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Unification on Subvarieties of Pseudocomple- mented lattices L.M. Cabrer Unification on Subvarieties of Introduction Algebraic Unification Pseudocomplemented lattices Fragments of Heyting algebras P-Lattices Definition Duality Leonardo Manuel Cabrer Main Result Other Results Università degli Studi di Firenze Dipartimento di Statistica, Informatica, Applicazioni “G. Parenti” Marie Curie Intra-European Fellowship – FP7 BLAST – 2013

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer [1] S. Ghilardi, Introduction Unification through projectivity, Algebraic Unification Journal of Logic and Comp. 7 (6) 733-752, 1997. Fragments of Heyting algebras P-Lattices Definition Duality Main Result Other Results

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer [1] S. Ghilardi, Introduction Unification through projectivity, Algebraic Unification Journal of Logic and Comp. 7 (6) 733-752, 1997. Fragments of Heyting algebras P-Lattices Definition Unification Problem: Finitely presented algebra A Duality Main Result Other Results

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer [1] S. Ghilardi, Introduction Unification through projectivity, Algebraic Unification Journal of Logic and Comp. 7 (6) 733-752, 1997. Fragments of Heyting algebras P-Lattices Definition Unification Problem: Finitely presented algebra A Duality Main Result Other Results Solution (Unifier): h : A → P P is projective

� � � Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer [1] S. Ghilardi, Introduction Unification through projectivity, Algebraic Unification Journal of Logic and Comp. 7 (6) 733-752, 1997. Fragments of Heyting algebras P-Lattices Definition Unification Problem: Finitely presented algebra A Duality Main Result Other Results Solution (Unifier): h : A → P P is projective Pre-order: h A P f h ′ P ′

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices Let A ∈ V a finitely presented algebra of a variety V and L.M. Cabrer U V ( A ) the pre-order of its unifiers. Then A is said to have unification type: Introduction Algebraic Unification Fragments of Heyting algebras P-Lattices Definition Duality Main Result Other Results

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices Let A ∈ V a finitely presented algebra of a variety V and L.M. Cabrer U V ( A ) the pre-order of its unifiers. Then A is said to have unification type: Introduction Algebraic Unification U V ( A ) Fragments of Heyting algebras P-Lattices Definition 1 Duality Main Result Other Results

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices Let A ∈ V a finitely presented algebra of a variety V and L.M. Cabrer U V ( A ) the pre-order of its unifiers. Then A is said to have unification type: Introduction Algebraic Unification U V ( A ) Fragments of Heyting algebras P-Lattices Definition 1 Duality Main Result Other Results n

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices Let A ∈ V a finitely presented algebra of a variety V and L.M. Cabrer U V ( A ) the pre-order of its unifiers. Then A is said to have unification type: Introduction Algebraic Unification U V ( A ) Fragments of Heyting algebras P-Lattices Definition 1 Duality Main Result Other Results n ∞

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices Let A ∈ V a finitely presented algebra of a variety V and L.M. Cabrer U V ( A ) the pre-order of its unifiers. Then A is said to have unification type: Introduction Algebraic Unification U V ( A ) Fragments of Heyting algebras P-Lattices Definition 1 Duality Main Result Other Results n ∞ 0

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer A variety V is said to have type: Introduction Algebraic Unification Fragments of Heyting algebras P-Lattices Definition Duality Main Result Other Results

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer A variety V is said to have type: Introduction Algebraic Unification Fragments of Heyting algebras ◮ 1 if every finitely presented A in V has unification P-Lattices type 1; Definition Duality Main Result Other Results

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer A variety V is said to have type: Introduction Algebraic Unification Fragments of Heyting algebras ◮ 1 if every finitely presented A in V has unification P-Lattices type 1; Definition Duality Main Result ◮ ω if every finitely presented A in V has finite Other Results unification type

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer A variety V is said to have type: Introduction Algebraic Unification Fragments of Heyting algebras ◮ 1 if every finitely presented A in V has unification P-Lattices type 1; Definition Duality Main Result ◮ ω if every finitely presented A in V has finite Other Results unification type and at least one finitely presented A 0 in V has not unification type 1;

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer A variety V is said to have type: Introduction Algebraic Unification Fragments of Heyting algebras ◮ 1 if every finitely presented A in V has unification P-Lattices type 1; Definition Duality Main Result ◮ ω if every finitely presented A in V has finite Other Results unification type and at least one finitely presented A 0 in V has not unification type 1; ◮ ∞ if every every finitely presented A of V has unification 1, n or ∞

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer A variety V is said to have type: Introduction Algebraic Unification Fragments of Heyting algebras ◮ 1 if every finitely presented A in V has unification P-Lattices type 1; Definition Duality Main Result ◮ ω if every finitely presented A in V has finite Other Results unification type and at least one finitely presented A 0 in V has not unification type 1; ◮ ∞ if every every finitely presented A of V has unification 1, n or ∞ and at least one finitely presented A 0 in V has unification has type ∞ ;

Unification on Introduction Subvarieties of Pseudocomple- Algebraic Unification mented lattices L.M. Cabrer A variety V is said to have type: Introduction Algebraic Unification Fragments of Heyting algebras ◮ 1 if every finitely presented A in V has unification P-Lattices type 1; Definition Duality Main Result ◮ ω if every finitely presented A in V has finite Other Results unification type and at least one finitely presented A 0 in V has not unification type 1; ◮ ∞ if every every finitely presented A of V has unification 1, n or ∞ and at least one finitely presented A 0 in V has unification has type ∞ ; ◮ 0 if at least one finitely presented A 0 in V has unification type 0.

Unification on Introduction Subvarieties of Pseudocomple- Fragments of Heyting algebras mented lattices L.M. Cabrer Introduction Algebraic Unification Fragment Type Fragments of Heyting algebras P-Lattices Heyting algebras (Ghilardi) Definition ω Duality Main Result Hilbert algebras 1 (Prucnal) Other Results Browerian semilattices 1 (Ghilardi) ( → , ¬ ) -Fragment (Cintula-Metcalfe) ω Bounded Distributive Lattices 0 (Ghilardi) Pseudocomplemented Lattices 0 (Ghilardi)

Unification on Introduction Subvarieties of Pseudocomple- Fragments of Heyting algebras: Bounded Distributive lattices mented lattices L.M. Cabrer [4] S. Bova and LMC, Introduction Unification and Projectivity in Algebraic Unification Fragments of Heyting De Morgan and Kleene Algebras algebras P-Lattices Order (published online June 2013). Definition Duality Main Result Other Results

Unification on Introduction Subvarieties of Pseudocomple- Fragments of Heyting algebras: Bounded Distributive lattices mented lattices L.M. Cabrer [4] S. Bova and LMC, Introduction Unification and Projectivity in Algebraic Unification Fragments of Heyting De Morgan and Kleene Algebras algebras P-Lattices Order (published online June 2013). Definition Duality Main Result Other Results Theorem Let L be a finitely presented (equivalently finite) bounded distributive lattice and H ( L ) be its Priestley dual. Then the unification type of L is: 1 iff H ( L ) is a lattice; finite iff for every x , y ∈ H ( L ) the interval [ x , y ] is a lattice; 0 otherwise.

Unification on Pseudocomplemented Lattices Subvarieties of Pseudocomple- Definition mented lattices L.M. Cabrer Introduction Algebraic Unification Fragments of Heyting algebras P-Lattices An algebra ( A , ∨ , ∧ , ¬ , 0 , 1 ) is a pseudocomplemented Definition Duality distributive lattice if ( A , ∨ , ∧ , 0 , 1 ) is a bounded distributive Main Result Other Results lattice and it satisfies a ∧ b = 0 ⇔ a ≤ ¬ b