Chajda, Ivan

Kühr, Jan

English

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

Mathematica SlovacaIvan Chajda; Jan KührA note on interval MV -algebrasMathematica Slovaca, Vol. 56 (2006), No. 1, 47--52Persistent URL: http://dml.cz/dmlcz/130876Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2006Institute 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.czMatherrKtfica Slovaca ©2006 „ ^ . Mathematical Institute Matli. SlOVaCa, 5 6 ( 2006 ) , NO. 1, 4 7 - 5 2 Slovák Academy of Sciences Dedicated to Professor Tibor Katrindk A NOTE ON INTERVAL i W - A L G E B R A S IVAN CHAJDA — J A N K U H R (Communicated by Sylvia Pulmannovd) ABSTRACT. We show that every interval of an MV-algebra M is an MV-al-gebra again; the operations on the interval are defined by means of certain poly-nomial functions over M. It is well known that every complemented modular lattice L = (L, V, A, 0,1) is relative complemented, i.e., if y G L is a complement of x G [a, 6] in L, then z = (y A b) V a is its relative complement in the interval [a, b]. Since any MF-algebra is a bounded distributive lattice with respect to its natu-ral order, the question arises whether every interval can be endowed with an M^-structure. Recall from [1] that an MV-algebra is an algebra M = (M, 0 , -.,0) of type (2,1,0) satisfying the identities (MV1) (x®y)®z = x@(y®z), (MV2) £ 0 2/ = 2 / 0 x , (MV3) £ 0 0 = x, (MV4) ->->£ = x, (MV5) £ 0 ^ 0 = ^ 0 , (MV6) -i(-ix ® y) ® y =-i(-iy ® x) @ x. MF-algebras were introduced by C. C. C h a n g in [2] as an algebraic counterpart of the Lukasiewicz many valued propositional calculus. Due to D. M u n d i c i ' s result [5], MF-algebras are intervals in abelian lattice-ordered groups: 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. Keywords : M17-algebra, interval MV-algebra. 47 IVAN CHAJDA —- JAN KUHR Given an abelian lattice-ordered group G = (G, +, —, 0, V, A) and 0 < u G G , the structure T(G,u) = ([0,iz],©,-«,0) is an MTV-algebra, where x © y := (x + y) A u and -<x := u — x for all x, y G [0, u], and also conversely, every MV-algebra is isomorphic to V(G, u) for some abelian lattice-ordered group G and 0 < u € G. For basic properties of MV-algebras we refer to [1]. Let M be an MV-algebra. If we put x < y if and only if -»x 0 y = 1, then < is a partial order on M , the natural order of M , and L(M) = (M, <) is a bounded distributive lattice in which x v y = -i (-ix 0 y) 0 y and x Ay = ->(-ix V -12/). An element x G M is called boolean if there is the complement x' of x in the lattice L (M) ; we use B(M) to denote the set of all boolean elements of M . As a matter of fact, B(M) = (B(M), V, A, 0, l ) is a boolean sublattice of L(M) in which x' = -<x. Moreover, x G B(M) if and only if x 0 x = x . It is worth adding that if x G B(M), then x 0 y = x V y for each y G M . We proved in [3] that MV-algebras are polynomially equivalent with the class of algebras A = (A, o, 0,1) of type (2,0,0) satisfying the identities (Al) x o l = l , l o x = x, (A2) (x o y) o y = (y o x) o x, (A3) x o (y o z) — y o (x o z). If M is an MV-algebra, then upon defining x o y := -.x 0 y, the algebra A ( M ) = (M, o, 0,1) fulfils (Al), (A2) and (A3). Conversely, let A be an algebra satisfying the axioms (Al), (A2) and (A3), and define M(A) = (A, 0,-.,O) by x 0 y := (x o 0) o y and ->x := x o 0; then M(A) is an MV-algebra. More generally, letting x < y if and only if x o y = 1 we define the induced order of A which makes every interval [a, 1] into a lattice with x\/y — (xoy)oy and x^y — ((xo?/)o(xoa)) oa. Hence for any a e A, M(a, 1) = ([a, 1], 0 a , ->a,a) is an MV-algebra, where x 0 a y := (x o a) oy and -iax := x o a. As an immediate consequence we obtain: PROPOSITION 1. ([3]) Let M be an MV-algebra, ae M. Define x 0 a y := -i(->x 0 a) 0 y, ^ a x := -<x 0 a. Then M(a, 1) = ([a, 1], 0 a , ->a, a) is an MV-algebra. MV-algebras in the sense of the above definition are right-MV-algebras since they are representable in the positive cones (the "right-hand side") of abelian lattice-ordered groups. But we can define a new binary operation 0 via x ©2/ := -i(->x0 -iy), 48 A NOTE ON INTERVAL MF-ALGEBRAS which leads to the concept of left-MV-algebras that are obtained from the neg-ative cones (the "left-hand side") of abelian lattice-ordered groups: A left-MV-algebra is an algebra M ' = (M, 0 , -., 1) of type (2,1,0) satisfying the axioms (MV17) (x 0 y) 0 z = x 0 (y 0 z), (MV2') x 0 y = y 0 x, (MV3') s © l = a ; , (MV4') -rnx = x, (MV5') z O - i l = - . l , (MV6') ->(->x Q y) O y =->(-iy O x) Q x. Let M ' be a left-MV-algebra and define x * y := x 0 -\y. It is easily seen that this binary operation has the following properties: (Al') x*0 = x, 0* .r = 0, (A2') x * (x * y) = y * (y * x), (A3') (x * y) * z = (x * z) *y. A tedious but straightforward calculation yields the analogue of Proposi-tion 1: PROPOSITION 2. Let A = (A, *, 0) be an algebra of type (2, 0) satisfying the identities (AV), (A27) and (A3'). Let a e A. Define x 0 a y := x * (a * y) and ->ax := a * x for x, y G [0, a] } Then M'(0, a) = ([0, a], 0 a , ~^a, a) is a left-MV-algebra. P r o o f . First observe that x * (a*y) = (a* (a* x)) * (a*y) = (a* (a * y)) * (a * x) = y * (a* x), which verifies (MV2 ;). (MV17): (x Ga y) 0 a z = (x * (a * y)) * (a* z) = (y * (a * x)) * (a* z) = (y * (a * z)) * (a * x) = x * (a * (y * (a* z))) = xOa(y Ga z). (MV37): By (AV) and (A2;) we have a * a = 0, and hence x 0 a a = x * (a * a) = x * 0 = x. 1 The induced order of A is defined by x < y if and only if x * y = 0. 49 IVAN CHAJDA — JAN KUHR (MV4'): -ia-iax = a * (a * x) = x . (MV5'): - ,aa 0 a x = (a * a) * (a * x) = 0 = - . a a . (MV6'): We have ->a(-iax 0 a 2/) 0 a y = y * (a * (a * (y * (a * (a * x ) ) ) ) ) = y * (y * x) and similarly -ia(-iay 0 a x) ©a x = x * (x * y) . D COROLLARY 3. If M' = (M, 0 , -., 1) 25 a left-MV-algebra and a G M . £be upon defining x Qa y := x 0 -.(a 0 ->y) and -»ax := a 0 -ix. £be structure M'(0, a) = ([0, a], 0 a , ^ a , a) is a left-MV-algebra. P r o o f . If M ' is a left- MV-algebra, then the algebra A(M' ) = (M, *,0) fulfils the above axioms, and so, by the proposition, M'(0, a) is a left- MV-al-gebra, where xQay = x*(a*y)=xQ ->(a 0 -.y) and ->°x = a*x = aQ-^x. D COROLLARY 4. Let M be an MV-algebra, a e M. Define M(0, a) = ([O,a],0a ,-ia ,O) via x ©a y := (x 0 y) A a ana7 -nax := -i(x ©->a). Then M(0, a) is an MV-algebra. P r o o f . For any x, y G [0, a] we have x 0 a y = V * ( ^ a x 0 a - a y ) = a 0 --(a 0 -nx 0 ->(a 0 -.(a 0 ->y))) = a A -i(-ix 0 ->(a A y)) = a A -i(-ix 0 ->y) = a A (x © y) and -iax = a 0 -ix = -i(->a © x) . D Combining Proposition 1 and Corollary 4, we get the promised description of interval MF-algebras: THEOREM 5. Let M be an MV-algebra and let a,b e M with a < b, where < is the natural order of M . Define M(a, b) = ([a, b], ©£, -.*, a) by x ©^ y := (->(->x © a) © y) A b and -£x := ->(x © -•&) © a . Tben M(a, b) is an MV-algebra. 50 A NOTE ON INTERVAL MV-ALGEBRAS P r o o f . Let x, y G [a, b]. Then x ®a y = (x ®a y) A 6 = (-i(->x ® a) ffi y) A b and = -i(-i(-ix © a) ® -16 ffi a) ffi a = n((xVfl) ffi -16) ffi a = -i(rcffi-i6) ffi a . • COROLLARY 6. Let M be an MV-algebra and let a,b e M such that a < b. where < is the natural order of M . If a,b G F?(M), tten in £be MV-algebra M(a, b) ufe fta^e x ffia 2/ = # © y and -<ax = (^x A b) V a. P r o o f . For any x, 2/ G [a, b], x ffia y = ("•(-»:£ ffi a) ffi y) A b = (-.(-ix V a) ffi 2/) A b = ((x A -ia) ffi y) A 6 = {x@y) A (->affiy) Ab = x ffi H since -<affi2/>- ,affia = l and x f f i 2 / < b © b = b. • COROLLARY 7. Le£ A = (A, o , l ) be an algebra satisfying (Al) , (A2) and (A3). Le£ a , 6 G . 4 s?/cb that a < b, where < is the induced order of A . Define x ®a 2/ := ((& ° ((# o a) o 2/)) o (b o a)) o a and ->ax := (b o x) o a . Tben M(a,b) = ([a, b], ®a ,->a ,a) is an MV-algebra. P r o o f . The structure ([a, 1], ®a , - i a , a) is an MT^-algebra, where x ®a y = (x o a) oy and -iax = x o a, and therefore, M(a, b) is an MF-algebra, where x ffia y = (x ffia 2/) A 6 = ((x o a) oy) Ab = ((b o ((x o a) o 2/)) o (b o a)) o a as [a, 1] is a lattice with x A y = ((2/ o x) o (y o a)) o a for any x, 2/ G [a, 1], and - a x = - . a (x f f i a - a b ) = ((x o a) o (bo a)) o a = (b o ((x o a) o a) j o a = (b o x) o a. • 51 IVAN CHAJDA — JAN KUHR REFERENCES [1] CIGNOLI, R. L. O .—D'OTTAVIANO, I. M. L.—MUNDICI, D . : Algebraic Foundations of Many-valued Reasoning, Kluwer Acad. Publ. , Dordrecht-Boston-London, 2000. [2] CHANG, C C : Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467-490. [3J CHAJDA, I. HALAS, R.—KUHR, J . : Distributive lattices with sectionally antitone in-volutions, Acta Sci. Math. (Szeged) 71 (2005), 19-33. [4] CHAJDA, I.—HALAS, R.—KUHR, J . : Implication in MV-algebras, Algebra Universalis 52 (2004), 377-382. [5] MUNDICI, D . : Interpretation of AF C* -algebras in Lukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15-63. Received May 24, 2004 Department of Algebra and Geometry Faculty of Science Palacký University Tomkova 40 CZ-779 00 Olomouc CZECH REPUBLIC E-mail: chajda@inf.upol.cz  kuhr@inf.upol.cz 52 

A note on interval $MV$-algebras

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A note on interval $MV$ -algebras

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