Rachůnek, Jiří

Šalounová, Dana

English

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaJiří Rachůnek; Dana ŠalounováDirect product factors in GMV-algebrasMathematica Slovaca, Vol. 55 (2005), No. 4, 399--407Persistent URL: http://dml.cz/dmlcz/130795Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2005Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca ©2005 . . , . < - . . _,_ /^r\r\r\ M A ^r\r\ Ar\-r Mathematical Institute Matli. SlOVaCa, 55 (2005), NO. 4, 399-407 Slovák Academy of Sciences DIRECT P R O D U C T FACTORS IN GM^-ALGEBRAS JIŘÍ RACHŮNEK* — DANA ŠALOUNOVÁ** (Communicated by Anatolij Dvurečenskij ) ABSTRACT. GMV-algebras are non-commutative generalizations of MV-alge­bras and by A. Dvurecenskij they can be represented as intervals of unital lattice ordered groups. Moreover, they are polynomially equivalent to dually residuated ^-monoids (DP£-monoids) from a certain variety of DH^-monoids. In the paper, using these correspondences, direct product factors in GMV-algebras are intro-duced and studied and the lattices of direct factors are described. Further, the polars of projectable GMV-algebras are described. 1. Introduction The Lukasiewicz infinite valued propositional logic is one of the most impor-tant logics behind the theory of fuzzy sets. It is well known that MV-algebras in-troduced by C. C. C h a n g in [2] are an algebraic counterpart of the Lukasiewicz logic. Recently the first author in [14] and, independently, G. G e o r g e s c u and A. I o r g u l e s c u in [7], have introduced non-commutative generalizations of MV-algebras (non-commutative MV-algebras in [14] and pseudo MV-algebras in [7]) which are equivalent. Here, we will use for these algebras the name gen-eralized MV-algebras, briefly GMV-algebras. By A. D v u r e c e n s k i j [4], GMV-algebras can be considered as intervals in unital lattice ordered groups (^-groups). Moreover, by [14], there is a mutual correspondence between GMV-algebras and dually residuated lattice ordered monoids (DR£-monoids) belonging to a certain variety of DR£-monoids. At the same time, the ideals of GMV-algebras correspond to the convex ^-subgroups of the corresponding unital ^-groups and also to the ideals of the induced 2000 M a t h e m a t i c s Sub j ec t C l a s s i f i c a t i o n : Primary 06D35; Secondary 03G25, 06F05, 06F15. Keywords : GMV-algebra, DH^-monoid, £-group, direct factor, ideal, polar. The first author was supported by the Council of Czech Government, J 14/98: 153100011. The second author was supported by the Council of Czech Government, J 17/98: 275100015. 399 Jlftl RACHUNEK — DANA SALOUNOVA Di?£-monoids. These correspondences are used in the paper to studying di-rect decompositions of GMV-algebras. Further, they make it possible to con-sider direct factors of GMV-algebras in the form of their ideals, although ideals, in general, are not subalgebras of GMV-algebras. Moreover, projectable GMV-algebras are described here. The necessary results concerning the theories of M"V-algebras and of ^-groups can be found in [3], [9], [6] and in [1], [8], respectively. 2. Basic notions, denotations and relations DEFINITION. Let A = (A;0 , -n ,~ ,0 ,1) be an algebra of type (2 ,1 ,1 ,0 ,0) . Set x 0 y = ~(-ix 0 -*y) for any x,y G A. Then A is called a generalized MV-algebra (briefly: GMV-algebra) if for any x, y, z G A the following condi-tions are satisfied: (Al) x 0 (y 0 z) = (x 0 y) 0 z\ (A2) x®0 = x = 0®x\ (A3) £ 0 1 = 1 = 1 0 2 ; ; (A4) -il = 0 = ~ l ; (A5) i ( ~ x © ~ 2 / ) = ~( - ix0- .2 / ) ; (A6) x 0 (y 0 ~ x) = y 0 (x 0 ~ y) = (~^y 0 x) 0 y = (~^x 0 y) 0 x; (A7) (-ix®y)Gx = yO(x®~y)] (A8) ~->x = x. If we put x < y if and only if -*x 0 y = 1, then (A, <) is a bounded distributive lattice (0 is the least and 1 is the greatest element) with x V y = x (& (y O ~x) and x A i / = a ; 0 ( t / 0 ~ x ) . Let G = (G\ + , V, A) be a lattice ordered group (^-group) and 0 < u e G. For any x,y E [0,u] = {x e G : 0 < x < u} put x(By = (x + y) Au, -^x = u — x and ~x = -x + u. Then T(G,u) = ([0,u]; ©,-i ,~,0,tz) is a GMU-algebra. By a unital £-group we will mean a pair (G,u), where G is an ^-group and u is a strong order unit in G. (Recall that 0 < u G G is a strong order unit in G if for any a € G there is n G N such that — nu < a < nn, i.e., the convex ^-subgroup of G generated by u is equal to G.) Unital ^-groups and GMV-algebras are in a very close connection because A. D v u r e c e n s k i j in [4] proved that for any GMV-algebra A there is a unital £-group (G,u) such that A is isomorphic to T(G,u). 400 DIRECT PRODUCT FACTORS IN GMV-ALGEBRAS DEFINITION. An algebra M = (M; +, 0, V, A, - - , v-) of type (2, 0, 2, 2,2, 2) is called a DR£-monoid if ( M ; + , 0 , V, A) is a lattice ordered monoid satisfying the conditions (x, H, r, s G M ) : 5 + y > x <£=> x —- H < 5 and H + r > x 4=> x T— ?/ < r ; ((x - - j/) V 0) + 7/ < x V H , y + ((x v- j/) V 0) < x V y , x ^ x > 0, X v - X > 0 . GMF-algebras and .DJW-monoids are also in a close connection. Indeed, if A = (A; 0 , -i, ~ , 0,1) is a GMV-algebra and if we put x —- H = ->H 0 x and Xv-H = x 0 ~ y f o r any x,H G A, then by [14], M(A) = (A; 0 , 0, V, A, ---, v-) is a bounded K)it^-monoid (with 1 the greatest element and 0 the least) satisfying the identities (i) (Vx G A)(l v- (1 - - x) = x = 1 - - (1 v- x ) ) , (ii) (Vx G A)(Vy G A) ( l - ((1 v- x) + (l v- y)) = 1 v- ((1 - x ) + (l - ? / ) ) ) . Conversely, if M = (M; +, 0, V, A, —-, v-) is a bounded i}i?£-monoid with a greatest element 1 satisfying (i) and (ii) and if we put ->x = 1 —- x and ~ x = 1 v- x for x G M , then by [14], A(M) = (M; + , -«, ~, 0,1) is a GMF-algebra. Recall that if A is a GMF-algebra and 0 ^ i i C A, then i i is called an ideal of A if H is closed under the operation 0 and y < x implies y G H for any x e H and y e A. An ideal is called normal if -ix 0 y G i i if and only if /̂ 0 ~ x G H for each x,H G A. The normal ideals are exactly the kernels of GMV-homomorphisms. For any 0 ^ i i C A we have that H is an ideal of A if and only if H is a con-vex sub-DRl-monoid of M(A). (Convex sub-J3i?^-monoids of a Dit^-monoid M are also called ideals of M.) Further, if M is a DR£-monoid and I is a con-vex sub-DR£-monoid of M , then I is called normal if and only if x + i = i + x for any x € M. One can prove that for a GMF-algebra A, an ideal H of A is normal if and only if H is a normal convex sub-DR£-monoid of M(A). (See [12].) We will use these relations when studying direct decompositions of GMV-algebras, because ideals of GMF-algebras, in contrast to convex sub-DiZ^-monoids of DR£-monoids, need not be subalgebras of GMF-algebras. If A is a GMV-algebra, denote by C(A) and M(A) the set of ideals and of normal ideals of A, respectively. Analogously, if M is a K)i?£-monoid, then C(M) and N(M) will denote the set of convex sub-DR£-monoids and of normal convex sub-DR£-monoids, respectively. It is obvious that (C(A), c ) , (AT(A), C ) , (C(M),C) and (Af(M),c) are complete lattices. Let A = T(G,u) be a GMF-algebra and let (C(G),c) and (A/"(G),c) be the complete lattices of convex ^-subgroups and of ^-ideals of C?, respectively. 401 Jlftl RACHUNEK — DANA SALOUNOVA Let us consider the mapping ip\ C(A) —> C(G) such that (p(H) = {x £ G : \x\ A u £ H] for any H £ C(A). By [15; Theorem 2], cp is an isomorphism of C(A) onto C(G) and the inverse isomorphism to tp is the mapping rp such that ij>(K) = K H [0, w] for each K £ C(G) . Moreover, by [5; Theorem 6.1], the restriction of (p on M(A) is an isomorphism between Af(A) and M(G). 3. Direct factors of GM^-algebras In this part we will deal with direct decompositions of GMF-algebras which we will introduce by means of direct decompositions of the induced Dit^-monoids. DEFINITION. We will say that a DRl-mono\d M is an inner direct product of its convex sub-DRl-monoids (i.e. ideals) M 1 and M2 if there is an isomorphism ip of M onto the (external) direct product Mx x M2 of DRl-monoids Ml and M2 such that for each x £ M1 and each y £ M2 the relations (p(x) = (x, 0) and ip(y) = (0,2/) are valid. In such a case, we will also write M = Mx x M2 and say that M is a direct product of its sub-DRl -monoids M1 and M 2 . DEFINITION. If A is a GMV-algebra and H1,H2 £ C(A), then ,4 will be called a direct product of the ideals H1 and H2 if M = M(A) = M(H^xM(H2), where M(H{) is the convex sub-JDi?£-monoid of M induced by H^ i = 1, 2. We will write A = H1 x H2 and say that H1 and i i 2 are direct factors of the GMV-algebra A. Remark. a) By [16; Theorem 6], if M^M2e C(M), then M = M1x M2 if and only if 1. M1 +M2=M, M1nM2 = {0}; 2. (Vx1>y i £ M 1 ) ( V x 2 , H 2 £ M 2 ) (xx +x2=yx+y2 = > (a?! = ^ & x2 = H2)). Moreover, if M = M1 x M 2 , then M l 5 M 2 £ J\f(M) and Mx = M2X and Ai2 = M j1 , where M2 and M:1 are the polars of M2 and Mx, respectively. That means, if M = M(A) for a GMF-algebra A, then M is bounded (with the least element 0), and hence, for instance, M1 = M2 = {x £ M : ( V b £ M 2 ) ( b A x = 0 ) } . b) If A is a GMF-algebra and H £ C(A), then we will not distinguish H and M(H). 402 DIRECT PRODUCT FACTORS IN GMV-ALGEBRAS THEOREM 1. Let A be a GMV-algebra and HX,H2 G C(A). Then A = Hx x H2 if and only if Hx and H2 satisfy condition 1. P r o o f . Let A = T(G, u) be a GMV-algebra and let H1,H2 e C(A) satisfy condition 1. If Kx = (p(Hx) and K2 = (f(H2), then (since HX,H2 G M(M) = M(A)) we get KvK2e Af(G). By [16; Proposition 7], Hl®H2=HlV H2 in C(M(A))=C(A). Hence we get: G = <p(A) = <p(Hx © tf2) = <p(H, V tf2) = <p{Hx) V <p(H2), and since (p(Hx), ^p(H2) ^ N(G), we have G = tp(H1)+<p(H2) = K1+K2. Moreover, from H1CiH2 = {0} it follows that K1f]K2 = {0}, thus G = KlxK2 (and so also Kx = K2 and K2 = Kx). Let xx,yx G Hx, x 2 ,y 2 G H2 and xx 0 x2 = yx 0 y2. Since xx A x2 = 0 = 2/T A y2 , xx 0 x2 = xx V x2 = x1 + x2 and y-_ © y2 = ^ V 2/2 = ^ + y2 , therefore xx + x2 = ^ + H2. Hence from G = Kx x K2 we obtain xx = yx and x2 = y2, i.e., Hx and H2 satisfy also condition 2 for direct factors in M , and therefore in A, too. The converse implication is trivial. D THEOREM 2. Let A = T(G,u) be a GMV-algebra and let G = KxxK2 be a direct decomposition of the £ -group G. If Hx = ip(Kx) and H2 = i^(K2), then A = HxxH2. P r o o f . Let a G A. Then there exist ax G Kx and a2 G K2 such that a = ax + a2 . Since 0 < a l 5 a 2 < a < ^ , we have ax 0 a2 = ax + a2 , and so a = ax 0 a2 . Hence A = Hx 0 H2. Condition Hx n H2 = {0} is satisfied too, and therefore, by Theorem 1, A = Hx x H2. D The following theorem is now an immediate consequence. THEOREM 3. If A = T(G,u) is a GMV-algebra, then H G C(A) is a direct factor of A (and also of the DR£-monoid M(A)) if and only if ip(H) is a direct factor of the £ -group G. Remark. Let A = T(G,u) be a GMV-algebra, H1,H2 G C(A) and let A be the direct product of Hx and H2. If u = x̂1 + u2 = ux 0 u2, where ux G Hx and u2 G H2, then u{ is the greatest element in Hi, i = 1,2, and thus ux and H2 are additively idempotent elements in A, i.e. H{ = C(u{) = [0 ,^J , i = 1,2. For any CMF-algebra A, the DR£-monoid M(A) induced by A satisfies the condition (MV) x^(x^-y) = x/\y = x^-(x-^y). 403 Jlftl RACHUNEK — DANA SALOUNOVA Hence the sub-DR£-monoid in M(A) induced by any ideal in A satisfies con-dition (MV), too. By [13], the bounded Di^-monoids satisfying (MV) are just those induced by GMV-algebras. Therefore A is isomorphic to the direct prod-uct of the GMV-algebras with underlying sets H1 and H2 . (The fact that if a is an idempotent element in a GMF-algebra A, then the interval [0,a] can be considered as a GAfV-algebra was proved in [10], and that the operations in the GMlZ-algebra [0, a] can be expressed explicitly as x ®a y = x ® 2/J ~ a x = n X / ^ a a n d ~ a x = ~ x A a (x,y G [0, a]) was proved in [14].) Therefore we now get as a consequence the following theorem, which was proved by different methods in [10; Sections 4, 5]. THEOREM 4. Let A, A1 and A2 be GMV-algebras. Then A is isomorphic to the direct product Ax x A2 if and only if there is an idempotent element a £ A such that Ax = C(a) and A2 = C(-.a) = C(~ a). Moreover, the remark after Theorem 3 together with the fact that the idem-potent elements in A form a subalgebra B(A) of A which is a Boolean algebra and in which -ia = ~ a = a' for each a G B(A) (see [14]) imply: THEOREM 5. The direct factors of a GMV-algebra A form a Boolean sublat-tice of the lattice C(A) and also of the lattice of polars in A, which is isomorphic to the Boolean lattice of idempotent elements in A. Now, we will describe even more exactly the connections between the direct factors of a GMF-algebra A = T(C7, u) and of those of the corresponding unital ^-group (G,u). PROPOSITION 6. Let A = T(G,u) be a GMV-algebra, let A = HY x H2 be a direct decomposition of A and let Ki = (p(H{), i = 1,2. If a G A and a = ax © a2, where ai G H{, i = 1,2, then a2/H1 = (a2/Kl) n A and a>i/H2 = (a1/K2) n A . P r o o f . Let x G / 4 . Then x G a2/H1 if and only if (x —- a2)(&(a2 —^x)GH1, wrhich holds if and only if (Or - a2) V 0) + ((a2 - x) V 0)) A u G Hx , hence if and only if ((x - a2) V 0) + ((a2 - x) V 0) G Kx , and this is equivalent to ((x - a2) + (a2 - x)) V (a2 - x) V (x - a2) V 0 G Kx . 404 DIRECT PRODUCT FACTORS IN CMV-ALGEBRAS Therefore x E a2/Hx if and only if \a2 — x\ E Kx, which is equivalent to a2 — x E Kx, that means, to x E a2 + Kx. The second equality is analogous. • The direct factors of a GMV-algebra are its normal ideals, hence we can construct corresponding factor GMF-algebras. Using Proposition 6, now we will easily prove the following theorem. THEOREM 7. If A is a GMV-algebra and if A = Hx x H2 is a direct decom-position of A, then Hx = A/H2 and H2 = A/Hx. P r o o f . Let A = T(G,u) and let Kx and K2 be as in Proposition 6. Let f: K2^r G/Kx be the isomorphism of £ -groups such that f(c) = c/Kx = c-\-Kx for each cE K2. Let / = f\H . Let us denote by / ' : H2 —> A/Hx the mapping such that f(x) = x/Hx for each x E H2. By Proposition 6, / ( # ) = /(H) if and only if f(x) = f(y) for any x,H E JI2 • Thus / is a bijection of if2 onto A/Hx. At the same time, / is a restriction of the natural homomorphism v: A —r ̂ 4/H! of GMF-algebras, hence / is an isomorphism of H2 onto ^4/H!. Therefore H2 = . 4 / ^ . The second assertion is analogous. • 4. P ro jec t ab le GMX^-algebras Projectable ^-groups form an important class of ^-groups. Recall that an I -group G is called projectable if the polar a x is a direct factor in G for each a E G. Now we will introduce an analogous notion also for GMV-algebras. DEFINITION. A GMV-algebra A is called projectable if A = a1- x a±L for each a E A. Remark. a) By Theorem 1, A is projectable if and only if A = a1- © a-1-1 for each ae A. b) If a GMV-algebra A is projectable, then every polar in A is a normal ideal in yl. Hence by [5], every projectable GMV-algebra, similarly as in the case of ^-groups, is representable. In the next theorem, we will show connections between principal ideals and polars in projectable GMV-algebras. THEOREM 8. Let A be a projectable GMV -algebra. Then every polar in A is an intersection of principal (normal) ideals of A generated by elements from the set B(A) of all idempotent elements of A. 405 Jlftl RACHUNEK — DANA SALOUNOVA P r o o f . If A is a projectable GMF-algebra, then for any element a G A there is an element b G B(A) such that a1 = C(b) and aL1- = C(-^b). Let C C A be a polar in A. Then c= nd±= n ^ ) > where cd is an element in 2?(.A) such that dL = C(cd). Thus every polar in 4̂ is an intersection of principal (normal) ideals generated by elements of B(A) (i.e., an intersection of intervals in the form [0,x] where x G B(A)). D LEMMA 9. Let A = T(G,u) be a GMV-algebra and let H G C(A). Then H is the principal ideal CA(a) in A generated by an element a G A = [0, u] if and only if (f(H) is the principal convex £-subgroup CG(a) in G generated by a. P r o o f . Let ae A, J € C(A) and a G J. Then obviously a G (p(J) G C(G). Conversely, if L G C(G) and a G L (thus CG(a) C L) , then a G F n [0,u] = -0(F), that means (^(a) C ^(L) = (7O~1(F). Therefore ^ ( ^ ( a ) ) = CG(a). D PROPOSITION 10. Let A = r(G,w) be a GMV-algebra. Then A is a pro-jectable GMV-algebra if and only if G is a projectable £-group. P r o o f . Let a G A. Then a±A is the pseudo-complement of the ideal CA(a) in the lattice C(A), and hence, (f(a±A) is by Lemma 9 the pseudo-complement of the convex ^-subgroup CG(a) in the lattice C(G). Therefore if(a-LA) — aL° . The assertion now follows from Theorem 3. D The following theorem is a consequence of Theorem 8, Lemma 9 and Propo-sition 10. THEOREM 1 1 . Let (G, u) be a projectable unital £ -group. Then every polar in G is an intersection of principal convex £-subgroups (which are £-ideals) of G generated by elements x G G + satisfying the condition (x -f x) A u = x. REFERENCES [1] BIGARD, A.—KEIMEL, K.—-WOLFENSTEIN, S.: Groupes et Anneaux Réticulés, Springer Verlag, Berlin-Heidelberg-New York, 1977. [2] CHANG, C C : Algebraic analysis of many valued logic, Trans. Amer. Math. Soc. 88 (1958), 467-490. [3] CIGNOLI, R. O. L . — D Ю T T A V I A N O , I. M. L.—-MUNDICI, D . : Algebraic Foundation of Many-Valued Reasoning, Kluwer Acad. Publ., Dordrecht-Boston-London, 2000. [4] D V U R E Č E N S K I J , A. : Pseudo MV-algebras are intervals in t-groups, J. Aust. Math. Soc. 70 (2002), 427-445. 406 DIRECT PRODUCT FACTORS IN GMV-ALGEBRAS [5] DVURECENSKIJ , A . : States on pseudo MV-algebras, Studia Logica 68 (2001), 301-327. [6] DVURECENSKIJ , A.—PULMANNOVA, S.: New Trends in Quantum Structures, Kluwer Acad. Publ. , Dordrecht-Bos ton-London, 2000. [7] GEORGESCU, G.—IORGULESCU, A. : Pseudo MV-algebras, Mult.-Valued Log. 6 (2001), 95-135. [8] GLASS, A. M. W . : Partially Ordered Groups, World Scientific, Singapore-New Jersey-London-Hong Kong, 1999. [9] HAJEK, P . : Metamathematics of Fuzzy Logic, Kluwer, Amsterdam, 1998. [10] JAKUBIK, J . : Direct product decompositions of pseudo MV-algebras, Arch. Math. (Brno) 37 (2001), 131-142. [11] KOVAR, T . : A General Theory of Dually Residuated Lattice Ordered Monoids. Thesis, Palacky Univ., Olomouc, 1996. [12] KUHR, J . : Ideals of noncommutative DRl-monoids, Czechoslovak Math. J. 55 (2005), 97-111. [13] KUHR, J . : A generalization of GMV-algebras, Mult.-Valued Log. (To appear) . [14] RACHUNEK, J . : A non-commutative generalization of MV-algebras, Czechoslovak Math. J. 52 (127 ) (2002), 255-273. [15] RACHUNEK, J . : Prime spectra of non-commutative generalizations of MV-algebras, Algebra Universalis 48 (2002), 151-169. [16] RACHUNEK, J .—SALOUNOVA, D . : Direct decompositions of dually residuated lattice ordered monoids, Discuss. Math. Gen. Algebra Appl. 24 (2004), 63-74. Received November 27, 2003 Revised June 21, 2004 ' Department of Algebra and Geometry Faculty of Sciences Palacký University Tomkova 40 CZ-779 00 Olomouc CZECH REPUBLIC E-mail: rachunek@inf.upol.cz ' Department of Mathematical Methods in Economy Faculty of Economics VSB-Technical University Ostrava Sokolská 33 CZ-701 21 Ostrava CZECH REPUBLIC E-mail: dana.salounova@vsb.cz 407 

Direct product factors in GMV-algebras

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Direct product factors in GMV-algebras

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Institute of Mathematics AS CR, v. v. i.