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Interpolation in \Lukasiewicz logic and amalgamation of MV-algebras Daniele Mundici Dept. of Mathematics “Ulisse Dini” University of Florence, Florence, Italy

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2-simplex 0-simplex 1-simplex 3-simplex we all know what a simplex in R n is

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polyhedron P= finite union of simplexes S i in R n P need not be convex, nor connected a polyhedron P = U S i is said to be rational if so are the vertices of every simplex S i

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our main themes: rational polyhedra and \Lukasiewicz logic Chapter 1: Local Deduction as a main ingredient of interpolation and amalgamation

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\Lukasiewicz logic L ∞ FORMULAS are exactly the same as in boolean logic any VALUATION V evaluates formulas into the real unit interval [0,1] via the inductive rules: V(¬F) = 1–V(F) V(F —> G) = min(1, 1–V(F)+V(G)) Therefore, every valuation V is uniquely determined by its values on the variables: V(X 1 ),...,V(X n ) CONSEQUENCE RELATION: F |– G means that every valuation satisfying F also satisfies G

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formulas yield functions f:[0,1] n —>[0,1] as boolean formulas yield f:{0,1} n —>{0,1} every formula F(X 1,...,X n ) determines a map f F : [0,1] n —>[0,1] by f Xi = the ith coordinate map f ¬F = 1 – f F f F —> G = min(1, 1 – f F + f G )

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definable functions of one variable the ONESET f F -1 (1) of f F is the set of valuations satisfying the formula F oneset(f F )=zeroset(¬f F ) for each formula F, its associated function f F is continuous, linear, and each linear piece has integer coefficients (for short, f F is a McNaughton function)

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oneset of f F = Mod(F) by induction on the number of connectives in F, the oneset of f F is a rational polyhedron, and so is the oneset of f ¬F and of f F —> G EACH ZEROSET AND EACH ONESET IS A RATIONAL POLYHEDRON IN [0,1] n

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(Local) Deduction Theorem Theorem. For any two formulas A and B, the following conditions are equivalent: 1. Every valuation satisfying A also satisfies B 2. For some m=1,2,... the formula A—>(A—>(A—>... —>(A—>(A—>B))... )) is a tautology 3. B is obtained from A and the tautologies via Modus Ponens PROOF. 2—>3 easy; 3—>1 induction; 1—>2 is proved geometrically

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assume oneset(f A ) contained in oneset(f B ) 1 1 let T be a triangulation of [0,1] such that the functions f A and f B both formulas A and B are linear over each interval of T fAfA fBfB

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f A & f A < f A 1 1 f A&A fBfB applying \Lukasiewicz conjunction to A, from the formula A&A we get obtain a minorant f A&A of f A, still with the same one set of f A Recall definition P&Q = ¬(P —> ¬Q)

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f A & f A & f A < f A & f A < f A 1 1 fkAfkA fBfB by iterated application of the \Lukasiewicz conjunction we obtain a function f k A = f A &f A &...&f A with the same oneset of f A, and with the additional property that f A k ≤ f B

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for large k this will hold at every simplex of T 1 1 fAfA fBfB in other words, we have the tautology A k —>B, which is the same as the desired tautology A—>(A—>(A—>... —>(A—>(A—>B))... ))

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Chapter 2: Interpolation (as a main tool to amalgamation)

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interpolation/amalgamation Craig interpolation theorem fails in \Lukasiewicz logic, because the tautology x ¬x—>y ¬y has no interpolant deductive interpolation is like Craig interpolation, with the |– symbol in place of the implication connective (more soon) over the last 25 years, several proofs have been given of deductive interpolation for \Lukasiewicz infinite-valued propositional logic deductive interpolation, together with local deduction, is a main tool to prove the amalgamation theorem for the algebras of \Lukasiewicz infinite-valued logic

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amalgamation: many proofs the first proof of amalgamation used the categorical equivalence between MV-algebras and unital lattice-ordered groups (relying on Pierce's amalgamation theorem). in the early eighties I heard from Andrzej Wro\’nski during one of his visits to Florence, that the Krakow group had a proof of the amalgamation property for MV-algebras without negation (i.e., Komori’s C algebras) recent proofs, like the proof by Kihara and Ono, follow by applying to MV-algebras results in universal algebra I will present a simple geometric proof of the amalgamation theorem, using Deductive Interpolation

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background literature F. Montagna, Interpolation and Beth's property in propositional many- valued logics: A semantic investigation, Annals of Pure and Applied Logic, 141: , This is based on: N.Galatos, H. Ono, Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica, 83: , For the proof of Theorem 5.8, the following is needed: A. Wro\'nski, On a form of equational interpolation property, In: Foundations of Logic and Linguistic, G.Dorn, P. Weingartner, (Eds.), Salzburg, June 19, 1984, Plenum, NY, 1985, For the proof of Theorem I on page 25, the following is needed: P.D. Bacisch, Amalgamation properties and interpolation theorems for equational theories, Algebra Universalis, 5:45-55, 1975.

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(Deductive) Interpolation If F |– G then there is a formula J such that F |– J, J |– G, and each variable of J is a variable of both F and G our proof will be entirely geometrical

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rational polyhedra are preserved under projection the projection of a (rational) polyhedron onto a (rational) hyperplane is a (rational) polyhedron we record this fact as the PROJECTION LEMMA

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rational polyhedra are preserved under perpendicular cylindrification we record this fact as the CYLINDRIFICATION LEMMA

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oneset of f F = Mod(F) recall: THE ZEROSET (AND THE ONESET) OF ANY \LUKASIEWICZ FORMULA IS A RATIONAL POLYHEDRON IN [0,1] n we now prove the converse: EACH RATIONAL POLYHEDRON IN [0,1] n IS THE ZEROSET OF SOME \LUKASIEWICZ FORMULA

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rational half-spaces in [0,1] n a rational line L in [0,1] 2 H is one of the half-planes bounded by L in the square [0,1] 2 PROBLEM: Does there exist a formula F such that the zeroset of f F coincides with H ? mx+ny+p=0, with m,n,p integers, m>0 H ANSWER: Yes, by induction on |m|+|n|

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then every rational polyhedron is a zeroset this blue half-space is a zeroset then so is this rational triangle (formulas can express intersections) and this rational polyhedron (formulas can express unions) ANY RATIONAL POLYHEDRON IN [0,1] n IS THE ZEROSET OF SOME f F

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this was known to McNaughton (1951) FOLKLORE LEMMA Rational polyhedra contained in the n-cube [0,1] n coincide with zerosets (and also coincide with onesets) of definable maps, i.e., functions of the form f F where F ranges over formulas in n variables we record the FOLKLORE LEMMA by writing: RATIONAL POLYHEDRA=ONESETS=MODELSETS

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Deductive interpolation PROOF. We may write var(F) = X u Z var(G) = Y u Z, for X,Y,Z pairwise disjoint sets of variables Mod(F) = f F -1 (1) = P, which by the Folklore Lemma is a rational polyhedron in [0,1] XuZ by the Projection Lemma, the projection of P onto R Z is a rational polyhedron Q contained in [0,1] Z

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Mod(G) = f G -1 (1) = R, a rational polyhedron in [0,1] YuZ Mod(F) = P X Y Z Mod(G)=R the hypothesis F |— G states that, in the space R XuYuZ Mod(F) is contained in Mod(G) Q

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Q Mod(F) = P We then obtain the first half of interpolation: F |— J X Y Z by the Folklore Lemma, there is a formula J(Z) such that Q=Mod(J) regarding J as a formula in the variables X,Z, then Mod(J) is this blue rectangle! = Mod(J)

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Mod(F) = P X Y Z Mod(G)=R in the space R YuZ, Mod(J) is contained in Mod(G) Q=Mod(J) regarding J as a formula in Y,Z, then Mod(J) is this blue rectangle! We then obtain the second half of interpolation: J |— G

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Chapter 3: Amalgamation of the algebras of \Lukasiewicz logic, i.e., Chang MV-algebras

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MV-algebras (in Wajsberg’s version) directly from \Lukasiewicz axioms A—>(B—>A) (A—>B)—>((B—>C)—>(A—>C)) ((A—>B)—>B)—> ((B—>A)—>A) (¬A—>¬B)—>(B—>A)

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the amalgamation property Z A B we have

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the usual setup Z A B D we want henceforth, all blue maps are one-one we have

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the embedding of Z into A let us focus attention on the embedding of Z into A without loss of generality, Z is a subalgebra of A thus the set A is the disjoint union of Z and some set X, A=Z U X Z A

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extending maps to homomorphisms the identity map z—>z uniquely extends to a homomorphism s Z of the free MV-algebra FREE Z onto Z similarly, the identity map a—> a uniquely extends to a homomorphism s A of FREE A onto A let ker s Z and ker s A denote the kernels of these maps Z FREE Z sZsZ

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Z A FREE XUZ ker(s A ) ker(s Z ) sAsA sZsZ all blue arrows are inclusions all red arrows are surjections intuitively, this trivial Largeness Lemma states that ker(s Z ) is as large as possible in ker(s A ).

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Z AB

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Z AB FREE Z FREE XUZ FREE YUZ ker(s A ) ker(s B ) ker(s Z ) sAsA szsz sAsA

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Z AB FREE Z FREE XUZ FREE YUZ ker(s A ) ker(s B ) ker(s Z ) sAsA szsz I = the ideal generated by ker(s A ) U ker(s B ) FREE XUYUZ sAsA

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Z AB FREE Z FREE XUZ FREE YUZ ker(s A ) ker(s B ) ker(s Z ) sAsA szsz I = the ideal generated by ker(s A ) U ker(s B ) FREE XUYUZ sAsA D

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Z AB FREE Z FREE XUZ FREE YUZ ker(s A ) ker(s B ) ker(s Z ) sAsA szsz i = the ideal generated by ker(s A ) U ker(s B ) FREE XUYUZ sAsA D µ µ(x/ ker(s A )) = x/i there remains to be proved that µ is one-one

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e/i = 0 means that e is an element of i. In other words, (theories ~ ideals) a, b |– e for some a in ker(s A ) and b in ker(s B ) Let e be an element of FREE XUYUZ such that e/i =0. We must prove e/ker(s A ) = 0

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end of the proof of amalgamation

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Chapter 4: Further geometric developments on projective MV- algebras

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why should we insist in giving many proofs of MV-amalgamation and interpolation? because MV-algebras provide a benchmark for other structures of interest in algebraic logic because interpolation and amalgamation are deeply related to many fundamental logical-algebraic-geometric notions: quantifier elimination, cut elimination, joint consistency, joint embedding, unification, projectives,... let us briefly review what is known about finitely generated projective MV-algebras, i.e., retracts of FREE n for some n this is joint work with Leonardo Cabrer, to appear in Communications in Contemporary Math., and based on earlier joint work on Algebra Universalis 62 (2009) 63–74.

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projectives are routinely characterized by duality Every n-generated projective MV-algebra A is finitely presented (essentially, Baker) A is finitely presented iff A=M(P) for some polyhedron P lying in some n-cube [0,1] n (Baker-Beynon duality) DEFINITION P is said to be a Z-retract if the MV- algebra M(P) is projective Problem: characterize Z-retracts, among all polyhedra

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this property is not easy to handle; thus, we must find equivalent conditions for a polyhedron P to be retract of [0,1] n a first property of Z-retracts: they are retracts of some cube [0,1] n

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The elements of the fundamental group π 1 (P) (introduced by Poincaré) of a connected polyhedron P are the equivalence classes of the set of all paths with initial and final points at a given basepoint p, under the equivalence relation of homotopy. The fundamental groups of homeomorphic spaces are isomorphic. to check if P is a retract it suffices to check that all homotopy groups of P are trivial

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equivalents for P to be a retract of [0,1] n THEOREM. For any polyhedron P in [0,1] n the following conditions are equivalent: (a) P is a retract of [0,1] n (b) P is connected and all homotopy groups π i (P) are trivial (c) P is contractible (can be continuously shrunk to a point). Proof. (a)—>(b) by the functorial properties of the homotopy groups π i. The implications (b)—>(a) and (b)— >(c) follow from Whitehead theorem in algebraic topology. (c)—>(b) is trivial. QED

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M(P) is not projective P is not a Z- retract, because it is not simply connected let P be this polyhedron

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M(P) is not projective a second property of Z-retracts: P must contain a vertex of [0,1] n P is not a Z-retract, because it does not contain any vertex of the unit square

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PROPOSITION If P is a Z-retract, then P has a triangulation Ω such that the affine hull of every maximal simplex in Ω contains some integer point of R n a third property: strong regularity M(P) is not projective this P is not a Z- retract: for, the affine hull of the vertical red segment does not contain any integer point 0 1 1

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projectiveness: 3 necessary conditions THEOREM ( L.Cabrer, D.M., 2009 ) If A is a finitely generated projective MV-algebra, then up to isomorphism, A=M(P) for some rational polyhedron lying in [0,1] n such that (i) P contains some vertex of [0,1] n, (ii) P is contractible, and (iii) P is strongly regular. are these three conditions also sufficient for an MV-algebra A to be finitely generated projective ?

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yes, when the maximal spectrum is one- dimensional THEOREM ( L.Cabrer, D.M. ) Suppose the maximal spectrum of A is one-dimensional. Then A is n- generated projective if and only if A is isomorphic to M(P) for some contractible strongly regular rational polyhedron in [0,1] n containing a vertex of [0,1] n. It is not known if these three conditions are sufficient in general. They become sufficient if contractibility is strengthened to collapsibility

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a sequence of collapses

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a sufficient condition for P to be a Z-retract, i.e., for M(P) to be projective THEOREM ( L.Cabrer, D.M., Communications in Contemporary Mathematics ) If P has a collapsible strongly regular triangulation containing a vertex of [0,1] n then M(P) is projective.

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A is finitely presented homomorphism isomorphism indecomposable A is free n-generated A is n-generated dim(maxspec(A))=d A=M(P) is projective A=M(P), P a polyhedron Z-map Z-homeomorphism P is connected P=unit cube [0,1] n P lies in [0,1] n dim(P)=d P is a Z-retract algebrageometry

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\Lukasiewicz logic and MV-algebras together are a rich source of geometric inspiration thank you

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