Free alg’s via functor on partial alg’s Dion Coumans and Free algebras via a functor Sam van Gool on partial algebras Free algebra step-by-step Free image-total functor Dion Coumans and Sam van Gool Application to KB Topology, Algebra and Categories in Logic (TACL) 26 – 30 July 2011 Marseilles, France 1 / 16
Free alg’s via functor on Logic via algebra partial alg’s Dion Coumans and Sam van Gool • Algebraic logic L , signature σ , variety V L of σ -algebras Free algebra step-by-step Free image-total functor Application to KB 2 / 16
Free alg’s via functor on Logic via algebra partial alg’s Dion Coumans and Sam van Gool • Algebraic logic L , signature σ , variety V L of σ -algebras Free algebra • Studying the logic L � Studying finitely generated free step-by-step V L -algebras Free image-total functor Application to KB 2 / 16
Free alg’s via functor on Logic via algebra partial alg’s Dion Coumans and Sam van Gool • Algebraic logic L , signature σ , variety V L of σ -algebras Free algebra • Studying the logic L � Studying finitely generated free step-by-step V L -algebras Free image-total functor Application to KB F σ ( x 1 , . . . , x m ) Language 2 / 16
Free alg’s via functor on Logic via algebra partial alg’s Dion Coumans and Sam van Gool • Algebraic logic L , signature σ , variety V L of σ -algebras Free algebra • Studying the logic L � Studying finitely generated free step-by-step V L -algebras Free image-total functor Application to KB F σ ( x 1 , . . . , x m ) Language [ · ] ⊣⊢ L F V L ( x 1 , . . . , x m ) Logic 2 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step • In many cases, a variety ( V L ) − of reducts is Free image-total well-understood and locally finite, e.g.: functor Application to KB 3 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step • In many cases, a variety ( V L ) − of reducts is Free image-total well-understood and locally finite, e.g.: functor Application to • Modal algebras = Boolean algebras + � , KB 3 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step • In many cases, a variety ( V L ) − of reducts is Free image-total well-understood and locally finite, e.g.: functor Application to • Modal algebras = Boolean algebras + � , KB • Heyting algebras = Distributive lattices + → , 3 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step • In many cases, a variety ( V L ) − of reducts is Free image-total well-understood and locally finite, e.g.: functor Application to • Modal algebras = Boolean algebras + � , KB • Heyting algebras = Distributive lattices + → , • . . . 3 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step • In many cases, a variety ( V L ) − of reducts is Free image-total well-understood and locally finite, e.g.: functor Application to • Modal algebras = Boolean algebras + � , KB • Heyting algebras = Distributive lattices + → , • . . . • Regard F V L ( x 1 , . . . , x n ) as colimit of a chain of finite algebras in the reduced signature, and add the additional operation(s) step-by-step: 3 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool Language Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB Logic 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool T 0 Language Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB Logic • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool T 0 Language Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB B 0 Logic • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool T 0 T 1 Language Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB B 0 Logic • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool T 0 T 1 Language Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB B 0 B 1 Logic • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool · · · T 0 T 1 Language Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB · · · B 0 B 1 Logic • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool · · · T 0 T 1 T n Language Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB · · · B 0 B 1 B n Logic • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool · · · · · · T 0 T 1 T n Language Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB · · · · · · B 0 B 1 B n Logic • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool · · · · · · T 0 T 1 T n Language f Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB · · · · · · B 0 B 1 B n Logic • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool · · · · · · T 0 T 1 T n Language f Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB · · · · · · B 0 B 1 B n Logic f • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool · · · · · · T 0 T 1 T n Language f f f Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB · · · · · · B 0 B 1 B n Logic f f f • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n 4 / 16
Free alg’s via functor on Free algebra as colimit of a chain partial alg’s Dion Coumans and Sam van Gool · · · · · · T 0 T 1 T n Language f f f Free algebra step-by-step Free image-total [ · ] ⊣⊢ L functor Application to KB · · · · · · B 0 B 1 B n Logic f f f • T n : formulas in variables x 1 , . . . , x m of rank ≤ n in operation f • B n : L -equivalence classes of formulas in T n F V L ( x 1 , . . . , x m ) = colim n ≥ 0 B n 4 / 16
Free alg’s via functor on Research Question partial alg’s Dion Coumans and Sam van Gool Free algebra · · · · · · B 0 B 1 B n step-by-step Free image-total functor Application to Can B n + 1 be obtained from B n by a uniform method? KB 5 / 16
Free alg’s via functor on Research Question partial alg’s Dion Coumans and Sam van Gool Free algebra · · · · · · B 0 B 1 B n step-by-step Free image-total functor Application to Can B n + 1 be obtained from B n by a uniform method? KB • Yes, if the variety is defined by pure rank 1 equations [N. Bezhanishvili, Kurz] 5 / 16
Free alg’s via functor on Research Question partial alg’s Dion Coumans and Sam van Gool Free algebra · · · · · · B 0 B 1 B n step-by-step Free image-total functor Application to Can B n + 1 be obtained from B n by a uniform method? KB • Yes, if the variety is defined by pure rank 1 equations [N. Bezhanishvili, Kurz] • Yes, in some particular cases outside this class: S4 modal algebras [Ghilardi], Heyting algebras [Ghilardi, N. Bezhanishvili & Gehrke]. 5 / 16
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