I d rather drive a truck. Ricky Nelson 10. Finite Lattices and their Congruence Lattices If memories are all I sing I d rather drive a truck. Ricky Nelson In this chapter we want to study the structure of finite lattices, and how it is reflected

More information

The Square of Opposition in Orthomodular Logic

The 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 information

LATTICES WITH INVOLUTION^)

LATTICES WITH INVOLUTION^) LATTICES WITH INVOLUTION^) BY J. A. KALMAN Introduction. By a "lattice with involution," or "i-lattice," we shall mean a lattice A together with an involution [l, p. 4] x >x' in A. A distributive *'- lattice

More information

Anna University, Chennai, November/December 2012

Anna University, Chennai, November/December 2012 B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2012 Fifth Semester Computer Science and Engineering MA2265 DISCRETE MATHEMATICS (Regulation 2008) Part - A 1. Define Tautology with an example. A Statement

More information

ON FILTERS IN BE-ALGEBRAS. Biao Long Meng. Received November 30, 2009

ON 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 information

Stone Algebra Extensions with Bounded Dense Set

Stone Algebra Extensions with Bounded Dense Set tone Algebra Extensions with Bounded Dense et Mai Gehrke, Carol Walker, and Elbert Walker Department of Mathematical ciences New Mexico tate University Las Cruces, New Mexico 88003 Abstract tone algebras

More information

(Rgs) Rings Math 683L (Summer 2003)

(Rgs) Rings Math 683L (Summer 2003) (Rgs) Rings Math 683L (Summer 2003) We will first summarise the general results that we will need from the theory of rings. A unital ring, R, is a set equipped with two binary operations + and such that

More information

Rings and groups. Ya. Sysak

Rings and groups. Ya. Sysak Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...

More information

Extensions of pure states

Extensions of pure states Extensions of pure M. Anoussis 07/ 2016 1 C algebras 2 3 4 5 C -algebras Definition Let A be a Banach algebra. An involution on A is a map a a on A s.t. (a + b) = a + b (λa) = λa, λ C a = a (ab) = b a

More information

2MA105 Algebraic Structures I

2MA105 Algebraic Structures I 2MA105 Algebraic Structures I Per-Anders Svensson http://homepage.lnu.se/staff/psvmsi/2ma105.html Lecture 12 Partially Ordered Sets Lattices Bounded Lattices Distributive Lattices Complemented Lattices

More information

Some Pre-filters in EQ-Algebras

Some 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 information

γ γ γ γ(α) ). Then γ (a) γ (a ) ( γ 1

γ γ γ γ(α) ). Then γ (a) γ (a ) ( γ 1 The Correspondence Theorem, which we next prove, shows that the congruence lattice of every homomorphic image of a Σ-algebra is isomorphically embeddable as a special kind of sublattice of the congruence

More information

Sets and Motivation for Boolean algebra

Sets and Motivation for Boolean algebra SET THEORY Basic concepts Notations Subset Algebra of sets The power set Ordered pairs and Cartesian product Relations on sets Types of relations and their properties Relational matrix and the graph of

More information

Finite homogeneous and lattice ordered effect algebras

Finite 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 information

Arithmetic Funtions Over Rings with Zero Divisors

Arithmetic Funtions Over Rings with Zero Divisors BULLETIN of the Bull Malaysian Math Sc Soc (Second Series) 24 (200 81-91 MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Arithmetic Funtions Over Rings with Zero Divisors 1 PATTIRA RUANGSINSAP, 1 VICHIAN LAOHAKOSOL

More information

MATH 326: RINGS AND MODULES STEFAN GILLE

MATH 326: RINGS AND MODULES STEFAN GILLE MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called

More information

EXAM 3 MAT 423 Modern Algebra I Fall c d a + c (b + d) d c ad + bc ac bd

EXAM 3 MAT 423 Modern Algebra I Fall c d a + c (b + d) d c ad + bc ac bd EXAM 3 MAT 23 Modern Algebra I Fall 201 Name: Section: I All answers must include either supporting work or an explanation of your reasoning. MPORTANT: These elements are considered main part of the answer

More information

DERIVATIONS. Introduction to non-associative algebra. Playing havoc with the product rule? BERNARD RUSSO University of California, Irvine

DERIVATIONS. Introduction to non-associative algebra. Playing havoc with the product rule? BERNARD RUSSO University of California, Irvine DERIVATIONS Introduction to non-associative algebra OR Playing havoc with the product rule? PART VI COHOMOLOGY OF LIE ALGEBRAS BERNARD RUSSO University of California, Irvine FULLERTON COLLEGE DEPARTMENT

More information

ON ORDER-CONVERGENCE

ON ORDER-CONVERGENCE ON ORDER-CONVERGENCE E. S. WÖLK1 1. Introduction. Let X be a set partially ordered by a relation ^ and possessing least and greatest elements O and I, respectively. Let {f(a), aed] be a net on the directed

More information

4.4 Noetherian Rings

4.4 Noetherian Rings 4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)

More information

REMARKS ON REFLEXIVE MODULES, COVERS, AND ENVELOPES

REMARKS ON REFLEXIVE MODULES, COVERS, AND ENVELOPES REMARKS ON REFLEXIVE MODULES, COVERS, AND ENVELOPES RICHARD BELSHOFF Abstract. We present results on reflexive modules over Gorenstein rings which generalize results of Serre and Samuel on reflexive modules

More information

Jónsson posets and unary Jónsson algebras

Jónsson posets and unary Jónsson algebras Jónsson posets and unary Jónsson algebras Keith A. Kearnes and Greg Oman Abstract. We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than P, and κ is a cardinal

More information

Weakly Semicommutative Rings and Strongly Regular Rings

Weakly Semicommutative Rings and Strongly Regular Rings KYUNGPOOK Math. J. 54(2014), 65-72 http://dx.doi.org/10.5666/kmj.2014.54.1.65 Weakly Semicommutative Rings and Strongly Regular Rings Long Wang School of Mathematics, Yangzhou University, Yangzhou, 225002,

More information

CURVATURE VIA THE DE SITTER S SPACE-TIME

CURVATURE VIA THE DE SITTER S SPACE-TIME SARAJEVO JOURNAL OF MATHEMATICS Vol.7 (9 (20, 9 0 CURVATURE VIA THE DE SITTER S SPACE-TIME GRACIELA MARÍA DESIDERI Abstract. We define the central curvature and the total central curvature of a closed

More information

Section 4.4 Functions. CS 130 Discrete Structures

Section 4.4 Functions. CS 130 Discrete Structures Section 4.4 Functions CS 130 Discrete Structures Function Definitions Let S and T be sets. A function f from S to T, f: S T, is a subset of S x T where each member of S appears exactly once as the first

More information

TESTING FOR A SEMILATTICE TERM

TESTING FOR A SEMILATTICE TERM TESTING FOR A SEMILATTICE TERM RALPH FREESE, J.B. NATION, AND MATT VALERIOTE Abstract. This paper investigates the computational complexity of deciding if a given finite algebra is an expansion of a semilattice.

More information

Computational Completeness

Computational Completeness Computational Completeness 1 Definitions and examples Let Σ = {f 1, f 2,..., f i,...} be a (finite or infinite) set of Boolean functions. Any of the functions f i Σ can be a function of arbitrary number

More information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Spring

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Spring c Dr Oksana Shatalov, Spring 2016 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element in the set

More information

ON THE CONGRUENCE LATTICE OF A FRAME

ON THE CONGRUENCE LATTICE OF A FRAME PACIFIC JOURNAL OF MATHEMATICS Vol. 130, No. 2,1987 ON THE CONGRUENCE LATTICE OF A FRAME B. BANASCHEWSKI, J. L. FRITH AND C. R. A. GILMOUR Recall that the Skula modification SkX of a topological space

More information

EP elements and Strongly Regular Rings

EP elements and Strongly Regular Rings Filomat 32:1 (2018), 117 125 https://doi.org/10.2298/fil1801117y Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat EP elements and

More information

The 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 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 information

A NOTE ON EXTENSIONS OF PRINCIPALLY QUASI-BAER RINGS. Yuwen Cheng and Feng-Kuo Huang 1. INTRODUCTION

A NOTE ON EXTENSIONS OF PRINCIPALLY QUASI-BAER RINGS. Yuwen Cheng and Feng-Kuo Huang 1. INTRODUCTION TAIWANESE JOURNAL OF MATHEMATICS Vol. 12, No. 7, pp. 1721-1731, October 2008 This paper is available online at http://www.tjm.nsysu.edu.tw/ A NOTE ON EXTENSIONS OF PRINCIPALLY QUASI-BAER RINGS Yuwen Cheng

More information

REGULAR IDENTITIES IN LATTICES

REGULAR IDENTITIES IN LATTICES TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 158, Number 1, July 1971 REGULAR IDENTITIES IN LATTICES BY R. PADMANABHANC) Abstract. An algebraic system 3i = is called a quasilattice

More information

Left almost semigroups dened by a free algebra. 1. Introduction

Left almost semigroups dened by a free algebra. 1. Introduction Quasigroups and Related Systems 16 (2008), 69 76 Left almost semigroups dened by a free algebra Qaiser Mushtaq and Muhammad Inam Abstract We have constructed LA-semigroups through a free algebra, and the

More information

GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS

GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS IJMMS 2003:11, 681 693 PII. S016117120311112X http://ijmms.hindawi.com Hindawi Publishing Corp. GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS SERGIO CELANI Received 14 November 2001 We introduce the

More information

Linear Algebra Lecture Notes-I

Linear Algebra Lecture Notes-I Linear Algebra Lecture Notes-I Vikas Bist Department of Mathematics Panjab University, Chandigarh-6004 email: bistvikas@gmail.com Last revised on February 9, 208 This text is based on the lectures delivered

More information

AAA616: Program Analysis. Lecture 3 Denotational Semantics

AAA616: Program Analysis. Lecture 3 Denotational Semantics AAA616: Program Analysis Lecture 3 Denotational Semantics Hakjoo Oh 2018 Spring Hakjoo Oh AAA616 2018 Spring, Lecture 3 March 28, 2018 1 / 33 Denotational Semantics In denotational semantics, we are interested

More information

STRICTLY ORDER PRIMAL ALGEBRAS

STRICTLY ORDER PRIMAL ALGEBRAS Acta Math. Univ. Comenianae Vol. LXIII, 2(1994), pp. 275 284 275 STRICTLY ORDER PRIMAL ALGEBRAS O. LÜDERS and D. SCHWEIGERT Partial orders and the clones of functions preserving them have been thoroughly

More information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2016 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from the set A to the set B is a correspondence that assigns to

More information

A Generalization of Boolean Rings

A Generalization of Boolean Rings A Generalization of Boolean Rings Adil Yaqub Abstract: A Boolean ring satisfies the identity x 2 = x which, of course, implies the identity x 2 y xy 2 = 0. With this as motivation, we define a subboolean

More information

ON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS. Christian Gottlieb

ON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS. Christian Gottlieb ON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS Christian Gottlieb Department of Mathematics, University of Stockholm SE-106 91 Stockholm, Sweden gottlieb@math.su.se Abstract A prime ideal

More information

Lattice Theory Lecture 5. Completions

Lattice Theory Lecture 5. Completions Lattice Theory Lecture 5 Completions John Harding New Mexico State University www.math.nmsu.edu/ JohnHarding.html jharding@nmsu.edu Toulouse, July 2017 Completions Definition A completion of a poset P

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 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 information

3. Abstract Boolean Algebras

3. Abstract Boolean Algebras 3. ABSTRACT BOOLEAN ALGEBRAS 123 3. Abstract Boolean Algebras 3.1. Abstract Boolean Algebra. Definition 3.1.1. An abstract Boolean algebra is defined as a set B containing two distinct elements 0 and 1,

More information

FONT * AND GONZALO RODRÍGUEZ PÉREZ*

FONT * AND GONZALO RODRÍGUEZ PÉREZ* Publicacions Matemátiques, Vol 36 (1992), 591-599. A NOTE ON SUGIHARA ALGEBRAS JOSEP M. FONT * AND GONZALO RODRÍGUEZ PÉREZ* Abstract In [41 Blok and Pigozzi prove syntactically that RM, the propositional

More information

Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma

Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma Chapter 2 Integral Extensions 2.1 Integral Elements 2.1.1 Definitions and Comments Let R be a subring of the ring S, and let α S. We say that α is integral over R if α isarootofamonic polynomial with coefficients

More information

On 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)

On 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 information

TROPICAL SCHEME THEORY

TROPICAL 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 information

A general Stone representation theorem

A general Stone representation theorem arxiv:math/0608384v1 [math.lo] 15 Aug 2006 A general Stone representation theorem Mirna; after a paper by A. Jung and P. Sünderhauf and notes by G. Plebanek September 10, 2018 This note contains a Stone-style

More information

On the filter theory of residuated lattices

On 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 information

THE LATTICE OF SUBVARIETIES OF SEMILATTICE ORDERED ALGEBRAS

THE LATTICE OF SUBVARIETIES OF SEMILATTICE ORDERED ALGEBRAS THE LATTICE OF SUBVARIETIES OF SEMILATTICE ORDERED ALGEBRAS A. PILITOWSKA 1 AND A. ZAMOJSKA-DZIENIO 2 Abstract. This paper is devoted to the semilattice ordered V-algebras of the form (A, Ω, +), where

More information

Spectrum of fuzzy prime filters of a 0 - distributive lattice

Spectrum of fuzzy prime filters of a 0 - distributive lattice Malaya J. Mat. 342015 591 597 Spectrum of fuzzy prime filters of a 0 - distributive lattice Y. S. Pawar and S. S. Khopade a a Department of Mathematics, Karmaveer Hire Arts, Science, Commerce & Education

More information

One-sided clean rings.

One-sided clean rings. One-sided clean rings. Grigore Călugăreanu Babes-Bolyai University Abstract Replacing units by one-sided units in the definition of clean rings (and modules), new classes of rings (and modules) are defined

More information