Proposition Let A be an MV chain. The zerosets of a polynomial p ( x 1 , . . . , x n , a 1 , . . . , a m ) ∈ A [ x 1 , . . . , x n ] coincide with finite unions of polyhedra of the form { x | a ( x ) ≥ 0 } , where a ( x ) is an affine function on A.
With the same kind of argument one can prove the following analogue of McNaughton Theorem itself for MV-chains. Call McNaughton function over A a function f : A n → A for which there is a covering of A n by finitely many polyhedra P 1 , . . . , P k of the form { x | a ( x ) ≥ 0 } , such that f on each polyhedron is affine.
Proposition Let A be an MV chain. Let p ∈ A [ x 1 , . . . , x n ] . Then p defines a McNaughton function from A n to A. Conversely, every McNaughton function from A n to A is definable by a polynomial.
Algebraic Sets : In this section we focus on Diophantine algebraic geometry: that is, we take the same algebra A both to define constants in polynomials and to evaluate polynomials. Definition Let A be an MV-algebra. Let S ⊆ A [ x 1 , . . . , x n ] , S � = ∅ . Consider the set { ( a 1 , . . . , a n ) ∈ A n | p ( a 1 , . . . . a n ) = 0 , ∀ p ( x 1 , . . . , x n ) ∈ S } . Denote this set by V ( S ), called the algebraic set determined by S . Note that algebraic sets are determined by ideals.
Definition Call an ideal J ⊆ A [ x 1 , . . . , x n ] singular if V ( J ) = ∅ . Otherwise call J non- singular .
Proposition Suppose we have a non-empty X ⊆ A n . Then let I ( X ) = { p ∈ A [ x 1 , . . . , x n ] | p (¯ y ) = 0 , ∀ ¯ y ∈ X } where ¯ y = ( y 1 , . . . , y n ) , y i ∈ A. Then I ( X ) is an ideal of A [ x 1 , . . . , x n ] .
Point ideals and point radicals : Call an ideal J ⊆ A [¯ x ] a point ideal if for some a = ( a 1 , . . . , a n ) ∈ A n we have J = I (¯ ¯ a ).
We consider the fixpoints of the adjunction ( I , V ): √ For an ideal I ⊆ A [¯ x ] let pt I = � { I (¯ a ) | I ⊆ I (¯ a ) } . √ We call pt I the point radical of I .
We have: Proposition √ For a non-singular ideal J , I ( V ( J )) = pt J.
We want to characterize those ideals J ⊆ A [ x 1 , . . . , x n ] such that I ( V ( J )) = J .
Nullstellensatz theorem : Theorem The ideals J such that I ( V ( J )) = J are exactly the point-radical ideals. Proposition There is a one-one correspondence between point − radicals and sets . algebraic
algebras : Coordinate Definition Let Z ⊆ A n be a non-empty algebraic set. By the co-ordinate MV-algebra of Z we mean the MV-algebra A [¯ x ] / I ( Z ) .
Proposition For a non-singular ideal J the co-ordinate MV-algebra of V ( J ) is √ A [¯ x ] / pt J. √ Let MV A = { A [ x 1 , . . . , x n ] / J | J = pt J , n = 0 , 1 , 2 . . . } . Then MV A is a category with morphisms the MV-homomorphisms. The category of Coordinate algebras .
Definition Let Z 1 ⊆ A n , Z 2 ⊆ A m be algebraic sets. A mapping ϕ : Z 1 → Z 2 is called a polynomial map iff there are polynomials p 1 , . . . , p m ∈ A [ x 1 , . . . , x n ] such that ϕ ( a 1 , . . . , a n ) = ( p 1 ( a 1 , . . . , a n ) , . . . , p m ( a 1 , . . . , a n )) for every ( a 1 , . . . , a n ) ∈ Z 1 .
Let Z ( A ) be the collection of all algebraic subsets of A n . Then with polynomial maps as morphisms, Z ( A ) becomes a category , The category of Algebraic Sets We have the following duality:
Let Z ( A ) be the collection of all algebraic subsets of A n . Then with polynomial maps as morphisms, Z ( A ) becomes a category , The category of Algebraic Sets We have the following duality:
Theorem The category of Coordinate algebras and The category of Algebraic Sets are dually isomorphic.
It can be proved that: two algebraic sets are isomorphic iff their corresponding coordinate algebras are isomorphic.
Logic of polynomials : ◮ The completeness theorem of � Lukasiewicz infinite valued logic can be phrased in several ways. ◮ One way is this, for [0 , 1] valued logic, if σ is a wff in the variables v 1 , . . . , v n , and if the value of σ for all values of the v i is always 1, ◮ then in the Lindenbaum algebra [ σ ] = 1, where [ σ ] is the class of σ .
Now [ σ ] can be interpreted as a function [ σ ] : [0 , 1] n → [0 , 1] by [ σ ]( r 1 , . . . , r n ) equals the value of σ with the assignment v i = r i . With this interpretation the completeness theorem can be phrased as: if the function [ σ ] equals 1 on [0 , 1] n , then [ σ ] = 1 in the Lindenbaum algebra.
We can apply this idea to our context and we get what we call polynomial completeness. We introduce the following notion:
Definition An MV algebra A is polynomially complete if for every n , the only polynomial in n variables inducing the zero function on A n is the zero polynomial.
The name polynomial completeness suggests that polynomial functions over A describe completely the polynomials of A , because if A is polynomially complete, then the evaluation homomorphism from A [ x 1 , ..., x n ] to PF n ( A ) is an MV-algebra isomorphism.
MV − chains : A characterization of polynomially complete We do not have a complete characterization of polynomially complete MV algebras, however we give one for MV chains. Theorem Let C be an MV chain. The following are equivalent: 1. C is polynomially complete; 2. every polynomial p ∈ C [ x 1 , . . . , x n ] which induces the zero function on C induces the zero function on DH ( C ) , where DH ( C ) is the divisible hull of C.
Corollary ◮ Every MV chain can be embedded in a polynomially complete MV chain. ◮ Every simple infinite MV chain is polynomially complete. ◮ No discrete MV chain A is polynomially complete. ◮ No MV chain A of finite rank is polynomially complete.
The finitely presented case : A study of finitely presented MV algebras is based on rational polyhedra in [0 , 1] n . Indeed in MV-algebras theory we have that the following are equivalent (Mundici): ◮ A is finitely presented ◮ For some rational polyhedron P , A is isomorphic to the MV-algebra of restrictions to P of McNaughton functions ◮ A is isomorphic to LIND θ for some satisfiable formula θ . We would like to extend the results of the theory of finitely presented MV-algebras as far as possible in a more general situation, where:
◮ formulas ϕ are replaced by polynomials p , ◮ polynomials evaluating to zero are preferred to formulas evaluating to one (this convention is somewhat a mismatch between algebraic geometry and logic), ◮ theories Φ are replaced by ideals J , ◮ finitely axiomatizable theories are replaced by principal ideals, ◮ polynomials may have constants out of an arbitrary MV algebra C , ◮ the function Mod on theories is replaced by the function V on ideals of polynomials, ◮ the function Th on algebraic subsets of [0 , 1] n is replaced by the function I on algebraic subsets of C n .
We can ask questions related to composed functions like Th ( Mod ( T )). W´ ojcicki’s Theorem (for MV-algebras) implies that if T is a finitely axiomatized theory in � Lukasiewicz logic, then Th ( Mod ( T )) coincides with T . In algebraic terms, this corresponds to I ( V ( p )) = id ( p ) for every polynomial p , which we called strong completeness. Actually this property of MV algebras is very strong: in fact, it can be seen that it holds only for simple divisible MV algebras.
Since W´ ojcicki’s Theorem does not help us when polynomials may have constants, we could consider weakenings of strong completeness. For instance, for what algebras the ideal I ( V ( p )) is principal for every polynomial p ? Logically, this corresponds to stating that for all finitely axiomatizable theory T , the theory Th ( Mod ( T )) is finitely axiomatized.
More generally, what are the ideals J such that I ( V ( J )) is principal? This corresponds to considering the theories T such that Th ( Mod ( T )) is finitely axiomatizable.
So let C be an MV algebra. If J is a nonsingular ideal of C [ x 1 , . . . , x n ], and p , q are elements of C [ x 1 , . . . , x n ], then we say p ≡ J q if for every zero v of J in C n , p ( v ) = q ( v ). The Lindenbaum MV-algebra of J is LIND J = C [ x 1 , . . . , x n ] / ≡ J .
We denote by TF n ( C ) the MV algebra of truncated funtions on Ξ( C ) as defined in section 2, and by TF n ( C ) | S the MV algebra of truncated functions restricted to S , where S ⊆ C n . Proposition Let p ∈ C [ x 1 , . . . , x n ] be a polynomial with at least one zero in C n . Then the MV algebra LIND p is isomorphic to TF n ( C ) | V ( p ) .
Polyhedra and McNaughton − functions over simple divisible MV − algebras : We have seen that simple divisible MV algebras are particularly suitable for studying algebraic geometry, because they enjoy Wojcicki’s property I ( V ( p )) = id ( p ). We note that V ( p ) is the zeroset of a polynomial. We wish to describe more explicitly zerosets of polynomials in simple divisible MV algebras. To this aim we generalize the notions of polyhedron and McNaughton function over a simple divisible MV algebra A .
An A − convex polyhedron with integer slopes and vertices in A is the intersection of finitely many half − spaces of the form { ( x 1 , . . . , x n ) ∈ A n | r + m 1 x 1 + . . . + m n x n ≥ 0 } , where m i are integers and r belongs to the group Ξ( A ), the inverse Mundici functor applied to A . A polyhedron with integer slopes and vertices in A is a finite union of A-convex ones.
We called McNaughton function over A a function from A n to A continuous and piecewise affine, whose affine pieces have the form r + m 1 x 1 + . . . + m n x n , where m i are integers and r belongs to the group Ξ( A ). McNaughton functions over A form a MV algebra called M A n . They characterize polynomials in the following sense: Theorem Let A be a divisible MV chain. Then: ◮ A [ x 1 , . . . , x n ] = M A n ; ◮ the zerosets of polynomials in A [ x 1 , . . . , x n ] coincide with polyhedra with integer slopes and vertices in A.
Corollary In every simple divisible MV algebra A, for every ideal J ⊆ A [ x 1 , . . . , x n ] , the ideal I ( V ( J )) is principal if and only if V ( J ) is a polyhedron with integer slopes and vertices in A.
Corollary In every simple divisible MV algebra A, the operator I is a bijection between polyhedra with integer slopes and vertices in A and principal ideals of A [ x 1 , . . . , x n ] .
� L ukasiewicz logic with constants : Like classical algebraic geometry, MV algebraic geometry can be studied by three different viewpoints: ◮ geometric (the algebraic sets), ◮ algebraic (coordinate algebras) and ◮ logical (theories and models). While the first two approaches are studied in the previous sections of this paper, we are left with giving the basics of logic for Diophantine MV algebraic geometry. We must define � Lukasiewicz logic with constants in a fixed MV algebra A , which, according to the Diophantine approach, will be both the MV algebra where the constants of polynomials are taken and the MV algebra where polynomials are evaluated.
In order to begin the study of � Lukasiewicz logic with constants in a fixed MV algebra A , denoted by L ∞ ( A ) , by adding constants denoting elements of A .
Like any other logic we must specify the syntax and semantics of L ∞ ( A ) . First, formulas are defined inductively as follows: ◮ variables X 1 , X 2 , . . . are formulas; ◮ constants c a for every a ∈ A are formulas; ◮ if α is a formula, then ¬ α is a formula; ◮ if α, β is a formula, then α → β is a formula.
The semantics of L ∞ ( A ) is given in terms of valuation functions v from variables to elements of A . The value of a formula α in a valuation v is an element v ( ϕ ) of A defined by: ◮ v ( X i ) when X i is a variable; ◮ a when the formula is the constant c a ; ◮ v ( ¬ α ) = ¬ v ( α ); ◮ v ( α → β ) = v ( α ) → v ( β ).
Now the notions of satisfaction, model, tautology, semantic consequence are defined like in the theory of � Lukasiewicz logic. In particular, a model of a formula α is a valuation v such that v ( α ) = 1. A formula α is a tautology if v ( α ) = 1 for every valuation v . A formula α is a semantic consequence of a set of formulas Θ if every model of Θ is also a model of α .
In L ∞ ( A ) we give also a deductive system, extending the one of of � Lukasiewicz logic, with axioms for constants. The axioms are: ◮ α → ( β → α ); ◮ ( α → β ) → (( β → γ ) → ( α → γ )); ◮ (( α → β ) → β ) → (( β → α ) → α ); ◮ ( ¬ α → ¬ β ) → ( β → α ); ◮ c a ∗ ⊕ b → ( c a → c b ); ◮ ( c a → c b ) → c a ∗ ⊕ b ; ◮ c a ∗ → ¬ c a ; ◮ ¬ c a → c a ∗ .
Proposition For every MV algebra A, the MV algebras Lind ( A ) A [ x 1 , x 2 , . . . ] are isomorphic.
For every A , every provable formula of L ∞ ( A ) is a tautology. The converse implication does not hold in general, but we have a characterization in terms of polynomial completeness: Theorem For every MV algebra A, the logic L ∞ ( A ) is complete if and only if A is polynomially complete.
Finally we mention that one can also consider a non Diophantine logic L ′ ∞ ( A ) , which is identical to L ∞ ( A ) , except that formulas are evaluated in an arbitrary extension of A , rather than A itself. This time we have: Proposition For every MV algebra A, L ′ ∞ ( A ) is complete.
We can summarize the main results as follows: ◮ We identify polynomial functions over any MV algebra with a kind of truncated functions, thus obtaining a generalized, ”topology free” McNaughton Theorem; ◮ we give a form of Nullstellensatz for A [ x 1 , . . . , x n ]; ◮ we give a universal algebraic duality between algebraic sets and their coordinate algebras; ◮ we introduce the definition of polynomial complete MV algebra (i.e. one where polynomials and polynomial functions coincide) and we give a characterization of polynomially complete MV chains;
◮ we define a suitable kind of polyhedron over any MV algebra A and we characterize zeros of polynomial functions by means of these polyhedra; ◮ we give a completeness criterion for � Lukasiewicz logic with constants in terms of polynomial completeness.
All these results are motivated by a desire of understanding polynomials and polynomial functions on MV-algebras in view of applications to � Lukasiewivz Logic. In particular it seems interesting to see what MV-polynomials functions become when we move from [0 , 1] (where they coincide with McNaughton functions) to other MV-algebras possibly non topologized.
The results obtained so far suggest that a study of non-Diophantine algebraic geometry for MV algebras deserves to be pursued.
Now we switch to Non-Diophantine MV-geometry
We emphasize that there is a strong connection between non-Diophantine geometry and certain natural extensions of � Lukasiewicz logic with constants . In fact, like we can write a polynomial in an MV algebra A and evaluate it in any extension B of A , we can write formulas of � Lukasiewicz logic plus constants out of an MV algebra A and give them semantics in an extension B of A . This gives an interesting interaction between geometry and logic in MV algebras.
Again we have that: Algebraic sets are determined by ideals of A [ x 1 , ... x n ].
Again: Point − ideals and point − radicals play a crucial role: Definition By A - algebra we mean a pair B = ( A ′ , h ), where A ′ is an MV-algebra and h is a homomorphism from A to A ′ .
Let A be an MV algebra and B = ( A ′ , h ) be an A -algebra. Call an ideal J ⊆ A [¯ x ] a B - point ideal if for some a = ( a 1 , . . . , a n ) ∈ A ′ n we have J = I B (¯ ¯ a ). We consider the fixpoints of the adjunction ( I B , V B ):
For an ideal I ⊆ A [¯ x ] and A -algebra B √ let B I = � { I B (¯ a ) | ¯ a ∈ B , I ⊆ I B (¯ a ) } . √ We call B I the B - radical of I . When B = A , we obtain the point-radicals of Diophantine case.
We get Nullstellensatz-like theorem: Theorem The ideals J such that I B ( V B ( J )) = J are exactly the B-radical ideals. Proposition There is a one-one correspondence between: B- radicals and algebraic sets .
Non − Diophantine Coordinate MV − algebras : Again let A be an MV algebra and B = ( A ′ , h ) is an A -algebra. Definition Let Z ⊆ A ′ n be an algebraic set. By the coordinate MV − algebra of Z we mean the MV-algebra A [¯ x ] / I B ( Z ).
Proposition If V B ( J ) is nonempty then the co-ordinate MV-algebra of V B ( J ) is √ A [¯ x ] / B J. The category of coordinate algebras : √ Let MV B = { A [ x 1 , . . . , x n ] / J | J = B J , n = 0 , 1 , 2 . . . } . Then MV B is a category with morphisms the MV-homomorphisms.
Polynomial maps between algebraic sets : Definition Let Z 1 ⊆ A ′ n , Z 2 ⊆ A ′ m be algebraic sets. A mapping ϕ : Z 1 → Z 2 is called a polynomial map iff there are polynomials p 1 , . . . , p m ∈ A [ x 1 , . . . , x n ] such that ϕ ( a 1 , . . . , a n ) = ( p 1 ( a 1 , . . . , a n ) , . . . , p m ( a 1 , . . . , a n )) for every ( a 1 , . . . , a n ) ∈ Z 1 .
The category of algebraic sets : Let Z B be the collection of all algebraic subsets of A ′ n . Then with polynomial maps as morphisms, Z B becomes a category. We have the following duality:
The category of algebraic sets : Let Z B be the collection of all algebraic subsets of A ′ n . Then with polynomial maps as morphisms, Z B becomes a category. We have the following duality: Theorem The categories of algebraic sets and of coordinate algebras are dually isomorphic.
The finitely presented case : We begin with a definition. Let A be an MV algebra, B = ( A ′ , h ) be an A -algebra. A 1 -algebraic set is a set Z ⊆ A ′ n such that Z = V B ( p ) for some single polynomial p . ( • ) So, 1-algebraic sets are particular cases of algebraic sets. An MV algebra A is called strongly complete if for every p ∈ A [ x 1 , . . . , x n ] we have I A ( V A ( p )) = id ( p ).
We have the following generalization of Wojcicki Theorem to the non Diophantine case: Proposition Let A be a strongly complete MV algebra and let B be any A-algebra. Let p ∈ A [ x 1 , . . . , x n ] . Then I B ( V B ( p )) = id ( p ) .
On the basis of the previous proposition we can say that: The duality between algebraic sets and coordinate algebras restricts to a duality between 1-algebraic sets and finitely presented A -algebras.
completeness : Polynomial We know that point radical ideals are the key concept of our Nullstellensatz result. So, one may be interested in understanding the structure of point radicals of ideals in an MV algebra of polynomials with constants in an MV algebra A . For example, one may ask what the point radical of zero looks like. The point radical of zero in A [ x 1 , . . . , x n ] is simply the set of all polynomials which induce the zero function on A . The following notion is useful in studying the point radical of zero.
An MV algebra A is said to be polynomially complete if for every n , the only polynomial in A [ x 1 , . . . , x n ] which induces the zero function on A is the zero polynomial.
Proposition An MV algebra A is polynomially complete if and only if A generates the variety MV A of MV algebras with coefficients in A.
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