On Lukasiewicz Logic with constants and non-Diophantine geometry - - PowerPoint PPT Presentation

Naples − Konstanz : Model Theory Days, 2013

On Lukasiewicz Logic with constants and non-Diophantine geometry on MV algebras

November 5, 2013

This talk is based on a joint work with Peter L. Belluce and Giacomo Lenzi.

The spirit of the work can be summarized by the following quotation, taken from (Lawere): In spite of its geometric origin, topos theory has in recent years some-times been perceived as a branch of logic, partly because of the contributions to the clarification of logic and set theory which it has made possible. However, the orientation of many topos theorists could perhaps be more accurately summarized by the

• bservation that what is usually called mathematical

logic can be viewed as a branch of algebraic geometry, and it is useful to make this branch explicit in itself.

We present a preliminary study of applying the concepts of algebraic geometry over fields to the theory of MV-algebras.

We present a preliminary study of applying the concepts of algebraic geometry over fields to the theory of MV-algebras. Rational polyhedra are the genuine algebraic varieties of the formulas of Lukasiewicz Logic, in a precise sense, indeed

A function f from [0, 1]n to [0, 1] is called a McNaughton function if it is continuous and there are k linear polynomials with integer coefficients such that for every y ∈ [0, 1]n there is j such that f (x) = pj(x). Then McNaughton Theorem says that McNaughton functions from [0, 1]n to [0, 1] form an MV algebra isomorphic to the free MV algebra on n generators and zerosets of McNaughton functions coincide with rational polyhedra.

Now, McNaughton functions are functions from [0, 1]n to [0, 1], so that in the theory of MV-algebras, the MV-algebra [0, 1] plays a fundamental role.

Now, McNaughton functions are functions from [0, 1]n to [0, 1], so that in the theory of MV-algebras, the MV-algebra [0, 1] plays a fundamental role. On the other hand, there are reasons to be interested in other MV algebras, because every MV-algebra can be viewed as the Lindenbaum algebra of some many-valued logic, and as such, it has logical relevance.

This is why we try to generalize somewhat the theory of McNaughton functions to MV-algebras as general as possible.

In order to develop our theory we proceed along lines of algebraic geometry over varieties in universal algebra.

We note that in algebraic geometry the central notion is the one of

• polynomial. One has three possibilities:

◮ considering coefficient-free algebraic geometry; this allows one

to evaluate polynomials in arbitrary fields;

◮ considering Diophantine algebraic geometry: this means that

the field where coefficients are taken coincides with the field where polynomials are evaluated;

◮ considering general, non-Diophantine algebraic geometry,

where polynomials take coefficients in a field K and are evaluated in an extension L of K.

It turns out that all these three possibilities can be extended to universal algebra.

It turns out that all these three possibilities can be extended to universal algebra. Since universal algebra subsumes the equational theory of MV algebras, we can consider what happens in universal algebraic geometry

◮ coefficient-free, ◮ Diophantine ◮ non-Diophantine

• ver MV algebras.

Our main source of inspiration is the Galois connection between theories and models fully, described for infinite valued Lukasiewicz logic. Because of the completeness theorem, we can say that all information for this connection is already provided by the MV-algebra [0, 1].

However, since we are interested in a Diophantine and Non-Diophantine approach to MV-algebraic geometry, we would like to go beyond [0, 1] and consider an MV algebra A. This corresponds to adding to Lukasiewicz logic the atomic diagram of A.

Of course in the generalization we lose something: for instance, we lose the tight connection between zeros of (single) polynomials and principal polynomial ideals given by W´

• jcicki’s Theorem in the

case of A = [0, 1].

However, many concepts still make sense, like

◮ the category of algebraic sets and Z-maps (here replaced by

polynomial maps) and

◮ the category of MV-algebras and homomorphisms, as well as ◮ the equivalence between them.

Preliminaries: We give a quick review of MV algebras. An MV-algebra is a structure (A, ⊕,∗ , 0), where ⊕ is a binary

• peration, ∗ is a unary operation and 0 is a constant such that the

following axioms are satisfied for any a, b ∈ A: i) (A, ⊕, 0) is an abelian monoid, ii) (a∗)∗ = a, iii) 0∗ ⊕ a = 0∗ iv) (a∗ ⊕ b)∗ ⊕ b = (b∗ ⊕ a)∗ ⊕ a.

One can construct two functors Γ and Ξ from the category of MV algebras to the category of lattice ordered groups with strong unit (ℓu-groups) and conversely, so that the pair (Γ, Ξ) is an equivalence.(Mundici equivalence)

Term algebras: Let X be a non-empty set of elements called variables, let F be a type of algebra. T(X, F) denotes the term algebra.

MV − algebras and polynomials: Let A be an MV algebra and n be a positive integer. Let FA be the language of MV-algebras plus a constant symbol ca for every a ∈ A. Define A[x1, . . . , xn] (the MV algebra of polynomials in n variables with constants in A) to be the quotient Tn(X, FA)/CA, where CA is the congruence generated by the axioms for MV-algebras and the complete diagram of A.

MV − Polynomial functions: In MV-algebras (and in universal algebra in general) it is crucial to distinguish polynomials and polynomial functions. Equal polynomials induce the same function everywhere, but two polynomials can induce the same function on some MV-algebra without being equal.

Given an MV algebra A and an MV term p(x1, . . . , xn) ∈ Tn(X, A) we may define a function pA : An → A as follows: 1) if p(x1, . . . , xn) = xi, then pA(a1, . . . , an) = ai; 2) if p(x1, . . . , xn) = ca for some a ∈ A, then pA(a1, . . . , an) = a; 3) if p(x1, . . . , xn) = p1(x1, . . . , xn) ⊕ p2(x1, . . . , xn), then pA(a1, . . . , an) = p1A(a1, . . . , an) ⊕A p2A(a1, . . . , an); 4) if p(x1, . . . , xn) = p1(x1, . . . , xn) ⊙ p2(x1, . . . , xn), then pA(a1, . . . , an) = p1A(a1, . . . , an) ⊙A p2A(a1, . . . , an); 5) p∗

A(a1, . . . , an) = (pA(a1, . . . , an))∗A.

We call pA the MV-polynomial function induced on A by p. We have the following characterization:

Proposition

Two terms p, q in n variables give the same polynomial on an MV-algebra A if and only if there is an extension A′ of A[x1, . . . , xn] such that p, q are congruent modulo ≡A′.

A topology free McNaughton Theorem: The aim of this section is to generalize the McNaughton Theorem for the MV algebra [0, 1] to arbitrary MV algebras. The point is that McNaughton functions are continuous in the standard topology of [0, 1]n, but if one tries to generalize McNaughton functions to arbitrary MV algebras, one faces the problem of choosing a topology to use. Unfortunately, in general (to our knowledge) there is no single “natural” topology for arbitrary MV algebras as it happens in [0, 1].

So in order to achieve a generalization, some other notion should be used. We choose what we call truncated functions.

The classical McNaughton Theorem recalled above implies that free MV algebras can be represented as MV polynomials on [0, 1]n. However, these polynomials can also be represented as truncated infima of suprema of affine functions from [0, 1]n to R with integer coefficients. This idea can be extended to any MV algebra A; we can relate truncated infima of suprema of affine functions from An to Ξ(A) (recall that (Γ, Ξ) is the Mundici functorial equivalence), with MV polynomial functions on A.

Let A be an MV algebra with associated ℓu-group (G, u). A (G, u)-affine term (with integer coefficients) over G is a term (in the language of groups) of the form f (x1, . . . , xn) = g0 + m1x1 + . . . + mnxn, where g0 ∈ G and m1, . . . , mn ∈ Z. We let TFn(G, u) be the set of all truncated (G, u) functions in n variables. We note that the set TFn(G, u) of truncated (G, u)-functions is an MV algebra.

Theorem

Let A be an MV algebra. Then the MV algebras TFn(A) and PF n(A) coincide.

McNaughton functions and McNaughton Theorem: We recall the notion of McNaughton functions and McNaughton Theorem. A function f from [0, 1]n to [0, 1] is called a McNaughton function if it is continuous and there are k linear polynomials with integer coefficients such that for every y ∈ [0, 1]n there is j such that f (x) = pj(x). Then McNaughton Theorem says that McNaughton functions form an MV-algebra isomorphic to the free MV-algebra on n generators.

Polyhedra and McNaughton functions for MV − chains: We can exploit McNaughton Theorem to give the following characterization of zerosets of polynomials in MV chains. Given an MV algebra A, an affine function on A is a function of the form Σjmjxj + r, where mj are integers and r ∈ Ξ(A).

Proposition

Let A be an MV chain. The zerosets of a polynomial p(x1, . . . , xn, a1, . . . , am) ∈ A[x1, . . . , xn] coincide with finite unions

• f 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 : An → A for which there is a covering of An by finitely many polyhedra P1, . . . , Pk 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[x1, . . . , xn]. Then p defines a McNaughton function from An to A. Conversely, every McNaughton function from An 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[x1, . . . , xn], S = ∅. Consider the set {(a1, . . . , an) ∈ An | p(a1, . . . . an) = 0, ∀p(x1, . . . , xn) ∈ 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[x1, . . . , xn] singular if V (J) = ∅. Otherwise call J non- singular.

Proposition

Suppose we have a non-empty X ⊆ An. Then let I(X) = {p ∈ A[x1, . . . , xn] | p(¯ y) = 0, ∀¯ y ∈ X} where ¯ y = (y1, . . . , yn), yi ∈ A. Then I(X) is an ideal of A[x1, . . . , xn].

Point ideals and point radicals: Call an ideal J ⊆ A[¯ x] a point ideal if for some ¯ a = (a1, . . . , an) ∈ An 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[x1, . . . , xn] 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 algebraic sets.

Coordinate algebras:

Definition

Let Z ⊆ An 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[x1, . . . , xn]/J | J = pt √ J, n = 0, 1, 2 . . .}. Then MV A is a category with morphisms the MV-homomorphisms. The category

• f

Coordinate algebras.

Definition

Let Z1 ⊆ An, Z2 ⊆ Am be algebraic sets. A mapping ϕ : Z1 → Z2 is called a polynomial map iff there are polynomials p1, . . . , pm ∈ A[x1, . . . , xn] such that ϕ(a1, . . . , an) = (p1(a1, . . . , an), . . . , pm(a1, . . . , an)) for every (a1, . . . , an) ∈ Z1.

Let Z(A) be the collection of all algebraic subsets of An. Then with polynomial maps as morphisms, Z(A) becomes a category, The category

• f

Algebraic Sets We have the following duality:

Let Z(A) be the collection of all algebraic subsets of An. Then with polynomial maps as morphisms, Z(A) becomes a category, The category

• f

Algebraic Sets We have the following duality:

Theorem

The category

• f

Coordinate algebras and The category

• f

Algebraic Sets are dually isomorphic.

It can be proved that: two algebraic sets are isomorphic iff their corresponding coordinate algebras are isomorphic.

Logic

• f

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 v1, . . . , vn, and if the value of σ for all values of the vi is always 1,

◮ then in the Lindenbaum algebra [σ] = 1, where [σ] is the class

• f σ.

Now [σ] can be interpreted as a function [σ] : [0, 1]n → [0, 1] by [σ](r1, . . . , rn) equals the value of σ with the assignment vi = ri. 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

• nly polynomial in n variables inducing the zero function on An 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[x1, ..., xn] to PFn(A) is an MV-algebra isomorphism.

A characterization

• f

polynomially complete MV − chains: 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[x1, . . . , xn] 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 LINDθ 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´

• jcicki’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´

• jcicki’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[x1, . . . , xn], and p, q are elements of C[x1, . . . , xn], 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 LINDJ = C[x1, . . . , xn]/ ≡J.

We denote by TFn(C) the MV algebra of truncated funtions on Ξ(C) as defined in section 2, and by TFn(C)|S the MV algebra of truncated functions restricted to S, where S ⊆ C n.

Proposition

Let p ∈ C[x1, . . . , xn] be a polynomial with at least one zero in C n. Then the MV algebra LINDp is isomorphic to TFn(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 {(x1, . . . , xn) ∈ An|r + m1x1 + . . . + mnxn ≥ 0}, where mi 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

• f A-convex ones.

We called McNaughton function over A a function from An to A continuous and piecewise affine, whose affine pieces have the form r + m1x1 + . . . + mnxn, where mi are integers and r belongs to the group Ξ(A). McNaughton functions over A form a MV algebra called MA

n . They

characterize polynomials in the following sense:

Theorem

Let A be a divisible MV chain. Then:

◮ A[x1, . . . , xn] = MA n ; ◮ the zerosets of polynomials in A[x1, . . . , xn] coincide with

polyhedra with integer slopes and vertices in A.

Corollary

In every simple divisible MV algebra A, for every ideal J ⊆ A[x1, . . . , xn], 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[x1, . . . , xn].

• Lukasiewicz

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

• f 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 X1, X2, . . . are formulas; ◮ constants ca 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(Xi) when Xi is a variable; ◮ a when the formula is the constant ca; ◮ 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); ◮ ca∗⊕b → (ca → cb); ◮ (ca → cb) → ca∗⊕b; ◮ ca∗ → ¬ca; ◮ ¬ca → ca∗.

Proposition

For every MV algebra A, the MV algebras Lind(A) A[x1, x2, . . .] 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[x1, . . . , xn]; ◮ 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

• f 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[x1, ...xn].

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 = (a1, . . . , an) ∈ A′n we have J = IB(¯ a). We consider the fixpoints of the adjunction (IB, VB):

For an ideal I ⊆ A[¯ x] and A-algebra B let B √ I = {IB(¯ a) | ¯ a ∈ B, I ⊆ IB(¯ 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 IB(VB(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]/IB(Z).

Proposition

If VB(J) is nonempty then the co-ordinate MV-algebra of VB(J) is A[¯ x]/B √ J. The category

• f

coordinate algebras: Let MVB = {A[x1, . . . , xn]/J | J = B √ J, n = 0, 1, 2 . . .}. Then MVB is a category with morphisms the MV-homomorphisms.

Polynomial maps between algebraic sets:

Definition

Let Z1 ⊆ A′n, Z2 ⊆ A′m be algebraic sets. A mapping ϕ : Z1 → Z2 is called a polynomial map iff there are polynomials p1, . . . , pm ∈ A[x1, . . . , xn] such that ϕ(a1, . . . , an) = (p1(a1, . . . , an), . . . , pm(a1, . . . , an)) for every (a1, . . . , an) ∈ Z1.

The category of algebraic sets: Let ZB be the collection of all algebraic subsets of A′n. Then with polynomial maps as morphisms, ZB becomes a category. We have the following duality:

The category of algebraic sets: Let ZB be the collection of all algebraic subsets of A′n. Then with polynomial maps as morphisms, ZB 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 = VB(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[x1, . . . , xn] we have IA(VA(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[x1, . . . , xn]. Then IB(VB(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.

Polynomial completeness: 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[x1, . . . , xn] 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[x1, . . . , xn] 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.

We do not have yet a characterization of polynomially complete MV algebras, but a characterization for MV − chains is given :

Theorem

(Analogous the the Diophantine case) For every MV chain A the following are equivalent:

• 1. A is polynomially complete;
• 2. A is order dense in its divisible hull.

The previous proposition characterizes MV − chains which generate their coefficient variety. In general we can say:

Every MV-algebra can be embedded in a polynomially complete MV-algebra.

• Lukasiewicz

logic with constants: The extension of Lukasiewicz logic with constants taken from any MV algebra A, is denoted by L∞(A). The formulas are built from propositional letters and constants ca for every a ∈ A by means of negation ¬ and implication →. Formulas are evaluated with respect to valuations of the propositional letters in A, so that the value of a formula is always an element of A. In this sense, and in the spirit of this talk, we can consider L∞(A) as a Diophantine logic, suitable for studying Diophantine algebraic geometry. Instead, non − Diophantine logics will be the logics L′

∞(A) and

• L′′

∞(A) which we will introduce later on.

Extended semantics and A-algebraic semantics for L∞(A) So far we considered the “Diophantine” logic L∞(A). It is natural to consider two “non-Diophantine” variants, that is L′

∞(A) and

L′′

∞(A) .

We let L′

∞(A) be the same logic of

L∞(A), except that formulas are evaluated in extensions of A. Moreover we define another similar logic L′′

∞(A) to be the same

logic of L∞(A), except that formulas are evaluated in A-algebras.

We conclude that the following are equivalent for every formula α ∈ L∞(A):

◮ α is provable in

L∞(A);

◮ α is valid in

L′

∞(A); ◮ α is valid in

L′′

∞(A).

From the point of view of completeness (equality of tautologies and provable formulas) the logic L′′

∞(A) is as well behaved as

• L′

∞(A), in fact:

Theorem

For every MV algebra A, the logic L′′

∞(A) is complete.

Summing up, we got:

◮ A duality theorem between A-algebraic sets and coordinate

A-algebras ;

◮ an intrinsic characterization of polynomially complete MV

chains;

◮ some completeness results for

Lukasiewicz logics with constants, both Diophantine ( L∞(A)) and non-Diophantine ( L′

∞(A) and

L′′

∞(A)).

The list of results itself suggests directions for future research. For instance, we do not know how to characterize polynomially complete MV algebras, or infinite, principally complete MV algebras. We are also confident that the analogy with classical algebraic geometry showing up e.g. in the Nullstellensatz-like Theorem will suggest other directions for future research.

Thank you.