GGenerators of projective MV-algebras
Francesco Lacava · Donato Saeli
Abstract
In the last decade, interest in projective MV-algebras has growngreatly; see [1], [5] e [6]. In this paper we establish a necessary and sufficientcondition for n elements of the free n -generator MV-algebra to generate a pro-jective MV-algebra. This generalizes the characterization of the n free generatorsproved in [7]. Using this, some classes of projective generators for bigeneratedMV-algebras, are given. In particular, some effective procedures to determine, byelementary methods, generators of projective MV-algebras are explained.Nell’ultimo decennio, l’interesse per le MV-algebre proiettive `e cresciuto sensibil-mente; cfr. [1], [5] e [6]. In questa nota viene dimostrata una condizione necessariae sufficiente perch`e n elementi di F ree n . siano generatori di MV-algebre proiet-tive, che generalizza la caratterizzazione degli n generatori liberi dimostrata in [7].Utilizzando tale condizione nel caso di MV-algebre bigenerate sono state determi-nate alcune classi di generatori proiettivi. In particolare vengono esposti alcuniprocedimenti effettivi per la determinazione dei generatori di algebre proiettive,con metodi elementari. Keywords
MV-algebras · projective · retract · piecewise linear map MSC · Francesco LacavaUniversit`a degli Studi di FirenzeDipartimento di Matematica “Ulisse Dini”Viale Morgagni, 67/a - 50134 Firenze ItalyTel.: +39-055-7877316Email: [email protected]fi.itDonato SaeliVia Giovanni XXIII, 29 - 85100 Potenza, ItalyTel.: +39-0971-51280Email: [email protected] a r X i v : . [ m a t h . R A ] N ov francesco lacava, donato saeli Throughout this paper, by an MV-algebra A we mean a semisimple MV-algebra, that is, a subalgebra of a direct power of the standardMV-algebra [0, 1]. Thus for some set T, without loss of generality we canassume that each a ∈ A is a [0, 1]-valued function defined on T. In particular, since the free n -generator MV-algebra F ree n is semisimple,throughout we will identify F ree n with the algebra of McNaughton functionson the n -cube [0, 1] n . An MV-algebra A is said to be essentially n -generated if it has a gener-ating set of n elements, but no generating set of n - 1 elements. We refer to[4], for all unexplained notation and terminology.Throughout this paper, whenever a generates A we will assume that a (0) = 0 . Let (cid:32)L n +1 denote the (cid:32)Lukasiewicz MV-chain with n+1 elements. Thenfollowing [7], we will introduce the partial map t : (cid:32)L n +1 → (cid:32)L n +1 by stipulatingthat, for each a ∈ (cid:32)L n +1 t ( a ) = ( ra ) (cid:48) if there is r ∈ Z , r > , such that ra < r + 1) a = 1 , undefined otherwise.In particular, the map t is undefined for a = 0 ,
1. We further define t ( a ) = a , t s +1 ( a ) = t ( t s ( a )). Definition. - An element a ∈ (cid:32)L n +1 is said to be a cyclic generator of (cid:32)L n +1 , if there is an integer k ≥ t k ( a ) = 1 /n. Proposition 1.1. - If p is a prime number, then every element a ∈ (cid:32)L p +1 ,a (cid:54) = 0 , , is a cyclic generator of (cid:32)L p +1 . This is [7, 2.4].For p a prime number, let the integers m and p be such that 0 < m < p. Let k be the smallest positive integer satisfying t k (cid:16) mp (cid:17) = 1 p .Let the sequence of elements of (cid:32)L p +1 , be given by t (cid:16) mp (cid:17) = (cid:16) n mp (cid:17) (cid:48) , t (cid:16) mp (cid:17) = (cid:16) n t (cid:16) mp (cid:17)(cid:17) (cid:48) , . . . , t k (cid:16) mp (cid:17) = (cid:16) n k t k − (cid:16) mp (cid:17)(cid:17) (cid:48) Then the term γ m,p in the variable x is defined by γ m,p ( x ) = (cid:16) n k (cid:0) n k − . . . (cid:0) n ( n x ) (cid:48) (cid:1) (cid:48) . . . (cid:1) (cid:48) (cid:17) (cid:48) . When interpreted in the
F ree algebra, the term γ m,p representsa McNaughton function g m,p whose graph has three linear pieces,two of which (eventually degenerate into points) are horizontal, respectivelyat level 0 and at level 1. Moreover, g m,p ( m/p ) = 1 /p and g m,p ( x ) (cid:54) = 1 /p foreach x ∈ [0 ,
1] different from m/p.
This is [7, 2.5]. enerators of projective mv-algebras g , g , Fig. 1For p, a prime number, we can define the term λ p as follows: λ p ( x ) = (cid:16) px ∧ p (cid:0) ( p − x (cid:1) (cid:48) (cid:17) (cid:48) If interpreted in the
F ree algebra, the term λ p represents a McNaughtonfunction l p such that l p ( x ) = 0 if and only if x = 1 /p. The last nontrivialstatement follows by noting that x > /p implies p [1 − ( p − x ] < .l Fig. 2For all integers 0 < m < p, with p a prime number, let η m,p ( x ) = λ p ( γ m,p ( x )) . If interpreted in the
F ree algebra, the term η m,p represents a McNaughtonfunction t m,p such that:(1.1) For all x ∈ [0 , , t m,p ( x ) = 0 if and only if x = m/p. (1.2) For every L ∈ Z , L > there is Q = Q ( L ) ∈ Z , Q > such that,for all x ∈ [0 , with | x − m/p | ≥ /L, we have t m,p ( x ) ≥ /Q. This is [7, 2.6]. francesco lacava, donato saeli t , t , Fig. 3Following [4, 6.2.3], we say that an element a of an MV-algebrais archimedean if there is an integer n ≥ na + na = na. Lemma 1.2. - Suppose the elements a , a , . . . , a n ( essentially ) generatethe MV-algebra A of [0 , -valued functions . Let m i < p i be integers > with each p i prime (1 ≤ i ≤ n ) . Then the element n (cid:95) i =1 η m i ,p i ( a i ) fails to be archimedean if and only if for every ε > there is ¯ x ∈ [0 , n such that for i = 1 , . . . , n, we have (cid:12)(cid:12)(cid:12) a i (¯ x ) − m i p i (cid:12)(cid:12)(cid:12) < ε . Proof. - This follows at once from point (1.1) aforesaid. (cid:78)
Proposition 1.3. - Suppose the MV-algebras
A and B of [0 , -valuedfunctions are essentially generated by sets { a , a , ..., a n } and { b , b , ..., b n } respectively. Then the following two conditions are equivalent: • The map f : a i (cid:55)→ b i is ( uniquely ) extendible to an isomorphism of Aonto B • For all integers < m i < p i , with each p i prime, the element n (cid:95) i =1 η m i ,p i ( a i ) is archimedean if and only if so is the element n (cid:95) i =1 η m i ,p i ( b i ) . Proof. - ( ⇒ ) The statement is obvious.( ⇐ ) Vice versa, by contradiction, if we suppose that f cannot be extendedto an isomorphism between A and B, then a term t exists so that t ( a , a , . . . , a n ) = ˆ0 and t ( b , b , . . . , b n ) (cid:54) = ˆ0 . It follows that there are x ∈ [0 , n and δ > x ∈ III ( x , δ ) (the δ -neighbourhood of x ) we have t ( b ( x ) , b ( x ) , . . . , b n ( x )) (cid:54) = 0 . Since the b i ( x ) are McNaughton functions, it is possible to choose a x such that a neighborhood of ( b ( x ) , b ( x ) , . . . , b n ( x )) is contained in theimage by b ( x ) = ( b ( x ) , b ( x ) , . . . , b n ( x )) of III ( x , δ ) . Furthermore there are ¯ x ∈ [0 , n ∩ Q and ¯ δ > enerators of projective mv-algebras x ∈ III ( ¯ x , ¯ δ ) we have t ( b ( x ) , b ( x ) , . . . , b n ( x )) (cid:54) = 0 , also b i ( ¯ x ) = m i p i with p i prime numbers. So clearly, the element n (cid:95) i =1 η m i ,p i ( b i ) is not archimedeanand, by hypothesis, n (cid:95) i =1 η m i ,p i ( a i ) is not archimedean either.Now by the lemma 1.2 there is a ¯ ε such that, for each y satisfying (cid:12)(cid:12)(cid:12) y i − m i p i (cid:12)(cid:12)(cid:12) < ¯ ε ( i = 1 , . . . , n ) , we have t ( y ) (cid:54) = 0 . Since there is also an x ∈ [0 , n with (cid:12)(cid:12)(cid:12) a ( x i ) − m i p i (cid:12)(cid:12)(cid:12) < ¯ ε ( i = 1 , . . . , n ) , we conclude t ( a ( x ) , a ( x ) , . . . , a n ( x )) (cid:54) = 0 , which is a contradiction. (cid:78) In particular, the previous proposition tells us that in the in the freeone-generator MV-algebra
F ree , two subalgebras A and B, generatedrespectively by a and b, are isomorphic if and only if max( a ) = max( b ) = l ,that is if and only if, whatever are 0 < m < p, with p prime numberand mp ≤ l, we have that η m,p ( a ) is archimedean if and only if η m,p ( b ) isarchimedean.Let f, g ∈ F ree , with f (0) = g (0) = 0 . Then there is a sequence a = 0 < a < ... < a k − < a k = 1 of rational numbers in [0 , , togetherwith linear functions with integer coefficients p , . . . , p k , q , . . . , q k : R → R such that:1) over each interval [ a i − , a i ] , f coincides with p i , and g coincideswith q i , ( i = 1 , . . . , k )2) for each j = 2 , . . . , k, either p j − is distinct from p j or else q j − is distinct from q j . Let the function c = ( f, g ) : [0 , → [0 , defined by c ( t ) = ( f ( t ) , g ( t ));the shape of range of ( f, g ) is of course the broken line in [0 , joining, inthe order, the points P ≡ ( f ( a ) , g ( a )) , P ≡ ( f ( a ) , g ( a )) , . . . , P k ≡ ( f ( a k ) , g ( a k )) . We will name these points, that are known as the nodes of the range of ( f, g ) , extremals of the pair f and g. Proposition 1.4. - If f, g and f , g are two pairs of F ree elementssuch that the range of ( f, g ) coincides with the range of ( f , g ) , then thealgebra generated by f, g is isomorphic to the algebra generated by f , g . Proof. - It follows straight from proposition 1.3. (cid:78) francesco lacava, donato saeliExample 1.5. - The functions f ( x ) = x for 0 ≤ x ≤ − x for 16 ≤ x ≤ x − ≤ x ≤
131 for 13 ≤ x ≤ g ( x ) = x for 0 ≤ x ≤ − x for 16 ≤ x ≤ x − ≤ x ≤
381 for 38 ≤ x ≤ and the functions f ( x ) = x for 0 ≤ x ≤
131 for 13 ≤ x ≤ g ( x ) = x for 0 ≤ x ≤
121 for 12 ≤ x ≤ Corollary 1.6. - If f, g and f , g are two pairs of F ree elementswith the same extremals, then the algebra generated by f, g is isomorphicto the algebra generated by f , g . We recall that:- A MV-algebra A is named projective if, given any MV-algebras B and C, for each epimorphism g from C on B and each omomorphism f from A in B, then an omomorphism h from A in C, such that h ◦ g = f, alwaysexists.- The MV-algebra B is said a retract of the MV-algebra A, if there area monomorphism χ : B → A and an epimorphism ε : A → B such that εχ : B → B is the identity; so if B is a retract of A , then χ ( B ) isa subalgebra of A such as the endomorphism χε of A restricted to χ ( B ) isthe identity. It is known that a retract of a free MV-algebra is projective. enerators of projective mv-algebras F ree n . First of all, we observe, that if A is a projective MV-algebraessentially n -generated then it is a retract of F ree n . More generally,if
A, n -generated, is a retract of
F ree m with m > n, then A is a retractof F ree n . Lemma 2.1. - If x , . . . , x n , u , . . . , u n are elements of a MV-algebra, t isa term and δ ( x, u ) is the Chang’s distance, then there is k ∈ N such that δ (cid:0) t ( x , . . . , x n ) , t ( u , . . . , u n ) (cid:1) ≤ k (cid:2) δ ( x , u ) + · · · + δ ( x n , u n ) (cid:3) . Proof. - By an easy induction from a theorem on elementary propertiesof the Chang’s distance [2, 3.14]. (cid:78)
Proposition 2.2. - Let B be a n -generated subalgebra of the free algebra A = F ree n and let be g , . . . , g n the generators of A. If ϕ is an epimor-phism from A on B, then ϕ ( g ) = d , . . . , ϕ ( g n ) = d n generate B and thekernel of ϕ is the ideal I generated by δ ( d , g ) , . . . , δ ( d n , g n ) . Proof. - If a ∈ A, where a = t ( g , . . . , g n ) , is such that ϕ ( a ) = 0 , then: a = δ ( a,
0) = δ (cid:0) t ( g , . . . , g n ) , t ( d , . . . , d n ) (cid:1) ≤ k (cid:2) δ ( g , d ) + · · · + δ ( g n , d n ) (cid:3) , for an appropriate k ∈ N , that is a ∈ I. (cid:78) Thus, in order that ϕ restricted to B is an injection, it is necessary (andsufficient) that if b ∈ B and b ≤ k (cid:2) δ ( g , d ) + · · · + δ ( g n , d n ) (cid:3) , then b = 0 . Finally with the notations and conditions set above (proposition 2.2), weintroduce a set K, that we say “ equalizer ” of B : Definition 2.3. - K = (cid:8) x ∈ [0 , n ::: d ( x ) = g ( x ) , . . . , d n ( x ) = g n ( x ) (cid:9) . Theorem 2.4. - Let B be a n -generated subalgebra of A = F ree n , ϕ an epimorphism from A on B and let g , . . . , g n be the generators of A. Set d = ϕ ( g ) , . . . , d n = ϕ ( g n ) , then the epimorphism ϕ restricted to B, is an isomorphism if and only if for every u ∈ [0 , n there is x ∈ K such that d ( u ) = d ( x ) , . . . , d n ( u ) = d n ( x ) . Proof. - If for b ∈ B, b = t ( d , . . . , d n ) , were b ≤ k (cid:2) δ ( d , g ) + · · · + δ ( d n , g n ) (cid:3) and b (cid:54) = 0 , then there would be u ∈ [0 , n such that: b ( u ) = t (cid:0) d ( u ) , . . . , d n ( u ) (cid:1) (cid:54) = 0; but by hypothesis, x ∈ K exists such that d ( u ) = d ( x ) , . . . , d n ( u ) = d n ( x )and so t (cid:0) d ( u ) , . . . , d n ( u ) (cid:1) = t (cid:0) d ( x ) , . . . , d n ( x ) (cid:1) = b ( x ) (cid:54) = 0 . But, since d ( x ) = g ( x ) , . . . , d n ( x ) = g n ( x ) we have: (cid:0) k (cid:2) δ ( d , g ) + · · · + δ ( d n , g n ) (cid:3)(cid:1) ( x ) = k (cid:2) δ (cid:0) d ( x ) , g ( x ) (cid:1) + · · · + δ (cid:0) d n ( x ) , g n ( x ) (cid:1)(cid:3) = 0and so would be 0 (cid:54) = b ( x ) ≤ (cid:0) k (cid:2) δ ( d , g ) + · · · + δ ( d n , g n ) (cid:3)(cid:1) ( x ) = 0 . Vice versa, let u ∈ [0 , n be such that for every x ∈ K,d i ( u ) (cid:54) = d i ( x ) for some i, ≤ i ≤ n, francesco lacava, donato saeli consequently there is ε > v ∈ I ( u , ε ) and x ∈ K it is d i ( v ) (cid:54) = d i ( x ) . In particular there areintegers 0 < m < p , . . . , < m n < p n , with p , . . . , p prime numbers, u ∈ I ( u , ε ) and (cid:37) > d ( u ) = m p , . . . , d n ( u ) = m n p n and I ( u , (cid:37) ) ⊆ I ( u , ε ) . Then we can choose h ∈ N so that the element b = (cid:16) h n (cid:95) i =1 η m i ,p i ( d i ) (cid:17) (cid:48) is nonzero only in I ( u , (cid:37) ) . Thus, for k ∈ N large enough,is b ≤ k (cid:2) δ ( g , d ) + · · · + δ ( g n , d n ) (cid:3) , but b (cid:54) = 0 also. (cid:78) Corollary 2.5. - Let A be a n -generated MV-algebra. A is projectiveif and only if is isomorphic to a subalgebra B of F ree n , whereby,however we choose u ∈ [0 , n there is x ∈ K such that d ( u ) = d ( x ) , . . . , d n ( u ) = d n ( x ) . Corollary 2.6 (Di Nola). - Every monogenerated subalgebra of
F ree is projective. We devote the next section to some constructive applicationsof theorem 2.4 for bigenerated projective algebras.
Since the equalizer K must be a finite union of triangular simplexes T , T , ..., T k and each triangular simplex T i is determinated by a system a i x + b i y + c i ≥ a i x + b i y + c i ≥ α i ) a i x + b i y + c i ≥ Theorem 3.1. - A subalgebra B of F ree generated by a and b, is projective if and only if the pairs (cid:0) m p , m p (cid:1) , with < m j < p j and p j prime ( j = 1 , , that satisfy one of the systems ( α ) , ( α ) , . . . , ( α k ) , are all and only those for wich the element η m ,p ( a ) ∨ η m ,p ( b ) is not archi-medean. Proof. - By corollary 2.5. (cid:78)
Therefore, it is important to establish wich are the triangular simplexesthat give rise the projective algebras. Certainly, the union of such simplexesmust be connected and it must contain . But it can be not convex, as it iseasy to see. enerators of projective mv-algebras Remark 3.2. - On the plane
Oxz, we consider the points O ≡ (0 , , P ≡ ( a, h ) , K ≡ (0 , k ) , Q ≡ ( a, l ) ,O (cid:48) ≡ ( b, , P (cid:48) ≡ ( c, h ) , K (cid:48) ≡ ( b, k ) and Q (cid:48) ≡ ( c, l ) , where the real numbers a, b and c, each nonzero, are pairwise different.Similarly, h, k and l are supposed to be all nonzero and pairwise different.If s and t ∈ R , the line x = s intersects the line through O P and theline through
K Q respectively at points S ≡ ( s, u ) and S (cid:48) ≡ ( s, u (cid:48) ); andthe line x = t intersects the line through O (cid:48) P (cid:48) and the line through K (cid:48) Q (cid:48) respectively at points T ≡ ( t, v ) and T (cid:48) ≡ ( t, v (cid:48) ) . If, for a given s, we choose t = b + [ s ( c − b ) /a ] ( or if, for a given t, we choose s = a ( t − b ) / ( c − b )) , then u = v and u (cid:48) = v (cid:48) . Fig. 50 francesco lacava, donato saeli
In the space R , we denote by ∆ U the triangle whose vertices belong tothe set U = (cid:8) ( t , v , z ) , ( t , v , z ) , ( t , v , z ) (cid:9) , wherethe points ( t , v , , ( t , v , , ( t , v ,
0) are not aligned. We denote by z = f U ( t, v ) the linear function corresponding to the plane settled by thepoints of U . Proposition 3.3.
Let the sets: A = (cid:8) ( x , y , a ) , ( x , y , a ) , ( x , y , a ) (cid:9) , B = (cid:8) ( x , y , b ) , ( x , y , b ) , ( x , y , b ) (cid:9) , C = (cid:8) ( x , y , , ( x , y , , ( x , y , (cid:9) , with the condition that the points of C are not aligned.Arbitrarily chosena set J = (cid:8) ( ξ , η , , ( ξ , η , , ( ξ , η , (cid:9) , whose points are not aligned,a 3-list ( i , i , i ) in { , , } , and consequently also the sets: L = (cid:8) ( ξ , η , a i ) , ( ξ , η , a i ) , ( ξ , η , a i ) (cid:9) and M = (cid:8) ( ξ , η , b i ) , ( ξ , η , b i ) , ( ξ , η , b i ) (cid:9) . Then, for every ( ξ , η , ∈ ∆ J there is ( x , y , ∈ ∆ C such that f L ( ξ , η ) = f A ( x , y ) and f M ( ξ , η ) = f B ( x , y ) . Proof. - Suppose first, that ( i , i , i ) is a permutation of { , , } . Let α the plane passing through the points ( ξ , η , a i ) , ( ξ , η , b i )and ( ξ , η , . This plan meets the plane β, passing through the points( ξ , η , a i ) , ( ξ , η , b i ) and ( ξ , η , a i ) , along a straight line r which intersects the side of the triangle ∆ L , of extremes ( ξ , η , a i ) and( ξ , η , a i ) and the side of the triangle ∆ M , of extremes ( ξ , η , b i ) and( ξ , η , b i ) , respectively, in points ( ξ , η , h ) and ( ξ , η , k ) (Fig. 6.0).For the remark 3.2, on the side of the triangle ∆ C , of extremes ( x i , y i , x i , y i , , there is a point ( x , y ,
0) such that f A ( x , y ) = h and f B ( x , y ) = k (Fig 6.1). Just now consider the plane γ passing throughthe points ( x i , y i , a i ) , ( x i , y i , b i ) e ( x , y ,
0) and the assertion followsby a further application of the remark 3.2 (Fig. 6.2).If ( i , i , i ) is not a permutation of { , , } , similar proofs hold. (cid:78) enerators of projective mv-algebras Here the 3-list is (2,3,1). For simplicity, points in plane z = 0 are denotedwith the first two coordinates, while the other points only with their heights. Fig. 6.0
First application of remark 3.2, this yields the point( x , y ,
0) corresponding to the point ( ξ , η , Fig. 6.12 francesco lacava, donato saeli
Second application of remark 3.2, this yields the point( x , y ,
0) corresponding to the point ( ξ , η , Fig. 6.2We conclude finally exposing some applications. i ) - A first very simple case arises if the equalizer is: K = (cid:8) ( x, y ) ∈ [0 , : g ( y ) ≤ x ≤ g ( y ) and f ( x ) ≤ y ≤ f ( x ) (cid:9) , where f ( x ) , f ( x ) , g ( y ) and g ( y ) are McNaughton functionsand the follow conditions are satisfied: f ( x ) ≤ f ( x ) , g ( y ) ≤ g ( y ); ( • ) y ≥ f (cid:0) g ( y ) (cid:1) , y ≤ f (cid:0) g ( y ) (cid:1) , y ≤ f (cid:0) g ( y ) (cid:1) , y ≥ f (cid:0) g ( y ) (cid:1) . Set : A = (cid:8) ( x, y ) ∈ [0 , : y ≤ f ( x ) (cid:9) ,B = (cid:8) ( x, y ) ∈ [0 , : y ≥ f ( x ) (cid:9) ,C = (cid:8) ( x, y ) ∈ [0 , : x ≤ g ( y ) (cid:9) ,D = (cid:8) ( x, y ) ∈ [0 , : x ≥ g ( y ) (cid:9) . Fig. 7 enerators of projective mv-algebras d ( x, y ) = x if ( x, y ) ∈ A ∪ K ∪ B,g ( y ) ” ( x, y ) ∈ C,g ( y ) ” ( x, y ) ∈ Dd ( x, y ) = y if ( x, y ) ∈ C ∪ K ∪ D,g ( y ) ” ( x, y ) ∈ A,g ( y ) ” ( x, y ) ∈ B generate, by the corollary 2.5 and the proposition 3.3, a projective subalgebraof F ree . Example 3.4. - The McNaughton functions f ( x ) = ≤ x ≤ − x for 13 ≤ x ≤ x − ≤ x ≤ − x for 23 ≤ x ≤ , f ( x ) = − x for 0 ≤ x ≤ x for 12 ≤ x ≤ g ( y ) = ≤ x ≤ y − ≤ y ≤ − y for 13 ≤ y ≤ y − ≤ y ≤ − y for 23 ≤ y ≤ , g ( y ) = ≤ y ≤ − y for 13 ≤ y ≤ y for 25 ≤ y ≤
121 for 12 ≤ y ≤ − y for 23 ≤ y ≤ y − ≤ y ≤
341 for 34 ≤ y ≤ satisfy the conditions ( • ); Fig. 84 francesco lacava, donato saeli Therefore d ( x, y ) = y − x ≤ y − and y ≤ / , − y ” x ≤ − y and y ≥ / , y − x ≤ y − and y ≤ / , − y ” x ≤ − y and y ≥ / , − y ” x ≥ − y and y ≤ / , y ” x ≥ y and y ≥ / , − y ” x ≥ − y and y ≤ / , y − x ≥ y − and y ≥ / ,x otherwise and d ( x, y ) = x if y ≤ x and x ≤ / , − x ” y ≤ − x and x ≥ / , x − y ≤ x − and x ≤ / , − x ” y ≤ − x and x ≥ / , − x ” y ≥ − x and x ≤ / ,x ” y ≥ x and x ≥ / ,y otherwise are the generators of a projective subalgebra of F ree . d ( x, y ) d ( x, y ) Fig. 9
Example 3.5. - We consider the subalgebra A of F ree generated by thefunctions: f ( x ) = x for 0 ≤ x ≤ − x for 13 ≤ x ≤ x − ≤ x ≤ g ( x ) = x for 0 ≤ x ≤ − x for 13 ≤ x ≤ x − ≤ x ≤ x − ≤ x ≤ . enerators of projective mv-algebras f ( x ) g ( x ) Fig. 10We can see easily that the extremals of f ( x ) and g ( x ) are the points P ≡ (0 , , P ≡ (1 , , P ≡ (1 / , / , P ≡ (0 ,
1) and P ≡ (1 , f, g ) consists in the diagonals of the square [0 , . However, these diagonals may be regarded as the set of the points of [0 , enclosed by the following four McNaughton functions f ( x ) = x for 0 ≤ x ≤ − x for 12 ≤ x ≤ , f ( x ) = − x for 0 ≤ x ≤ x for 12 ≤ x ≤ ,g ( y ) = y for 0 ≤ y ≤ − y for 12 ≤ y ≤ , g ( y ) = − y for 0 ≤ y ≤ y for 12 ≤ y ≤ , that obviously satisfy the conditions ( • ) . It follows that the functions d ( x, y ) = y if 0 ≤ y ≤ and y ≥ x, or12 ≤ y ≤ and y ≤ x, − y if 0 ≤ y ≤ and y ≥ − x, or12 ≤ y ≤ and y ≤ − x,x otherwise and d ( x, y ) = x if 0 ≤ x ≤ and x ≥ y, or12 ≤ x ≤ and x ≤ y, − x if 0 ≤ x ≤ and x ≥ − y, or12 ≤ x ≤ and x ≤ − y,y otherwise francesco lacava, donato saeli d ( x, y ) d ( x, y ) Fig. 11are the generators of a projective subalgebra of
F ree ;but this, by the proposition 1.3, is isomorphic to A.ii ) - Now, let’s consider the case that the equalizer is: K = (cid:8) ( x, y ) ∈ [0 , : g ( y ) ≤ x ≤ g ( y ) and f ( x ) ≤ y ≤ f ( x ) (cid:9) , where f ( x ) , f ( x ) , g ( y ) and g ( y ) are broken lines, for wichthe conditions: ≤ f ( x ) ≤ x for 0 ≤ x ≤ , ≤ f ( x ) ≤ − x for 12 ≤ x ≤ , − x ≤ f ( x ) ≤ ≤ x ≤ ,x ≤ f ( x ) ≤ ≤ x ≤ , ≤ g ( y ) ≤ y for 0 ≤ y ≤ , ≤ g ( y ) ≤ − y for 12 ≤ y ≤ , − y ≤ g ( y ) ≤ ≤ y ≤ ,y ≤ g ( y ) ≤ ≤ y ≤ enerators of projective mv-algebras A = (cid:8) ( x, y ) ∈ [0 , : y ≤ f ( x ) (cid:9) ,B = (cid:8) ( x, y ) ∈ [0 , : y ≥ f ( x ) (cid:9) ,C = (cid:8) ( x, y ) ∈ [0 , : x ≤ g ( y ) (cid:9) .D = (cid:8) ( x, y ) ∈ [0 , : x ≥ g ( y ) (cid:9) , are convex. Fig. 12For short, only a special case is examined; from wich, by extensionand symmetry, it follows easily how to treat the general case. Let f ( x ) = axb for 0 ≤ x ≤ bcad + bc ,c (1 − x ) d for bcad + bc ≤ x ≤ , with integers a ≥ , b ≥ a, c ≥ d ≥ c ; f ( x ) = 1 , g ( y ) = 0 , g ( y ) = 1 . If on the plane z = 0 , we consider the points P ≡ (cid:16) bcad + bc , acad + bc (cid:17) and Q ≡ (0 , , then the segments OP and P Q represent f ( x ) . In the sheaves z − y + λ ( by − ax ) = 0 and z − y + µ ( dy + cx − c )we should choose respectively the planes z = (1 − b ) y + ax (1)and z = (1 − d ) y + c (1 − x ) , (2)wich intersect each other along the straight line (cid:26) ( d − b ) y + ( a + c ) x = c (1 − b ) y + ax = z. The plane z = x intersects the plane (1) on the straight line (cid:26) ( b − y = ( a − xz = x. On the plane z = 0 , the straight lines ( b − y = ( a − x and ( d − b ) y + ( a + c ) x = c intersect each other in a point S ≡ ( x S , y S ) , where x S = c ( b − b − a + c ) + ( d − b )( a − . francesco lacava, donato saeli Therefore, we must distinguish three cases, according to x S . If x S ≤ , the planes z = 1 − x and z = x intersect the plane (2),respectively on the straight lines (cid:26) ( d − y + ( c − x = c − z = 1 − x and (cid:26) ( d − y + ( c + 1) x = cz = x ;so that on the plane z = 0 , the straight line ( d − y + ( c + 1) x = c intersects the straight line ( d − y + ( c + 1) x = c in the point T ≡ (cid:18) , · c − d − (cid:19) and obviously, the straight line ( b − y + ( a − x = 0 in the point S. So, if R ≡ (0 , /
2) indicates the middle point of
OQ, the triangle
OP Q constists of two quadrilaterals
OST R and
P ST Q and of two triangles
OSP and
QRT (Fig. 13). Fig. 13And again by corollary 2.5 and proposition 3.3, it follows that the functions d ( x, y ) = x and d ( x, y ) = (1 − b ) y + ax if ( x, y ) ∈ OSP, (1 − d ) y + c (1 − x ) ” ( x, y ) ∈ P ST Q,x ” ( x, y ) ∈ OST R, − x ” ( x, y ) ∈ QRT,y ” ( x, y ) ∈ K generate a projective subalgebra of F ree . enerators of projective mv-algebras x S = 12 , then the points S and T coincide; so that, for the definitionof d ( x, y ) , the four triangles OSP P SQ OSR and
QRS are to be con-sidered (Fig. 14). Fig. 14That is d ( x, y ) = (1 − b ) y + ax if ( x, y ) ∈ OSP, (1 − d ) y + c (1 − x ) ” ( x, y ) ∈ P SQ,x ” ( x, y ) ∈ OSR, − x ” ( x, y ) ∈ QRS,y ” ( x, y ) ∈ K . Finally, if x S ≥ , then on the plane z = 0 , the straight lines( d − y + ( c − x = c − d − b ) y + ( a + c ) x = c intersect eachother in a point U ≡ ( x U , y U ) where x U = c ( b −
1) + d − b ( d − a + c ) − ( d − b )( c −
1) ;it is easy to check that x S ≥
12 implies x U ≥ . On the plane z = 0 , the straight line ( b − y − ( a + 1) x = − , projection of the straight line intersection of the plane (1) with the plane z = 1 − x, intersects the straight line ( b − y − ( a − x = 0 at thepoint V ≡ (cid:18) , · a − b − (cid:19) and the straight line ( d − y + ( c − x = c − U (Fig. 15).We have: d ( x, y ) = (1 − b ) y + ax if ( x, y ) ∈ OV U P, (1 − d ) y + c (1 − x ) ” ( x, y ) ∈ P U Q,x ” ( x, y ) ∈ OV R, − x ” ( x, y ) ∈ QU V R,y ” ( x, y ) ∈ K . francesco lacava, donato saeliExample 3.6. - Let f ( x ) = x ≤ x ≤ , − x )8 for 2137 ≤ x ≤ ,f ( x ) = 1 , g ( y ) = 0 , g ( y ) = 1 . We have: P ≡ (cid:16) , (cid:17) , x S = 1831 , x U = 1933 , V ≡ (cid:16) , (cid:17) ;Fig. 15 d ( x, y ) = x,d ( x, y ) = x − y ) if y < = 2 x/ and y < = 3 − x and y > = x/ and y > = (3 x − / , − x ) − y ” y > = 3 − x and y < = 3(1 − x ) / and y > = 2(1 − x ) / ,x ” y < = x/ and x < = 1 / , − x ” x > = 1 / and y < = (3 x − / and y < = 2(1 − x ) / ,y ” y > = 2 x/ or y > = 3(1 − x ) / . enerators of projective mv-algebras d ( x, y ) Fig. 16 iii ) - Now we examine the case that equalizer K is identified by a generictriangle with a vertex at the origin.Fig. 17Let ax + by = 0 , a x + b y = 0 and a x + b y + c = 0 be respectivelythe equations of the straight lines for OA, OB, AB (Fig. 17),with a, a , a , b, b , b ∈ Z , a, a , c > , b, b < , and( a, b ) = ( a , b ) = ( a , b , c ) = 1 . Then: K = { ( x, y ) : ax + by ≤ a x + b y ≥ a x + b y + c ≥ } . francesco lacava, donato saeli We can apply theorem 2.4, taking into account the proposition 3.3; moreprecisely, we proceded to the construction of the generators d and d as-suming that they coincide, respectively, with g = x and g = y on theequalizer K. Then, we connect these “basic pieces” of d and d with theplane z = 0 , by appropriate planes, so that the hypothesis of the theorem2.4 are satisfied.For the side OA, we consider two planes belonging respectively to the sheaves z = x + k ( ax + by ) and z = y + h ( ax + by ) . As the only point of K, whereone of the coordinates x, y takes zero value is the origin of axes, then thesetwo planes must intersect necessary the plane z = 0 on the same straightline. This implies ( ka + 1)( hb + 1) − hakb = 0 , that is ka + hb + 1 = 0 . So, the planes of the relative sheaves that satisfy such condition are: z = − hbx + kby (1) z = hax − kay (2)and their intersection straight line with the plane z = 0 is y = hk x. (3)If k , h is a particular solution of the diophantine equation ak + bh = − , the general solution has the form: k = k − sb, h = h + sa, with s ∈ Z wichever;so that hk = − ab + 1 /b s − k /b and for s < k b + 1 ab we have: 0 < hk < − ab . In a similar manner, we proceed for the side OB ; fixing the planes z = − h (cid:48) b x + k (cid:48) b y (4) z = h (cid:48) a x − k (cid:48) a y (5)as well as their intersection straight line with the plane z = 0 y = h (cid:48) k (cid:48) x (6)with the conditions k (cid:48) a + h (cid:48) b + 1 = 0 and h (cid:48) k (cid:48) > − a b . Now, we indicate with A (cid:48) and A (cid:48)(cid:48) , respectively, the intersection pointsof the vertical for A with the planes z = x and z = y ; with B (cid:48) and B (cid:48)(cid:48) the intersection points of the vertical for B with the same planes. enerators of projective mv-algebras δ = ba − ab and δ = b a − a b , we have: A ≡ (cid:16) − bcδ , acδ (cid:17) and B ≡ (cid:16) − b cδ , a cδ (cid:17) , with δ, δ >
0; consequently A (cid:48) ≡ (cid:16) − bcδ , acδ , − bcδ (cid:17) , A (cid:48)(cid:48) ≡ (cid:16) − bcδ , acδ , acδ (cid:17) and B (cid:48) ≡ (cid:16) − b cδ , a cδ , − b cδ (cid:17) , B (cid:48)(cid:48) ≡ (cid:16) − b cδ , a cδ , a cδ (cid:17) . We consider the sheaves of planes z − x + l ( a x + b y + c ) = 0 and z − y + m ( a x + b y + c ) = 0 , generated by the plane a x + b y + c = 0 and, respectively, by the planes z = x and z = y ; these sheaves intercept on the plane z = 0 the sheaves ofstraight lines x = l ( a x + b y + c ) and y = m ( a x + b y + c ) . So the coordinates of the point P, intersection of a straight line of the firstsheaf with one straight line of the second, must satisfy the system (cid:26) l ( a x + b y + c ) = xm ( a x + b y + c ) = y. It follows, immediately, that the point P must lie on the straight line throughthe origin of equation mx = ly and that P ≡ (cid:16) lc − la − mb , mc − la − mb (cid:17) . In order that, the straight line through OP lies between the straight linesthrough OA and OB (Fig. 18), it must be − ab < ml < − a b , and the parameters m and l must have the same sign.If m < l < − la − mb <
0; so that is a lc − la − mb + b mc − la − mb + c = c − la − mb < , therefore P is an outer point to the equalizer K (it is in zone 3) and when m and l in absolute value increase, its distance from the side AB of K decreases.The equation of a generic straight line of the sheaf generated by a straight4 francesco lacava, donato saeli line of the first sheaf with one of the second, (that is a straight line of theplane z = 0 passing through P ) is: h (cid:2) l ( a x + b y + c ) − x (cid:3) + k (cid:2) m ( a x + b y + c ) − y (cid:3) = 0 , with h, k ∈ Z , wich can be put in the form (cid:2) a ( hl + km ) − h (cid:3) x + (cid:2) b ( hl + km ) − k (cid:3) y + c ( hl + km ) = 0 . Let ¯ ax + ¯ by + ¯ c = 0 (7)and ˆ ax + ˆ by + ˆ c = 0 (8)be the equations for two straight lines belonging to this sheaf.The equation of a McNaughton generic plane of the sheaf whose axis is thestraight line (7), can be written in the form z = λ (¯ ax + ¯ by + ¯ c ) = λ (cid:110) ¯ h (cid:2) l ( a x + b y + c ) − x (cid:3) + ¯ k (cid:2) m ( a x + b y + c ) − y (cid:3)(cid:111) , with λ integer, and in order that the point A (cid:48) lies on this plane, it must be: − bcδ = λ (cid:48) (cid:110) ¯ h (cid:104) bcδ (cid:105) + ¯ k (cid:104) − acδ (cid:105)(cid:111) , from wich − b = λ (cid:48) (¯ hb − ¯ ka );while, if we require that the point A (cid:48)(cid:48) lies on a plane of the same sheaf,it must be: acδ = λ (cid:48)(cid:48) (cid:110) ¯ h (cid:104) bcδ (cid:105) + ¯ k (cid:104) − acδ (cid:105)(cid:111) , from wich a = λ (cid:48)(cid:48) (¯ hb − ¯ ka ) . As ( a, b ) = 1 , it follows necessarily¯ hb − ¯ ka = ∓ , λ (cid:48) = ± b and λ (cid:48)(cid:48) = ∓ a. If we choose ¯ hb − ¯ ka = 1 , then λ (cid:48) = − b and λ (cid:48)(cid:48) = a and the equations oftwo planes of the sheaf, one passing through A (cid:48) , the other through A (cid:48)(cid:48) , arerespectively: z = − b ¯ ax − b ¯ by − b ¯ c (9)and z = a ¯ ax + a ¯ by + a ¯ c. (10)These planes intersect z -axis in two points with z (cid:48) = − b ¯ c = − bc (¯ hl + ¯ km ) and z (cid:48)(cid:48) = a ¯ c = ac (¯ hl + ¯ km );so that z (cid:48) and z (cid:48)(cid:48) > hl + ¯ km > . Now the equation ¯ hb − ¯ ka = 1 admits the integer solutions:¯ h = h + at, ¯ k = k + bt, with arbitrary integer parameter t ; so the condition¯ hl + ¯ km > t ( la + mb ) > − mk − lh and this is satisfiedfor t > − mk + lh la + mb (being la + mb > . enerators of projective mv-algebras B (cid:48) and B (cid:48)(cid:48) , are: z = − b ˆ ax − b ˆ by − b ˆ c (11)and z = a ˆ ax + a ˆ by + a ˆ c ; (12)with the integer parameters ˆ h and ˆ k (wich determine ˆ a, ˆ b and ˆ c ) satisfyingthe conditions ˆ hb − ˆ ka = 1 , and ˆ hl + ˆ km > . Finally, we note that the planes through the points A (cid:48) , B (cid:48) , P and A (cid:48)(cid:48) , B (cid:48)(cid:48) , P have, respectively, equations: z = (1 − la ) x − lb y − lc (13)and z = − ma x + (1 − mb ) y − mc. (14)We denote, also respectively, by Q and R the intersection points betweenthe straight lines (3), (7) and (6), (8).Fig. 18So the generators d and d , zero along and outside of the broken line O Q P R O, coincide: d with planes (1), (9), (13), (11), (4) and z = x and d with planes (2), (10), (14), (12), (5) and z = y, respectively, on the triangles O Q A, Q A P, A P B, P B R, B R O and K (Fig. 18).It follows immediately, by proposition 3.3, that the generators d and d , francesco lacava, donato saeli so defined, satisfy the conditions of the theorem 2.4.These considerations can be easily extended also to the case thatequalizer K is decomposable into triangles, all with a vertex in the origin.Infact, if the construction, above described, is repeated for each triangle O A B , O A B , . . . , O A n B n , we obtain in particular respectively the quadrilaterals O Q P R , O Q P R , . . . , O Q n P n R n , whose union is a polygon O Q P S . . . S m P n R n O that containes K. As each segment of the perimeter of this polygon belongsto one of the quadrilaterals
O Q i P i R i , it is possible to use convenient piecesof planes determinated for each triangle, to define the generators (Fig. 19).Fig. 19In conclusion, using the methods described in § b ) , in the case of equalizerscontaining the diagonals and whose boundary consists of continuous andpiecewise linear functions, projective generators are obtained. Similarly, thesame result is obtained in § c ) , in the case of equalizers formed by triangleswith a common vertex coinciding with one of the vertices of [0 , . Apart enerators of projective mv-algebras § References L. M. Cabrer - D. Mundici , Projective MV-algebras and rational polyhedra ,Algebra Universalis , no. 1, 63-74 (2009).2. C. C. Chang , Algebraic analysis of many valued logics , Trans. Amer. Math.Soc., , 467-490 (1958).3. C. C. Chang , A new proof of the completeness of the (cid:32)Lukasiewicz axioms ,Trans. Amer. Math. Soc., , 74-80 (1959).4. R. Cignoli - I. D’Ottaviano - D. Mundici , Algebraic foundations of many-valued reasoning, Trends in Logic, Kluwer, Dordrecht 2000.5.
A. Di Nola - R. Grigolia - A. Lettieri , Projective MV-algebras , Internat.J. Approx. Reason. , no. 3, (2008) 323-332.6. A. Di Nola - R. Grigolia , Projective MV-algebras and their automorphismgroups , J. Mult.-Valued Logic Soft Comput. , no. 3, 291-317 (2003).7. F. Lacava , A characterization of free generating sets in MV-algebras , AlgebraUniversalis57