On lattice-valued frames: the completely distributive case Javier Gutiérrez García (joint work with Ulrich Höhle and M. Angeles de Prada Vicente) Chengdu, April 6, 2010 On lattice-valued frames Dedicated to Professor Liu on his 70th birthday On lattice-valued frames On lattice-valued frames On lattice-valued frames On lattice-valued frames Origin of the work On lattice-valued frames • Visit of Ulrich Höhle to Bilbao, April 4 to 11, (2008) • 29th Linz Seminar on Fuzzy Set Theory: “Foundations of Lattice-Valued Mathematics with Applications to Algebra and Topology” Linz, (Austria), February 4 to 9, (2008) Attendants: Rodabaugh, Zhang, Höhle, Kubiak, Šostak, de Prada Vicente,. A. Pultr, S.E. Rodabaugh, Category theoretic aspects of chain-valued frames: Part I: Categorical and presheaf theoretic foundations, Part II: Applications to lattice-valued topology, Fuzzy Sets and Systems 159 (2008) 501–528 and 529–558. On lattice-valued frames • Visit of Ulrich Höhle to Bilbao, April 4 to 11, (2008) • 29th Linz Seminar on Fuzzy Set Theory: “Foundations of Lattice-Valued Mathematics with Applications to Algebra and Topology” Linz, (Austria), February 4 to 9, (2008) Attendants: Rodabaugh, Zhang, Höhle, Kubiak, Šostak, de Prada Vicente,. A. Pultr, S.E. Rodabaugh, Category theoretic aspects of chain-valued frames: Part I: Categorical and presheaf theoretic foundations, Part II: Applications to lattice-valued topology, Fuzzy Sets and Systems 159 (2008) 501–528 and 529–558. On lattice-valued frames • 29th Linz Seminar on Fuzzy Set Theory: “Foundations of Lattice-Valued Mathematics with Applications to Algebra and Topology” Linz, (Austria), February 4 to 9, (2008) Attendants: Rodabaugh, Zhang, Höhle, Kubiak, Šostak, de Prada Vicente,. • Visit of Ulrich Höhle to Bilbao, April 4 to 11, (2008) A. Pultr, S.E. Rodabaugh, Category theoretic aspects of chain-valued frames: Part I: Categorical and presheaf theoretic foundations, Part II: Applications to lattice-valued topology, Fuzzy Sets and Systems 159 (2008) 501–528 and 529–558. On lattice-valued frames • L-valued (Fuzzy) topology • Pointfree (Pointless) topology, Frame (Locale) theory Introduction On lattice-valued frames • L-valued (Fuzzy) topology Introduction • Pointfree (Pointless) topology, Frame (Locale) theory On lattice-valued frames Introduction • Pointfree (Pointless) topology, Frame (Locale) theory • L-valued (Fuzzy) topology On lattice-valued frames Pointfree topology On lattice-valued frames POINTFREE TOPOLOGY S S A \ Bi = (A \ Bi ) i2I i2I f −1 preserves S and \ (Y ; OY ) (OY ; ⊆) Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o / Pointfree topology Motivation (X; OX) /o /o /o /o /o /o /o / (OX; ⊆) On lattice-valued frames POINTFREE TOPOLOGY f −1 preserves S and \ (Y ; OY ) (OY ; ⊆) Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o / Pointfree topology Motivation S S (X; OX) /o /o /o /o /o /o /o / (OX; ⊆) A \ Bi = (A \ Bi ) i2I i2I On lattice-valued frames POINTFREE TOPOLOGY f −1 preserves S and \ (OY ; ⊆) Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o / Pointfree topology Motivation S S (X; OX) /o /o /o /o /o /o /o / (OX; ⊆) A \ Bi = (A \ Bi ) i2I i2I f (Y ; OY ) On lattice-valued frames POINTFREE TOPOLOGY f −1 preserves S and \ Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o / Pointfree topology Motivation S S (X; OX) /o /o /o /o /o /o /o / (OX; ⊆) A \ Bi = (A \ Bi ) i2I i2I f (Y ; OY ) /o /o /o /o /o /o /o / (OY ; ⊆) On lattice-valued frames POINTFREE TOPOLOGY f −1 preserves S and \ Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o / Pointfree topology Motivation S S (X; OX) /o /o /o /o /o /o /o / (OX; ⊆) A \ Bi = (A \ Bi ) O i2I i2I f f −1 (Y ; OY ) /o /o /o /o /o /o /o / (OY ; ⊆) On lattice-valued frames POINTFREE TOPOLOGY Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o / Pointfree topology Motivation S S (X; OX) /o /o /o /o /o /o /o / (OX; ⊆) A \ Bi = (A \ Bi ) O i2I i2I f f −1 f −1 preserves S and \ (Y ; OY ) /o /o /o /o /o /o /o / (OY ; ⊆) On lattice-valued frames POINTFREE TOPOLOGY Pointfree topology Motivation S S (X; OX) /o /o /o /o /o /o /o / (OX; ⊆) A \ Bi = (A \ Bi ) O i2I i2I f f −1 f −1 preserves S and \ (Y ; OY ) /o /o /o /o /o /o /o / (OY ; ⊆) Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o / On lattice-valued frames Pointfree topology Motivation S S (X; OX) /o /o /o /o /o /o /o / (OX; ⊆) A \ Bi = (A \ Bi ) O i2I i2I f f −1 f −1 preserves S and \ (Y ; OY ) /o /o /o /o /o /o /o / (OY ; ⊆) Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o / POINTFREE TOPOLOGY On lattice-valued frames in which ∗ complete lattices L W W ∗ a ^ i2I ai = fa ^ ai : i 2 Ig for all a 2 L and fai : i 2 Ig ⊆ L. • Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and finite meets. • O : Top ! Frm is a contravariant functor with X 7−! OX and f f −1 X ! Y 7−! OY −! OX. Pointfree topology the category of frames Frm • The objects in Frm are frames, i.e. On lattice-valued frames in which W W ∗ a ^ i2I ai = fa ^ ai : i 2 Ig for all a 2 L and fai : i 2 Ig ⊆ L. • Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and finite meets. • O : Top ! Frm is a contravariant functor with X 7−! OX and f f −1 X ! Y 7−! OY −! OX. Pointfree topology the category of frames Frm • The objects in Frm are frames, i.e. ∗ complete lattices L On lattice-valued frames • Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and finite meets. • O : Top ! Frm is a contravariant functor with X 7−! OX and f f −1 X ! Y 7−! OY −! OX. Pointfree topology the category of frames Frm • The objects in Frm are frames, i.e. ∗ complete lattices L in which W W ∗ a ^ i2I ai = fa ^ ai : i 2 Ig for all a 2 L and fai : i 2 Ig ⊆ L. On lattice-valued frames • O : Top ! Frm is a contravariant functor with X 7−! OX and f f −1 X ! Y 7−! OY −! OX. Pointfree topology the category of frames Frm • The objects in Frm are frames, i.e. ∗ complete lattices L in which W W ∗ a ^ i2I ai = fa ^ ai : i 2 Ig for all a 2 L and fai : i 2 Ig ⊆ L. • Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and finite meets. On lattice-valued frames Pointfree topology the category of frames Frm • The objects in Frm are frames, i.e. ∗ complete lattices L in which W W ∗ a ^ i2I ai = fa ^ ai : i 2 Ig for all a 2 L and fai : i 2 Ig ⊆ L. • Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and finite meets. • O : Top ! Frm is a contravariant functor with X 7−! OX and f f −1 X ! Y 7−! OY −! OX. On lattice-valued frames • Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards. • O : Top ! Loc is now a covariant functor with X 7−! OX and f f −1 X ! Y 7−! OX −! OY . Advantage: Loc can be thought of as a natural extension of (sober) spaces. Disadvantage: Morphisms thought in this way may obscure the intuition. Pointfree topology the dual category Loc=Frmop • The objects in Loc are frames, from now on, also called locales. On lattice-valued frames • O : Top ! Loc is now a covariant functor with X 7−! OX and f f −1 X ! Y 7−! OX −! OY . Advantage: Loc can be thought of as a natural extension of (sober) spaces. Disadvantage: Morphisms thought in this way may obscure the intuition. Pointfree topology the dual category Loc=Frmop • The objects in Loc are frames, from now on, also called locales. • Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards. On lattice-valued frames Advantage: Loc can be thought of as a natural extension of (sober) spaces. Disadvantage: Morphisms thought in this way may obscure the intuition. Pointfree topology the dual category Loc=Frmop • The objects in Loc are frames, from now on, also called locales. • Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards. • O : Top ! Loc is now a covariant functor with X 7−! OX and f f −1 X ! Y 7−! OX −! OY . On lattice-valued frames Disadvantage: Morphisms thought in this way may obscure the intuition. Pointfree topology the dual category Loc=Frmop • The objects in Loc are frames, from now on, also called locales. • Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards. • O : Top ! Loc is now a covariant functor with X 7−! OX and f f −1 X ! Y 7−! OX −! OY . Advantage: Loc can be thought of as a natural extension of (sober) spaces. On lattice-valued frames Pointfree topology the dual category Loc=Frmop • The objects in Loc are frames, from now on, also called locales. • Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards. • O : Top ! Loc is now a covariant functor with X 7−! OX and f f −1 X ! Y 7−! OX −! OY . Advantage: Loc can be thought of as a natural extension of (sober) spaces. Disadvantage: Morphisms thought in this way may obscure the intuition.