ukasiewicz logic is nullary
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The unication type ukasiewicz logic is nullary Topology, Algebra, and Categories in Logic http://logica.dmi.unisa.it/lucaspada University of Salerno Department of Mathematics Luca Spada Based on a joint work with V. Marra The

  1. The uni�cation type of Łukasiewicz logic Example: and with integer coefficients. piece-wise linear, continuous, . . which is A McNaughton function is a function Example 2: McNaughton functions Main result polyhedra Rational problems uni�cation Related functions McNaughton MV-algebras logic Łukasiewicz MV-algebras Luca Spada is nullary . f : [0 , 1] n → [0 , 1] u ℓ -groups

  2. The uni�cation type of Łukasiewicz logic Example: and with integer coefficients. piece-wise linear, continuous, . . which is A McNaughton function is a function Example 2: McNaughton functions Main result polyhedra Rational problems uni�cation Related functions McNaughton MV-algebras logic Łukasiewicz MV-algebras Luca Spada is nullary . f : [0 , 1] n → [0 , 1] u ℓ -groups

  3. The uni�cation type of Łukasiewicz logic Example: and with integer coefficients. piece-wise linear, continuous, . . which is A McNaughton function is a function Example 2: McNaughton functions Main result polyhedra Rational problems uni�cation Related functions McNaughton MV-algebras logic Łukasiewicz MV-algebras Luca Spada is nullary . f : [0 , 1] n → [0 , 1] u ℓ -groups

  4. found the following characterisation of free MV-algebras: The uni�cation type . . One can endow the set of McNaughton functions in n variables with the structure of an MV-algebra by taking point-wise operation: MV-algebra) These functions are named after McNaughton, who first . Theorem (McNaughton 1951) . . . . . . . . The free MV-algebra over generators is isomorphic to the MV-algebra of McNaughton functions over of Łukasiewicz logic . . Rational is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras McNaughton functions Related uni�cation problems polyhedra . Main result The free MV-algebra . Example . . . . ( ) f ⊕ g = f ⊕ g ( x ) = f ( x ) ⊕ g ( x ) (recall that [0,1] is an u ℓ -groups

  5. The uni�cation type . of Łukasiewicz logic . One can endow the set of McNaughton functions in n variables with the structure of an MV-algebra by taking point-wise operation: MV-algebra) These functions are named after McNaughton, who first . Theorem (McNaughton 1951) . . . . . . . . The free MV-algebra over generators is isomorphic to the MV-algebra of McNaughton functions over . . . Rational is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras McNaughton functions Related uni�cation problems . . polyhedra Main result The free MV-algebra . Example . . ( ) f ⊕ g = f ⊕ g ( x ) = f ( x ) ⊕ g ( x ) (recall that [0,1] is an u ℓ -groups found the following characterisation of free MV-algebras:

  6. The uni�cation type . of Łukasiewicz logic . . One can endow the set of McNaughton functions in n variables with the structure of an MV-algebra by taking point-wise operation: MV-algebra) These functions are named after McNaughton, who first Theorem (McNaughton 1951) . . . . . . . . . . . . . is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras McNaughton functions Related uni�cation problems polyhedra Rational Main result The free MV-algebra . Example . ( ) f ⊕ g = f ⊕ g ( x ) = f ( x ) ⊕ g ( x ) (recall that [0,1] is an u ℓ -groups found the following characterisation of free MV-algebras: The free MV-algebra over κ generators is isomorphic to the MV-algebra of McNaughton functions over [0 , 1] κ .

  7. The uni�cation type . n -times If one truncates the operations of an u -group to the interval g , the result is an MV-algebra. . Theorem (Mundici 1986) . . such that . . . . . The category of MV-algerbas is equivalent to the category of Abelian u -groups (with -morphisms preserving the strong of Łukasiewicz logic unit). Related MV-algebras is nullary Luca Spada MV-algebras Main result Łukasiewicz polyhedra Rational problems uni�cation Example 3: laice ordered groups logic functions McNaughton A u ℓ -group is a lattice-ordered group G with an element g for any g ′ ∈ G there exists n ∈ N such that g + ... + g ≥ g ′ . � �� � u ℓ -groups

  8. The uni�cation type such that Abelian u -groups (with -morphisms preserving the strong The category of MV-algerbas is equivalent to the category of . . . . . . . . Theorem (Mundici 1986) . n -times of Łukasiewicz logic unit). Example 3: laice ordered groups Related is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras McNaughton functions uni�cation Rational problems polyhedra Main result A u ℓ -group is a lattice-ordered group G with an element g for any g ′ ∈ G there exists n ∈ N such that g + ... + g ≥ g ′ . � �� � u ℓ -groups If one truncates the operations of an u ℓ -group to the interval [0 , g ] , the result is an MV-algebra.

  9. The uni�cation type Example 3: laice ordered groups The category of MV-algerbas is equivalent to the category of . . . . . . . . Theorem (Mundici 1986) . n -times of Łukasiewicz logic such that unit). Main result uni�cation is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras McNaughton functions Related problems Rational polyhedra A u ℓ -group is a lattice-ordered group G with an element g for any g ′ ∈ G there exists n ∈ N such that g + ... + g ≥ g ′ . � �� � u ℓ -groups If one truncates the operations of an u ℓ -group to the interval [0 , g ] , the result is an MV-algebra. Abelian u ℓ -groups (with ℓ -morphisms preserving the strong

  10. finite-valued extension of Basic Logic by Dzik. The uni�cation type problems possible values of evaluations to some subchain Finite-valued logics are obtained by restricting the This was re-proved explicitly and generalised to any has unitary type. Ghilardi himself noticed that finite-valued Łukasiewicz logic Unitarity of �nite-valued Łukasiewicz logic Main result of Łukasiewicz logic Rational polyhedra uni�cation Luca Spada functions McNaughton MV-algebras logic Łukasiewicz MV-algebras Related is nullary u ℓ -groups     { 0 , 1 n , .., n − 1 n , 1 } of the [0,1] algebra.

  11. The uni�cation type uni�cation possible values of evaluations to some subchain Finite-valued logics are obtained by restricting the has unitary type. Ghilardi himself noticed that finite-valued Łukasiewicz logic Unitarity of �nite-valued Łukasiewicz logic Main result of Łukasiewicz logic Rational problems polyhedra Related Luca Spada functions McNaughton MV-algebras logic Łukasiewicz MV-algebras is nullary This was re-proved explicitly and generalised to any u ℓ -groups finite-valued extension of Basic Logic by Dzik.     { 0 , 1 n , .., n − 1 n , 1 } of the [0,1] algebra.

  12. The uni�cation type . . . In the theory of -groups all system of equations are solvable. In the light of the Beynon’s and Ghilardi’s results, one easily gets: . Corollary . . of Łukasiewicz logic . . . . . The unification type of the theory of -groups is unitary. In a forthcoming paper with V. Marra, we exploit a geometrical duality for -groups to give an algorithm that, taken any (system of) term in the language of -groups, . . . uni�cation is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras McNaughton functions Related problems . Rational polyhedra Main result . Theorem (Beynon 1977) . . outputs its most general unifier. Commutative laice-ordered groups ( ℓ -groups) Finitely generated projective ℓ -groups are exactly the finitely presented ℓ -groups. u ℓ -groups

  13. The uni�cation type . . . . In the light of the Beynon’s and Ghilardi’s results, one easily gets: . Corollary . . . . . . . . In a forthcoming paper with V. Marra, we exploit a geometrical duality for -groups to give an algorithm that, taken any (system of) term in the language of -groups, of Łukasiewicz logic . . . is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras McNaughton functions Related uni�cation problems Rational polyhedra Main result . Theorem (Beynon 1977) . outputs its most general unifier. Commutative laice-ordered groups ( ℓ -groups) Finitely generated projective ℓ -groups are exactly the finitely presented ℓ -groups. In the theory of ℓ -groups all system of equations are solvable. u ℓ -groups The unification type of the theory of ℓ -groups is unitary.

  14. The uni�cation type Corollary of Łukasiewicz logic . . . In the light of the Beynon’s and Ghilardi’s results, one easily gets: . . . . . . . . . . In a forthcoming paper with V. Marra, we exploit a . . . uni�cation is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras McNaughton functions . Related problems Rational polyhedra Main result . Theorem (Beynon 1977) outputs its most general unifier. Commutative laice-ordered groups ( ℓ -groups) Finitely generated projective ℓ -groups are exactly the finitely presented ℓ -groups. In the theory of ℓ -groups all system of equations are solvable. u ℓ -groups The unification type of the theory of ℓ -groups is unitary. geometrical duality for ℓ -groups to give an algorithm that, taken any (system of) term in the language of ℓ -groups,

  15. at least not unitary. The uni�cation type of Łukasiewicz logic the substitution x x (hence it must be ) or (hence it is the substitution x x , then it must unify either x is a unifier for x Indeed if This entails the unification type of Łukasiewicz logic to be ( Exercise : prove this by using McNaughton representation) Łukasiewicz logic has a weak disjunction property. Namely: Non unitarity of the uni�cation Main result polyhedra McNaughton is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras functions Rational Related uni�cation problems ). if φ ∨ ¬ φ is derivable then either φ or ¬ φ must be derivable . (In other words the rule φ ∨¬ φ φ, ¬ φ is admissible .) u ℓ -groups

  16. The uni�cation type polyhedra the substitution x x (hence it must be ) or (hence it is the substitution x x , then it must unify either x is a unifier for x Indeed if This entails the unification type of Łukasiewicz logic to be ( Exercise : prove this by using McNaughton representation) Łukasiewicz logic has a weak disjunction property. Namely: Non unitarity of the uni�cation of Łukasiewicz logic Main result Rational McNaughton is nullary Luca Spada MV-algebras Łukasiewicz logic MV-algebras ). problems functions Related uni�cation if φ ∨ ¬ φ is derivable then either φ or ¬ φ must be derivable . (In other words the rule φ ∨¬ φ φ, ¬ φ is admissible .) u ℓ -groups at least not unitary.

  17. The uni�cation type uni�cation This entails the unification type of Łukasiewicz logic to be ( Exercise : prove this by using McNaughton representation) Łukasiewicz logic has a weak disjunction property. Namely: Non unitarity of the uni�cation Main result polyhedra of Łukasiewicz logic problems Rational Related Luca Spada functions McNaughton MV-algebras logic Łukasiewicz MV-algebras is nullary if φ ∨ ¬ φ is derivable then either φ or ¬ φ must be derivable . (In other words the rule φ ∨¬ φ φ, ¬ φ is admissible .) u ℓ -groups at least not unitary. Indeed if σ is a unifier for x ∨ ¬ x , then it must unify either x (hence it is the substitution x �→ 1 ) or ¬ x (hence it must be the substitution x �→ 0 ).

  18. The uni�cation type . n . . . A rational polytope is the convex hull of a finite set of rational points. . . m . . n m . p q . . . . of Łukasiewicz logic MV-algebras is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. Consequences . Co-uni�cation Main result Rational polyhedral geometry McNaughton functions are only a first scratch on the surface of a stronger link between Łukasiewicz logic and Geometry. . De�nition . .

  19. The uni�cation type . n . . . A rational polytope is the convex hull of a finite set of rational points. . . m . . n m . p q . . . . of Łukasiewicz logic MV-algebras is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. Consequences . Co-uni�cation Main result Rational polyhedral geometry McNaughton functions are only a first scratch on the surface of a stronger link between Łukasiewicz logic and Geometry. . De�nition . .

  20. The uni�cation type . n . . . . A rational polytope is the convex hull of a finite set of rational points. m of Łukasiewicz logic . n m . p q . . . . . Consequences is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. MV-algebras . . Co-uni�cation Main result Rational polyhedral geometry McNaughton functions are only a first scratch on the surface of a stronger link between Łukasiewicz logic and Geometry. . De�nition • . .

  21. The uni�cation type . n . . . . A rational polytope is the convex hull of a finite set of rational points. m of Łukasiewicz logic . n m . p q . . . . . Consequences is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. MV-algebras . . Co-uni�cation Main result Rational polyhedral geometry McNaughton functions are only a first scratch on the surface of a stronger link between Łukasiewicz logic and Geometry. . De�nition • . .

  22. The uni�cation type . n . . . . A rational polytope is the convex hull of a finite set of rational points. m of Łukasiewicz logic . n m . p q . . . . . Consequences is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. MV-algebras . . Co-uni�cation Main result Rational polyhedral geometry McNaughton functions are only a first scratch on the surface of a stronger link between Łukasiewicz logic and Geometry. . De�nition • . .

  23. The uni�cation type . n . . . . A rational polytope is the convex hull of a finite set of rational points. m of Łukasiewicz logic . n m . p q . . . . . Consequences is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. MV-algebras . . Co-uni�cation Main result Rational polyhedral geometry McNaughton functions are only a first scratch on the surface of a stronger link between Łukasiewicz logic and Geometry. . De�nition • . .

  24. The uni�cation type . . . . rational polytopes. A rational polyhedron is the union of a finite number of . . . . . . . De�nition of Łukasiewicz logic . Rational polyhedral geometry [Cont.d] Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary .

  25. The uni�cation type . . . . rational polytopes. A rational polyhedron is the union of a finite number of . . . . . . . De�nition of Łukasiewicz logic . Rational polyhedral geometry [Cont.d] Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary .

  26. The uni�cation type . . . . rational polytopes. A rational polyhedron is the union of a finite number of . . . . . . . De�nition of Łukasiewicz logic . Rational polyhedral geometry [Cont.d] Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary .

  27. The uni�cation type . . . . rational polytopes. A rational polyhedron is the union of a finite number of . . . . . . . De�nition of Łukasiewicz logic . Rational polyhedral geometry [Cont.d] Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary .

  28. The uni�cation type . . . . rational polytopes. A rational polyhedron is the union of a finite number of . . . . . . . De�nition of Łukasiewicz logic . Rational polyhedral geometry [Cont.d] Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary .

  29. m spaces to the n -dimensional . n spaces. . De�nition . . The uni�cation type form the m -dimensional . . . . A -map is a continuous piecewise linear function with . The only difference here being the fact that we can operate of Łukasiewicz logic Duality for f.p. is nullary Luca Spada MV-algebras Rational polyhedra De�nitions MV-algebras MV-algebras. Consequences Co-uni�cation Main result Another way of looking at the McNaughton theorem is as a characterisation of definable functions in the language of integer coefficients. Z -maps

  30. The uni�cation type The only difference here being the fact that we can operate -map is a continuous piecewise linear function with A . . . . . . . . De�nition . spaces. MV-algebras. of Łukasiewicz logic Duality for f.p. is nullary Luca Spada MV-algebras Rational polyhedra De�nitions MV-algebras characterisation of definable functions in the language of Consequences Co-uni�cation Main result Another way of looking at the McNaughton theorem is as a integer coefficients. Z -maps form the m -dimensional R m spaces to the n -dimensional R n

  31. The uni�cation type of Łukasiewicz logic . . . . . . . . De�nition . spaces. The only difference here being the fact that we can operate MV-algebras. characterisation of definable functions in the language of De�nitions is nullary Luca Spada MV-algebras Rational polyhedra Duality for f.p. Another way of looking at the McNaughton theorem is as a MV-algebras Consequences Co-uni�cation Main result integer coefficients. Z -maps form the m -dimensional R m spaces to the n -dimensional R n A Z -map is a continuous piecewise linear function with

  32. . Let in algebraic geometry that associate ideals with varieties. of f.p. MV-algebras with These functors operate very similarly to the classical ones fp and fp I will define a pair of (contravariant) functors: -maps between them. of rational polyhedra and be the category . their homomorphisms. The uni�cation type of Łukasiewicz logic Rational polyhedra and MV-algebras Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary . .Let MV fp be the category

  33. in algebraic geometry that associate ideals with varieties. Rational polyhedra and MV-algebras These functors operate very similarly to the classical ones fp and fp I will define a pair of (contravariant) functors: of rational polyhedra and . their homomorphisms. of f.p. MV-algebras with of Łukasiewicz logic The uni�cation type Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary . .Let MV fp be the category . Let P Z be the category Z -maps between them.

  34. in algebraic geometry that associate ideals with varieties. Main result These functors operate very similarly to the classical ones and I will define a pair of (contravariant) functors: of rational polyhedra and . their homomorphisms. of f.p. MV-algebras with of Łukasiewicz logic Rational polyhedra and MV-algebras The uni�cation type Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary . .Let MV fp be the category . Let P Z be the category Z -maps between them. V : MV fp → P Z I : P Z → MV fp .

  35. The uni�cation type Main result These functors operate very similarly to the classical ones and I will define a pair of (contravariant) functors: of rational polyhedra and . their homomorphisms. of f.p. MV-algebras with of Łukasiewicz logic Rational polyhedra and MV-algebras . Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary .Let MV fp be the category . Let P Z be the category Z -maps between them. V : MV fp → P Z I : P Z → MV fp . in algebraic geometry that associate ideals with varieties.

  36. n such The uni�cation type of Łukasiewicz logic A is a rational polyhedron, so we set The set V t p for all s t s p that be the collection of all real points p in Let V F.p. MV-algebras and rational polyhedra: objects Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary V Let A = Free n ∈ MV fp . θ

  37. The uni�cation type Consequences A is a rational polyhedron, so we set The set V that F.p. MV-algebras and rational polyhedra: objects of Łukasiewicz logic Co-uni�cation Main result MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary V Let A = Free n ∈ MV fp . θ Let V ( θ ) be the collection of all real points p in [0 , 1] n such s ( p ) = t ( p ) for all ( s , t ) ∈ θ

  38. The uni�cation type MV-algebras that F.p. MV-algebras and rational polyhedra: objects Main result of Łukasiewicz logic Consequences Co-uni�cation Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary Let A = Free n ∈ MV fp . θ Let V ( θ ) be the collection of all real points p in [0 , 1] n such s ( p ) = t ( p ) for all ( s , t ) ∈ θ The set V ( θ ) is a rational polyhedron, so we set V ( A ) = V ( θ ) .

  39. The uni�cation type Let I P be the collection of all pair MV-terms s t such . I P Free n P generators, so it makes sense to set I P is a congruence of the free MV-algebras on n P t x for all x s x that F.p. MV-algebras and rational polyhedra: objects of Łukasiewicz logic Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary . Skip arrows Let P ∈ P Z .

  40. The uni�cation type of Łukasiewicz logic . I P Free n P generators, so it makes sense to set I P is a congruence of the free MV-algebras on n that F.p. MV-algebras and rational polyhedra: objects Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary . Skip arrows Let P ∈ P Z . Let I ( P ) be the collection of all pair MV-terms ( s , t ) such s ( x ) = t ( x ) for all x ∈ P

  41. The uni�cation type Consequences . generators, so it makes sense to set that F.p. MV-algebras and rational polyhedra: objects of Łukasiewicz logic Co-uni�cation Main result MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary . Skip arrows Let P ∈ P Z . Let I ( P ) be the collection of all pair MV-terms ( s , t ) such s ( x ) = t ( x ) for all x ∈ P I ( P ) is a congruence of the free MV-algebras on n I ( P ) = Free n I ( P ) .

  42. t i p The uni�cation type of Łukasiewicz logic A is a B h Then, the function A i I h A p as Suppose that h sends the generators of A into the elements F.p. MV-algebras and rational polyhedra: arrows Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary -map. Let h : A → B be a diagram in MV fp . { t i } i ∈ I of B , then define V ( h ): V ( B ) → V ( A )

  43. The uni�cation type Main result A is a B h Then, the function as Suppose that h sends the generators of A into the elements of Łukasiewicz logic F.p. MV-algebras and rational polyhedra: arrows -map. Co-uni�cation Rational MV-algebras Duality for f.p. De�nitions is nullary Luca Spada polyhedra Consequences MV-algebras Let h : A → B be a diagram in MV fp . { t i } i ∈ I of B , then define V ( h ): V ( B ) → V ( A ) V ( h ) p ∈ V ( A ) �− → ⟨ t i ( p ) ⟩ i ∈ I ∈ V ( A ) .

  44. The uni�cation type MV-algebras as Suppose that h sends the generators of A into the elements F.p. MV-algebras and rational polyhedra: arrows of Łukasiewicz logic Co-uni�cation Consequences Main result Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary Let h : A → B be a diagram in MV fp . { t i } i ∈ I of B , then define V ( h ): V ( B ) → V ( A ) V ( h ) p ∈ V ( A ) �− → ⟨ t i ( p ) ⟩ i ∈ I ∈ V ( A ) . Then, the function V ( h ): V ( B ) → V ( A ) is a Z -map.

  45. The uni�cation type of Łukasiewicz logic Then, the function P f Q f as Define F.p. MV-algebras and rational polyhedra: arrows Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary is a homomorphism of MV-algebras. Let ζ : P → Q be a diagram in P Z . I ( ζ ): I ( Q ) → I ( P )

  46. The uni�cation type Consequences Then, the function as Define F.p. MV-algebras and rational polyhedra: arrows of Łukasiewicz logic Co-uni�cation Main result MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary is a homomorphism of MV-algebras. Let ζ : P → Q be a diagram in P Z . I ( ζ ): I ( Q ) → I ( P ) I ( ζ ) f ∈ I ( Q ) �− → f ◦ ζ ∈ I ( P ) .

  47. The uni�cation type MV-algebras as Define F.p. MV-algebras and rational polyhedra: arrows Main result of Łukasiewicz logic Consequences Co-uni�cation Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary Let ζ : P → Q be a diagram in P Z . I ( ζ ): I ( Q ) → I ( P ) I ( ζ ) f ∈ I ( Q ) �− → f ◦ ζ ∈ I ( P ) . Then, the function I ( ζ ) is a homomorphism of MV-algebras.

  48. The uni�cation type of Łukasiewicz logic constitutes a contravariant equivalence between the two and The pair of functors . . . . . . . . Theorem (Folklore) . Duality for �nitely presented MV-algebras Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary categories. I : MV fp → P Z V : P Z → MV fp .

  49. projective MV-algebra is a retraction of the n-dimensional The uni�cation type . . . The rational polyhedron associated to the free algebra over n . Corollary . . . . . . . . The rational polyhedron associated to any n-generated cube . . of Łukasiewicz logic Consequences is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. MV-algebras Co-uni�cation . Main result Corollaries As a corollaries of the above duality one immediately gets . Corollary . . n . generators is the n-dimensional cube [0 , 1] n .

  50. The uni�cation type . . . . The rational polyhedron associated to the free algebra over n . Corollary . . of Łukasiewicz logic . . . . . The rational polyhedron associated to any n-generated projective MV-algebra is a retraction of the n-dimensional . . . MV-algebras is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. Consequences . Co-uni�cation Main result Corollaries As a corollaries of the above duality one immediately gets . Corollary . generators is the n-dimensional cube [0 , 1] n . cube [0 , 1] n .

  51. The uni�cation type . . . . The rational polyhedron associated to the free algebra over n . Corollary . . of Łukasiewicz logic . . . . . The rational polyhedron associated to any n-generated projective MV-algebra is a retraction of the n-dimensional . . . MV-algebras is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. Consequences . Co-uni�cation Main result Corollaries As a corollaries of the above duality one immediately gets . Corollary . generators is the n-dimensional cube [0 , 1] n . cube [0 , 1] n .

  52. The uni�cation type . . . . . . . . . Let P an injective rational polyhedron corresponding to a n -generated MV-algebra, then . . P . n . P . n . . . Proof. trivial. of Łukasiewicz logic Main result is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. MV-algebras Consequences Co-uni�cation Corollaries [Con.d] The fundamental group of any injective rational polyhedra is . Corollary . . . . . . . . id

  53. The uni�cation type . . Proof. . . . . . . . Let P an injective rational polyhedron corresponding to a The fundamental group of any injective rational polyhedra is n -generated MV-algebra, then . . . . . . . . of Łukasiewicz logic trivial. . Co-uni�cation is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. MV-algebras . Consequences Main result Corollaries [Con.d] . Corollary . . . . . . id π 1 ( P ) { π 1 ([0 , 1] n ) } π 1 ([0 , 1] n ) π 1 ( P )

  54. The uni�cation type . . Proof. . . . . . . . Let P an injective rational polyhedron corresponding to a The fundamental group of any injective rational polyhedra is n -generated MV-algebra, then . . . . . . . . of Łukasiewicz logic trivial. . Co-uni�cation is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. MV-algebras . Consequences Main result Corollaries [Con.d] . Corollary . . . . . . id π 1 ( P ) { π 1 ([0 , 1] n ) } π 1 ([0 , 1] n ) π 1 ( P )

  55. The uni�cation type . of Łukasiewicz logic Proof. . . . . . . . Let P an injective rational polyhedron corresponding to a The fundamental group of any injective rational polyhedra is n -generated MV-algebra, then . . . . . . . . trivial. . . Co-uni�cation is nullary Luca Spada MV-algebras Rational polyhedra De�nitions Duality for f.p. . Consequences MV-algebras Main result Corollaries [Con.d] . . . . id . . Corollary . {∗} ∥ π 1 ( P ) { π 1 ([0 , 1] n ) } π 1 ([0 , 1] n ) π 1 ( P )

  56. The uni�cation type Since Ghilardi’s approach is purely categorical one can speak P -map from P to Q , u 2. u is a 1. P is an injective rational polyhedron, An co-unifier for the problem Q as a pair P u where A co-unification problem as a rational polyedron Q . sense to define: the dual category of finitely presented MV-algebras, it makes As we have seen that rational polyhedra are equivalent to category. of co-unification to refer to the dual problem in the dual Co-uni�cation of Łukasiewicz logic Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary Q .

  57. The uni�cation type Since Ghilardi’s approach is purely categorical one can speak P -map from P to Q , u 2. u is a 1. P is an injective rational polyhedron, An co-unifier for the problem Q as a pair P u where A co-unification problem as a rational polyedron Q . sense to define: the dual category of finitely presented MV-algebras, it makes As we have seen that rational polyhedra are equivalent to category. of co-unification to refer to the dual problem in the dual Co-uni�cation of Łukasiewicz logic Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary Q .

  58. The uni�cation type Since Ghilardi’s approach is purely categorical one can speak P -map from P to Q , u 2. u is a 1. P is an injective rational polyhedron, An co-unifier for the problem Q as a pair P u where sense to define: the dual category of finitely presented MV-algebras, it makes As we have seen that rational polyhedra are equivalent to category. of co-unification to refer to the dual problem in the dual Co-uni�cation of Łukasiewicz logic Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary Q . ◮ A co-unification problem as a rational polyedron Q .

  59. The uni�cation type of Łukasiewicz logic 1. P is an injective rational polyhedron, sense to define: the dual category of finitely presented MV-algebras, it makes As we have seen that rational polyhedra are equivalent to category. of co-unification to refer to the dual problem in the dual Since Ghilardi’s approach is purely categorical one can speak Co-uni�cation Main result Co-uni�cation Consequences MV-algebras Duality for f.p. De�nitions polyhedra Rational MV-algebras Luca Spada is nullary ◮ A co-unification problem as a rational polyedron Q . ◮ An co-unifier for the problem Q as a pair ( P , u ) where 2. u is a Z -map from P to Q , u : P − → Q .

  60. The uni�cation type Indeed the most general form of a co-unification problem is . . . . The unification type of the 1-variable fragment of Łukasiewicz logic is finitary. In particular the proof shows that there are at most two most general unifiers, for any given formula. . . . 0 . 1 . A . B . . of Łukasiewicz logic A sequence of is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement uni�ers . A crucial lemma Strictness and unboundness Finitarity result . Proposition With A or B possibly empty or restricted to a point.

  61. The uni�cation type Indeed the most general form of a co-unification problem is . . . . The unification type of the 1-variable fragment of Łukasiewicz logic is finitary. most general unifiers, for any given formula. . . . 0 . 1 . A . B . . of Łukasiewicz logic A sequence of is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement uni�ers . A crucial lemma Strictness and unboundness Finitarity result . Proposition With A or B possibly empty or restricted to a point. In particular the proof shows that there are at most two

  62. The uni�cation type Indeed the most general form of a co-unification problem is . . . . The unification type of the 1-variable fragment of Łukasiewicz logic is finitary. most general unifiers, for any given formula. . . . 0 . 1 . A . B . . of Łukasiewicz logic A sequence of is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement uni�ers . A crucial lemma Strictness and unboundness Finitarity result . Proposition With A or B possibly empty or restricted to a point. In particular the proof shows that there are at most two

  63. The uni�cation type x . 1 . A . B . x . . . . . . x x y y 0 . of Łukasiewicz logic A sequence of is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement uni�ers geometrical phenomenon: A crucial lemma Strictness and unboundness From 1-variable to the full calculus The reason of the absence of a good unification theory for Łukasiewicz logic has to be found in the following .

  64. The uni�cation type x . 1 . A . B . x . . . . . . x x y y 0 . of Łukasiewicz logic A sequence of is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement uni�ers geometrical phenomenon: A crucial lemma Strictness and unboundness From 1-variable to the full calculus The reason of the absence of a good unification theory for Łukasiewicz logic has to be found in the following .

  65. The uni�cation type . 0 . 1 . A . B . . . . . . x x y y of Łukasiewicz logic . geometrical phenomenon: Łukasiewicz logic has to be found in the following is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement A sequence of uni�ers A crucial lemma Strictness and unboundness From 1-variable to the full calculus The reason of the absence of a good unification theory for . x ∨ x ∗ • •

  66. The uni�cation type A . . 0 . 1 . . Łukasiewicz logic has to be found in the following B . . . . . . of Łukasiewicz logic geometrical phenomenon: The reason of the absence of a good unification theory for Prologue is nullary Luca Spada MV-algebras Rational polyhedra From 1-variable to the full calculus Main result Statement A sequence of uni�ers A crucial lemma Strictness and unboundness . x ∨ x ∗ • • x ∨ x ∗ ∨ y ∨ y ∗

  67. The uni�cation type . . 0 . 1 . A B geometrical phenomenon: . . . . . . . of Łukasiewicz logic . Łukasiewicz logic has to be found in the following Statement is nullary Luca Spada MV-algebras Rational polyhedra Main result The reason of the absence of a good unification theory for Prologue A sequence of uni�ers A crucial lemma Strictness and unboundness From 1-variable to the full calculus x ∨ x ∗ • • x ∨ x ∗ ∨ y ∨ y ∗ S 1

  68. The uni�cation type . . 0 . 1 . A B geometrical phenomenon: . . . . . . . of Łukasiewicz logic . Łukasiewicz logic has to be found in the following Statement Luca Spada polyhedra is nullary Main result The reason of the absence of a good unification theory for Prologue A sequence of MV-algebras uni�ers A crucial lemma Strictness and unboundness From 1-variable to the full calculus Rational x ∨ x ∗ • • S 0 x ∨ x ∗ ∨ y ∨ y ∗ S 1

  69. The uni�cation type Proof. It is sufficient to exhibit one problem with nullary . . . . . . The full Łukasiewicz logic has nullary unification type. type. As seen above the co-unification problem associated to . x x y y is the rational polyhedron . . . Theorem of Łukasiewicz logic Statement is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue A sequence of . uni�ers A crucial lemma Strictness and unboundness Nullarity of Łukasiewicz logic A

  70. The uni�cation type type. . . . . . . The full Łukasiewicz logic has nullary unification type. As seen above the co-unification problem associated to of Łukasiewicz logic x x y y is the rational polyhedron . . . . Theorem Statement is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue A sequence of . uni�ers A crucial lemma Strictness and unboundness Nullarity of Łukasiewicz logic A Proof. It is sufficient to exhibit one problem with nullary

  71. The uni�cation type of Łukasiewicz logic . . is the rational polyhedron type. As seen above the co-unification problem associated to The full Łukasiewicz logic has nullary unification type. . . . . . . . . Theorem . Nullarity of Łukasiewicz logic unboundness is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement A sequence of uni�ers A crucial lemma Strictness and A Proof. It is sufficient to exhibit one problem with nullary ( x ∨ x ∗ ∨ y ∨ y ∗ , 1)

  72. i is a n for some n , so the pairs i are The uni�cation type . of Łukasiewicz logic . . Consider the following sequence of pair of maps and rational polyhedra, . . . . . . . It can be proved (cfr. Cabrer and Mundici) that each retract of i . . . uni�ers is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement A sequence of . co-unifiers for A . A crucial lemma Strictness and unboundness Proof Cont.'d . Step 1. . t 1 t 2 t 3 ζ 1 ζ 2 ζ 3

  73. i are The uni�cation type . . of Łukasiewicz logic . . . Consider the following sequence of pair of maps and rational polyhedra, . . . . . . . so the pairs i . . . Statement is nullary Luca Spada MV-algebras Rational polyhedra Main result Step 1. Prologue A sequence of Strictness . Proof Cont.'d unboundness and co-unifiers for A . lemma A crucial uni�ers t 1 t 2 t 3 ζ 1 ζ 2 ζ 3 It can be proved (cfr. Cabrer and Mundici) that each t i is a retract of [0 , 1] n for some n ,

  74. The uni�cation type Consider the following sequence of pair of maps and rational . of Łukasiewicz logic . . . . polyhedra, . . . . . . . . . . Step 1. Prologue is nullary Luca Spada MV-algebras Rational polyhedra . Main result co-unifiers for A . Statement Strictness Proof Cont.'d unboundness A sequence of and lemma A crucial uni�ers t 1 t 2 t 3 ζ 1 ζ 2 ζ 3 It can be proved (cfr. Cabrer and Mundici) that each t i is a retract of [0 , 1] n for some n , so the pairs ( t i , ζ i ) are

  75. ij such that the following diagram commutes. . j . i ij is just the embedding of i in The uni�cation type . . . The sequence is increasing, i.e. for any i j , there exists . . . A . j . i . ij Indeed . . of Łukasiewicz logic A sequence of is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement uni�ers . A crucial lemma Strictness and unboundness Proof Cont.'d . Step 2. . j

  76. ij is just the embedding of i in The uni�cation type . of Łukasiewicz logic . . . . . . A . . . . . Indeed . . . A sequence of is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Step 2. Statement uni�ers A crucial lemma Strictness and unboundness Proof Cont.'d . j The sequence is increasing, i.e. for any i < j , there exists ι ij such that the following diagram commutes. t j ζ j ι ij ζ i t i

  77. The uni�cation type . . of Łukasiewicz logic . . . . . Step 2. . A . . . . . . . . Statement is nullary Luca Spada MV-algebras Rational polyhedra Main result Proof Cont.'d Prologue A sequence of uni�ers A crucial lemma Strictness and unboundness The sequence is increasing, i.e. for any i < j , there exists ι ij such that the following diagram commutes. t j ζ j ι ij ζ i t i Indeed ι ij is just the embedding of t i in t j

  78. i and an . i The uni�cation type commute. . . . . For any co-unifier u P of A , there exists some arrow u (called the lift of u ) making the following diagram . . . P . A . i . u . . . of Łukasiewicz logic A sequence of is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement uni�ers . A crucial lemma Strictness and unboundness Proof Cont.'d . Step 3: The lifting of de�nable functions. u

  79. i and an . i The uni�cation type commute. . . . . there exists some arrow u (called the lift of u ) making the following diagram . . of Łukasiewicz logic P . A . i . u . . . . A sequence of is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement uni�ers . A crucial lemma Strictness and unboundness Proof Cont.'d . Step 3: The lifting of de�nable functions. u For any co-unifier ( u , P ) of A ,

  80. The uni�cation type . . . . . and an arrow u (called the lift of u ) making the following diagram commute. . of Łukasiewicz logic P . A . . . u . . . . . is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement A sequence of uni�ers A crucial lemma Strictness and unboundness Proof Cont.'d . Step 3: The lifting of de�nable functions. u For any co-unifier ( u , P ) of A , there exists some t i t i ζ i

  81. The uni�cation type . of Łukasiewicz logic . . . . u (called the lift of u ) making the following diagram commute. . . P . A . . . u . . . . Step 3: The lifting of de�nable functions. is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement A sequence of uni�ers A crucial lemma Strictness and unboundness Proof Cont.'d . u For any co-unifier ( u , P ) of A , there exists some t i and an arrow ˜ t i ˜ ζ i

  82. a finite portion of the piece-wise linear cover . The uni�cation type “ Lifting of functions ” Lemma , widely used in algebraic . . . . The above lemma is the piecewise linear version of the The crucial fact here is that we can always factorize through topology. . As a matter of fact, for general reasons when such a map exists is unique up to translations . So the fact that in our setting such a map is actually a -map is a quite pleasant discovery. . . of Łukasiewicz logic Prologue is nullary Luca Spada MV-algebras Rational polyhedra Main result Statement Considerations A sequence of uni�ers A crucial lemma Strictness and unboundness The Definable Lifting Lemma has two important corollaries.

  83. The uni�cation type “ Lifting of functions ” Lemma , widely used in algebraic . . . . . The above lemma is the piecewise linear version of the topology. of Łukasiewicz logic The crucial fact here is that we can always factorize through As a matter of fact, for general reasons when such a map exists is unique up to translations . So the fact that in our setting such a map is actually a -map is a quite pleasant discovery. . . Considerations unboundness is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement A sequence of uni�ers A crucial lemma Strictness and The Definable Lifting Lemma has two important corollaries. a finite portion of the piece-wise linear cover .

  84. The Definable Lifting Lemma has two important corollaries. of Łukasiewicz logic discovery. exists is unique up to translations . So the fact that in our As a matter of fact, for general reasons when such a map The crucial fact here is that we can always factorize through topology. “ Lifting of functions ” Lemma , widely used in algebraic The above lemma is the piecewise linear version of the . . . . . . . The uni�cation type unboundness and is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement A sequence of uni�ers A crucial lemma Strictness Considerations a finite portion of the piece-wise linear cover . setting such a map is actually a Z -map is a quite pleasant

  85. The uni�cation type of Łukasiewicz logic discovery. exists is unique up to translations . So the fact that in our As a matter of fact, for general reasons when such a map The crucial fact here is that we can always factorize through topology. “ Lifting of functions ” Lemma , widely used in algebraic The above lemma is the piecewise linear version of the . . . . . . . Considerations unboundness and is nullary Luca Spada MV-algebras Rational polyhedra Main result Prologue Statement A sequence of uni�ers A crucial lemma Strictness a finite portion of the piece-wise linear cover . setting such a map is actually a Z -map is a quite pleasant The Definable Lifting Lemma has two important corollaries.

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