Bounded Distributive Lattices with Fusión and

Southeast Asian Bulletin of Mathematics (2004) 28: 099 1010 Southeast Asían

Bullctin of Mathematics ©SKAMS. 2004

Bounded Distributive Lattices with Fusión and Implicatíoii Sergio Celan i Dcpartarneiitü de Matemática, (''acuitad de Ciencias Exactas, Universidad Nacional del Centro and CONICET, Pinto ;«){), 7000 Tandil, Argentina., Email: scolani'1 i'cxa.uniccn.edu.ar

AMS Mathematics Subjcct Classification(2000): 03G25, 06D50 Abstrar.t. We introduce and stndy thc variety oí' DLFI-latí ices, i.e. bonnded distributivp lattices with an opcratiori of implication —», and a binary operation o, called fusión. These algebras are a roninion generaliza! ions of the Relevant algebras studied by A. Urquhart [10] ano! sume known ordercd algebraic strnctures connccted with inanyvalued logics, as for example MV-algebras and BL-algebras. Wc develop a Priestley dnality for thc DLFI-lattices. This duality is used to give a characterization of the congruences by meaiiH of some closed siibsetu of the dual spare, and (,o give a characterization of thc subalgebras by means of reflexive and transitive relations dehned on the dual space. Kcywords: Bounded distributive lattices w i t h fusión and implication; Pricstley duality; Congruences; Subalgebras.

1. Iiitroduction

Residuated boundcd distributive lattices (RDL-lattices) are algebras of the form (j4, V, A, —*,o,0, 1} whorr the rcduct (A, V. A, O, 1} ÍK ¡i bounded diytributive lattice and the pair {o, —>) is an ndjoint pair, Le., thc foliowirig property is valid: (R) X o y < z ¡fí X < (/ —* Z.

The RDL-latt.icos appear, in one forui or in other, in jn-actically all algebraic structures associated with niany-valucd logic and relevance logic. For instance, BL-algebras, MV-algebras and Relevance algebras are residuated distribuí.ive lattices that correspond to the algebraic scinaiitics of basic propositional logic, Lukasiewicz ínfinite-valued propositional logic and relevant logics respectively (see [5, (i, 7, 2, 11, 1(1]).

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The aim of this paper is to eonsider the bounded distributivo lattices with an iniplication —> and a fusión o, but these operations are not necessarily connectcd by means of the property (R). These algebras are called bounded distributivo latticcs with fusión and iniplication (DLFI-latticos). We will develop a duality theory by extending the duality botween bounded distributiva lattices and Priestley spaccs. Our spaces liave two tornary relations, a relation for the iniplication, and another one for the fusión. The two relations coincide when the property (11} liolds in the algebra. This duality is an extensión of Urquhart's duality in [10]. In Section 2, we shall give a representation theorem hy mcans of relational structuros (also called iraníes) for DLFI-lattices. In Sect.ion li, we develop the Priestley duality for the DLFI-lattices. In the varieties of the MV-algebras and BL-algebras, and also in some varieties of relevance algebras, the congruences are determined by a particular class of filters (seo for exaniple, [7, 5]). But a similar charactcrization is not known for some more general classos of algebras. In Soction 4, we give a characterization of the congruences of a DLFI-lattices by me;ins of closed subsets of it.s dual spaco. It is known that there exists a duality betwoen Boolean subalgebras of a givcn Boolean algebra A and certain equivalenco relations dofincd on the Boolean spaeo of A. This result was extended to bounded distributivo latticcs in [3], proving that there exists a duality betwecn subalgebras of a given bounded distributivo lattice and lattice preorder of the associated Priestley space. In Sectioii 5, wo shall prove that. thcre exists a duality bctween subalgebras of a DLFI-lattices A and a particular type of lattice preonler dcíined on the dual space of A.

2. DLFI-lattices Definition 2.1. A bounded distributive lattice with fusión (DLF). or DliF-lattice, is an alfitbra A = {A, V, A, o, 0,1} where {.A, V, A, 0,1} is a bounded distributive lattice and ( F l ) a o(bV c) = (a o b) V (a o c), (F2) (« V b) o r = (a o c) V (6 o c), (F3) « o O = U o ( í = 0, A bounded distributive lattice with implication (DLI), or DLI-lattice, in an algebra A = (A, V, A, —*, O, 1} whrrc (A, V, A. 0,1} is a bounded distributive. lattir.e muí (11) a — l = l. (12) (a -» b) A (a -> c) = a -» (b A c), (III) (a -^ c) A (b -> c) = (a V b) -+ c, A bounded distributive lattice with fusión and implication (DLFI), or DLFIlattice, is an algebra A — {,4, V, A, o, —» 0,1) such that (A, V, A, o, 0,1} ¿s a DLF and (A, V, A, —*, 0,1) is a DLI. A DLFI A is called a residuated bounded distribulaftice (RDL). or HDL-algebra, if for all a, b, e £ A, a o b < c O a < b —>• c.

Bounded Distributive LaÜkvs witli Fusión ;uid Inijjlicat i b, b —* x < a —* x, aox < bo x, and x o a < x o 6. Let A be a bounded distributive lattice. The filter (ideal) generated by a subset X C A will be denoted by F ( X ) ( I ( X ) ) . and the colleel.ion oí'all filters (ideáis) will be denoted by Fi(A) (Id(A)}. Tlie set of prime h'lters of A we be denote by ^(^4). The set complemont of a snbset Y C A will be denotcd by Yc or A - Y. Let A be a DLFI-lattice and let F, C € Fi(A). We defino the following subsets of A: FoG = {x£A:3(f. G = {x e ,4 : 3{/, 3) e F x C

fog y] < g -* (.r V ;/y). lint, as q A (a V b) S Q and r/ e (í, then ;r V y e P, wliich is a contradiction. TlniK, Q is a i>riine filtcr of A. Lot us considor tho sct Tí = {// e F¿(^) : C C H and Q -> íí C P}. Since G € 7t, then H / 0. By thc Zorn's Lemma, thcrc cxists a maxirnal olcmont D in W. By a similar argumcnt to tho above it is show that D € X(A). For the proof of (ii). sec [10]. • Lct >1 be a DLFI-lattice. Lct us define in X(A) the ternary relations RA and 2U by (P, Q, D) e /?,, «• P o Q C D, and (P, Q, D) e 'A Ó P -> Q C /;. We note that if ,4 is a residuated bounde e X(X) Q o D C P, „. e g and 6 e D. (ii) n -> 6 e P «• VCM> e ^(.4) P -> Q C L> «TÍÍÍ a e Q, iTrip/ie» í/m¿ 6 e 7). Proof. For thc proof of (i), see [10]. (ii) Let us suppost- that a —* 6 ^ P. Let us consider the filter F(a) and thc ideal I(b). Then, (P —» F(a)) n I(b) — 0, because in coiitrary caae, there exist x € A and p £ P such thíit p < a —* x and x < b. So, p < a —* b e P. wliidí is a contradiction. Thus, thcre exists D € ^(^4) such that P —* F(«) C D and fe Í D. ]}y Theorcni 2.4, thcre exists Q € A"(A) such that P -* Q C D and

rt^g. Thíi proof of tho converse is easy.

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Let (X, /i(rf) - h(q -» í¿) € P. So, b) Q' C D', a e Q' , and 6 ^ D' . Then, by Theorem 2.4, there cxist Q.D £ X ( A i ) such that P -> Q C D. Q' ^ h~1(Q), and h~l(D) C D'. It follows that /¿(a) 6 Q, and by assumption, h(b) e D', which is impossiblc. The proof of h(a —> b) < h(a) —* /i(6) is similar and left to the reader. • Let us denote by 'DCTI the category of DLFI-lattices with homomorphisma of DLFI-lattices and let us denote by £T1 the category of DLFI-spaces with /^morphisms. By the above results and by the Priestley duality we can say that the catcgories £ TX and 'DLTT are dually eqnivalents.

4. Congrucnces Let us recall that if A in a hounded distributive lattice and Con(A) denotes the set of all c-ongruences of A, then for each closed snbset Y of X(A), the relation 0(Y) = {(a,6) e Ax A : fi(a)r\ = ft(b)C\Y] 6 Con(A), and the correspondence Y —* ft(Y) establishes an anti-isomorphisni from the lattice of closed snbsets of X oiito Con(A) (see for instance [1]). Definition 4.1. Let A be. a DLFl-ltiüia: and O C A x A a lattice-congruence. We shall fitiy that 6 is a DLI-congruence if for all («,¿) e O and for all (c, d) e 8, (a —*• c, b —+ d) G 9. Similarly wc. define a DI*F-congruence. We denote by Con_(J4) and Con0(A) the lattice of DLI-congrnences and the lattice of DLF-congruences of A, respectively.

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Let {X, ((a A b) V f7),í/ —> (a V d)) e í'(V). This implies that í/ —» ((a A b) V r/) e P, and since q € Q and P —> Q C D, we nave (a A 6) V d e D , which is a contradiction. Thus, D e Y. Sufficic.nr.y of (i). Let a,b,;r,u € A such that (a,b) e 0(K) and (x, y) € é/(y). Let P e K and let us suppose that a —> x 6 P and b —> y QCD.btQ and .y i D. Lot Q' . D' e A" (A) such that g C Q', D' C D, (g', D') e D(P}. By assumptiou, Q' € Y. Since í> e g' and y ^ D, then a e g' and .r ^ D. But as a —> x € P and a G g', theii x e £>'. Since D' e y and (x,y) € 0(y), y £ 0' C D, whioh is a coritradiction. If wo riow suppose that x —* a e P and y —* 6 ^ P, then as tlio proof abovo we arrivo to contradiction. Tlius, ft(Y) is a DLI-congruence. Necessity of (ii). Leí V be a dosed subsot of X(A) such that 0(y) e Con 0 (A). Lct D e y, and let (P,g) e MaxR~^(D). Supposo that P A b) o 17 e D,

wliich is a contradiction. Thus P £ Y. Sirnilarly we can ]>rove that g e y. Sufficiency of (ii). Let (a,b) e 0(Y) and (./-,(/) e 0(y). Let P e y such that a o x e P. Then thero exist g, D e X(A) such t lial Q o D C P, a € Q and x e D. Lct g',D' e Tl/fu;/?- 1 ^) such that g C Q1 and D C D'. By assurnption, g' e y and D' e Y. Since a G g' and o: e ü', í> £ Q' and /y e D'. Thus, boycP. m

5. Subalgebras of DLFI-latticcs In this section, we shall characterize t h o binary relations associated with tho subalgebras of a DLFI-lat tice. Theso rosulls are basod 011 tlio dua.lity givcti in [3] betwecn (0,l)-sublattices of a boundod distributivo lattice L and the preordor latticc rclations defhícd on tho Priestley apace X(L). Let A e DLFI. Let A/ C A bo a 0.1 )-sublatticc of A. We símil say A/ is a DLF-subalt)tbra of A (Al is a DLI-sulmlycbra of A) if for all a, 6 € A/, a o b € Ai (a —> b 6 Ai), and A/ is a ^ubalgcbíu M of ,4 is a DLF-subalgebra of A arid a DLI-snbalgobra of A. Let A be a boundod distributivo la.tt.ioo and lot, M be a (U.l)-sublaLtÍce of -A. We define the binary relation RM = {(P,Q) £ X(A)2 : QnM C P}, and for each binary relation R of X(A) we dofíne tho set MR = {U e D(X(A}} : R~l(U) C t/}. Tho relation 7?j\ is rofloxivo. transitivo and it verifies the following coiidition:

(1) For overy P, g e X(A) if (P,Q) £ RM thore exists U e AIH such that P e (7, g i t/, and (e(P},e(g)) ^ fíA/wWo shall say that, a relation Ii dofinod in a Priostley spao.o X is a lattict: preorder if it is reflexive, transitivo and satisfies t.ho condition (1). In [H] it waa shown that Ala is a (0,l)-sublattice of D(,V(.4)}. U\¡ is a lattioo pníordor d f l i n c d m i A ' l - 4 ) . ; i n d l l i ; i l t ! i r ct HTcspom I r n c c \

• / i ' \ ostnblishes ; n i ; i l i l i

isomorphism betweon the lattioo oí' (0,l)-sublattices of a bonndod distributivo lattice A and tlie lattice of lattioo preorders dolined in tlio l'riestloy space X(A).

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Theorem 5.1. Let A be a DLFl-latticu and let AI be. a (0,l)-sublatticc of A. Then (i) M '¿,s a D~LF-subalt/c.bra of A if and only if it verifies the following condition: (o) forallPl,P2,Q,P e X(A), if P]oP 2 C P and PC\M C Q, then there exist Qi,Q2 fíuch that P^Af C Q l 5 P 2 n A / C Q 2 , andQioQ.¿ C Q. {ii) M is a lílil-subalgebra of A if and only if it verifies the following condiiion: (-») for all Q^Qi.Q.P e X(A). if Q -» Ql C Q2 ar¿íf F n A/ C Q T íhen there exist P],P 2 such tkat Qi n M C P,, P2 n A/ C Q2 and P^Pi CP 2 . Thuti, AI is a subalgebra of A if and onl.i¡ if lí¡\¡ verijics Un- coiidil.ioitK ( « ) and

Prouj. Necessüy uf (i). Lct P^P^Q.P € X(A) such that PI o P¿ C P, P n M C Q. and Ict. us consider thc ñlcrs F(P\ M), and F(P-¿ n A/). Since M is closcd uiider o, thcii it is oasy to seo that F(Pi n M) o F(P2 n A/) C Q. By Thtíorüiii 2.4, thorc (íxist Qi , Q2 e X(,4) snch that P: n A/ C Q j , P2 n A/ C Q 2 , and Q\ Q2 C Q.

Sufficiency of (i). Let a, 6 G A/ and let us suppose that a o & ^ M. So, it is easy to check that F(F(a o 6) n M) n /(o o b) = 0. By the Prime Filter Theorem llicrc i'xísts Q e X(A) such that F ( o o 6 ) n A / C Q and aob < q\» a and a 1) such that a^be 1> and /(a -> fe) n A/ n P = O-

(2)

Also, I (a ->• fe) n F(A/ n P) = 0, because if there cxist.s .r e A/ n P such that a: < a —» fe. then :r e 7(a -» fe) n A/ C P which cuntradict (2). Then there exists Q e -V(-A) such that Pn M C Q and o -* fe ^ Q. So, there exist Q\,Q-¿ e -^(-A) such that Q —* Qi Q Q^i" G Qi and fe ^ Qs- By assumption, there exist Pi-Pa e X ( A ) such that P -> P, C P2. y, n A/ C /',. and P2 n AA C Qi. Henee, a e Qi D M C P( and a —• fe € P, then fe e P2 n A/ C Q 2 , which is a m i i t r a d i r t i n i i . Therefore. ti

- b e A/.

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References [l] Cignuíi, R.: Ditil.rihaiivf LdUirt-' Congruences and 1'rÍKstlcy Spac.es, Actas del Primer Congreso A n t o n i o Monl.eíro. Universidad Nacional drl Sur. Bahía Blanca. «1 84 (1991). [2] Cignoli, R., D'Ottaviaiio. I.M.L.. Mundici. D.: Algehraic foundatioiis of rnanyvalued reasoning, Tmnds in Logir 7, Kluwer Academic Publishers, 2000. [3] Cignoli, R., Lafalce, S., Petrovieh, A.: Remarks 011 Priestley dnality for distributive lattices, Oder 8, 299-315 (1991). [4] Dwinger. R... Balbi-H. P.: Dislnlmiivc L