The quantale of Galois connections

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1 The quantale of Galois connections Jorge Picado Mathematics Department - University of Coimbra PORTUGAL June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 1

2 GALOIS CONNECTIONS [Birkhoff 1940, Ore 1944] Posets A, B A f B g June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 2

3 GALOIS CONNECTIONS [Birkhoff 1940, Ore 1944] Posets A, B A f B g f Ant(A, B) and g Ant(B, A) a A a gf(a), b B b fg(b) f( S) = f(s), g( S) = g(s). June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 2

4 GALOIS CONNECTIONS [Birkhoff 1940, Ore 1944] Posets A, B A f B g f Ant(A, B) and g Ant(B, A) a A a gf(a), b B b fg(b) f( S) = f(s), g( S) = g(s). f Gal(A, B) if f + : B A (necessarily unique) such that (f, f + ) is a Galois connection. June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 2

5 GALOIS CONNECTIONS [Birkhoff 1940, Ore 1944] Posets A, B A f g B f Ant(A, B) and g Ant(B, A) a A a gf(a), b B b fg(b) f( S) = f(s), g( S) = g(s). f Gal(A, B) if f + : B A (necessarily unique) such that (f, f + ) is a Galois connection. f( S) = f(s). (when A, B CLat) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 2

6 HOW TO COMPOSE GALOIS MAPS? (Gal(A, A d ), ) complete semigroup }{{} QUANTALE of residuated maps f( S) = f(s) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 3

7 HOW TO COMPOSE GALOIS MAPS? (Gal(A, A d ), ) complete semigroup }{{} QUANTALE of residuated maps f( S) = f(s) (Gal(A, A),?) Galois maps f( S) = f(s) HOW TO COMPOSE? June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 3

8 CATEGORICAL MOTIVATION [Freyd, Scedrov 1990] In any category with finite limits and images: R 1 > m π 1 1 X X π 2 X R 2 > m π 1 2 X X π 2 X June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 4

9 CATEGORICAL MOTIVATION [Freyd, Scedrov 1990] In any category with finite limits and images: R 1 R 2 m 1 m 2 X X X X π 1 π 2 π 1 π 2 X X X June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 4

10 CATEGORICAL MOTIVATION [Freyd, Scedrov 1990] In any category with finite limits and images: R 1 R 2 R 1 R 2 m 1 m 2 X X X X π 1 π 2 π 1 π 2 X X X June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 4

11 CATEGORICAL MOTIVATION [Freyd, Scedrov 1990] In any category with finite limits and images: X X R 1 R 2 π 1 π 2 R 1 R 2 m 1 m 2 X X X X π 1 π 2 π 1 π 2 X X X June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 4

12 CATEGORICAL MOTIVATION [Freyd, Scedrov 1990] In any category with finite limits and images: X X f R 1 R 2 π 1 π 2 R 1 R 2 m 1 m 2 X X X X π 1 π 2 π 1 π 2 X X X June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 4

13 CATEGORICAL MOTIVATION [Freyd, Scedrov 1990] In any category with finite limits and images: X X f R 1 R 2 June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 4

14 CATEGORICAL MOTIVATION [Freyd, Scedrov 1990] In any category with finite limits and images: X X f R 1 R 2 The composition of the relations m 1 and m 2 is the image of f: f(r 1 R 2 ) > m X X June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 4

15 APPLICATION TO THE CATEGORY OF LOCALES [Picado, Ferreira, Quaest. Math., to appear] Loc Frm d POINTFREE TOPOLOGY frames: complete Heyting algebras morphisms: preserve, June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 5

16 APPLICATION TO THE CATEGORY OF LOCALES [Picado, Ferreira, Quaest. Math., to appear] Loc Frm d POINTFREE TOPOLOGY frames: complete Heyting algebras morphisms: preserve, f Gal(L, L) special binary relations on L June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 5

17 APPLICATION TO THE CATEGORY OF LOCALES [Picado, Ferreira, Quaest. Math., to appear] Loc Frm d POINTFREE TOPOLOGY frames: complete Heyting algebras morphisms: preserve, f Gal(L, L) special binary relations on L g f Gal(L, L) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 5

18 BINARY (TENSOR) PRODUCTS Category of complete -semilattices The tensor product A B of A and B: R = R G-ideals of A B {x} U 2 R (x, U 2 ) R U 1 {y} R ( U 1, y) R (left) (right) a b := (a, b) (1,0) (0,1) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 6

19 BINARY (TENSOR) PRODUCTS Category of locales The product A B of A and B: R = R G-ideals of A B {x} U 2 R (x, U 2 ) R U 1 {y} R ( U 1, y) R (left) (right) a b := (a, b) (1,0) (0,1) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 6

20 THE GALOIS COMPOSITION f Gal(A, B), g Gal(B, C) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 7

21 THE GALOIS COMPOSITION f Gal(A, B), g Gal(B, C) R f,g := {a c A C b B \ {0} : f(a) b and g(b) c}. June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 7

22 THE GALOIS COMPOSITION f Gal(A, B), g Gal(B, C) R f,g := {a c A C b B \ {0} : f(a) b and g(b) c}. The Galois composition g f : A C is defined by (g f)(a) := {c C (a, c) R f,g }. June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 7

23 THE GALOIS COMPOSITION f Gal(A, B), g Gal(B, C) R f,g := {a c A C b B \ {0} : f(a) b and g(b) c}. The Galois composition g f : A C is defined by (g f)(a) := {c C (a, c) R f,g }. g f Gal(A, C) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 7

24 THE GALOIS COMPOSITION f Gal(A, Ant(A, B), g Gal(B, Ant(B, C) R f,g := {a c A C b B \ {0} : f(a) b and g(b) c}. The Galois composition g f : A C is defined by (g f)(a) := {c C (a, c) R f,g }. g f Gal(A, C) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 7

25 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 1: THE QUANTALE OF ANTITONE MAPS f Ant(A, B), g Ant(B, C) A g f C g f Ant(A, C) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 8

26 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 1: THE QUANTALE OF ANTITONE MAPS f Ant(A, B), g Ant(B, C) A g f C g f Ant(A, C) (g f)(a) := {c C b B \ {0} : f(a) b and g(b) c} June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 8

27 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 1: THE QUANTALE OF ANTITONE MAPS f Ant(A, B), g Ant(B, C) A g f C g f Ant(A, C) (g f)(a) := {c C b B \ {0} : f(a) b and g(b) c} associative g ( f i ) = (g f i ) ( g i ) f = (g i f) on frames June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 8

28 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 1: THE QUANTALE OF ANTITONE MAPS f Ant(A, B), g Ant(B, C) A g f C g f Ant(A, C) (g f)(a) := {c C b B \ {0} : f(a) b and g(b) c} associative g ( f i ) = (g f i ) ( g i ) f = (g i f) on frames THEOREM: If L is a frame then (Ant(L, L), ) is a quantale June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 8

29 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 2: Gal(L, L) IS A QUANTIC QUOTIENT of Ant(L, L) f Ant(L, L) j 0 (f)(a) := {b L S L : S = a and b f(s) s S}. June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 9

30 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 2: Gal(L, L) IS A QUANTIC QUOTIENT of Ant(L, L) f Ant(L, L) j 0 (f)(a) := {b L S L : S = a and b f(s) s S}. If L is a frame then j 0 (f) Ant(A, B) j 0 is a quantic prenucleus on Ant(L, L) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 9

31 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 2: Gal(L, L) IS A QUANTIC QUOTIENT of Ant(L, L) quantic prenucleus j 0 : Q Q preclosure operator j 0 (x) y j 0 (x y) x j 0 (y) j 0 (x y) Quantale (Q, ) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 10

32 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 2: Gal(L, L) IS A QUANTIC QUOTIENT of Ant(L, L) A GENERAL PROCEDURE ON QUANTALES quantic prenucleus j 0 : Q Q preclosure operator j 0 (x) y j 0 (x y) x j 0 (y) j 0 (x y) Quantale (Q, ) j(x):= {y Q j0 x y} Q j0 :=Fix(j 0 ) quantic nucleus j : Q Q { closure operator j(x) j(y) j(x y) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 10

33 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 2: Gal(L, L) IS A QUANTIC QUOTIENT of Ant(L, L) A GENERAL PROCEDURE ON QUANTALES quantic prenucleus j 0 : Q Q preclosure operator j 0 (x) y j 0 (x y) x j 0 (y) j 0 (x y) Quantale (Q, ) j(x):= {y Q j0 x y} Q j0 :=Fix(j 0 ) quantic nucleus j : Q Q { closure operator j(x) j(y) j(x y) For a jb := j(a b) (Q j, j) is a quantale Q j Q j June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 10

34 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 2: Gal(L, L) IS A QUANTIC QUOTIENT of Ant(L, L) f Ant(L, L) j 0 (f)(a) := {b L S L : S = a and b f(s) s S} Fix(j 0 )=Gal(L,L) j(f) = {g Gal(L, L) f g} June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 11

35 TOWARDS THE QUANTALE OF GALOIS CONNECTIONS [Alg. Univ., 2004] STEP 2: Gal(L, L) IS A QUANTIC QUOTIENT of Ant(L, L) f Ant(L, L) j 0 (f)(a) := {b L S L : S = a and b f(s) s S} Fix(j 0 )=Gal(L,L) j(f) = {g Gal(L, L) f g} Then Fix(j) = Gal(L, L) g jf = j(g f) = g f THEOREM: (Gal(L, L), ) is a quantale June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 11

36 RELATION QUANTALES A, B complete lattices [Erné, Picado, preprint] Gal(A, B) F 1 A B G-ideals G 1 [Shmuely, 1974] June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 12

37 RELATION QUANTALES A, B complete lattices [Erné, Picado, preprint] f F 1 {(a, b) f(a) b} {b (a, b) R} G 1 R June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 12

38 RELATION QUANTALES A, B complete lattices F 1 Ant(A, B) D(A B) G 1 [Erné, Picado, preprint] Gal(A, B) F 1 G 1 A B G-ideals [Shmuely, 1974] June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 12

39 RELATION QUANTALES A, B complete lattices F 1 Ant(A, B) D(A B) G 1 Ant(A, B) (A B) l [Erné, Picado, preprint] left G-ideals Gal(A, B) F 1 G 1 A B G-ideals June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 12

40 RELATION QUANTALES A, B complete lattices f( Ï S) = Î f(s) S Z F 1 Ant(A, B) D(A B) G 1 Ant(A, B) (A B) l [Erné, Picado, preprint] left G-ideals Z-Ant(A, B) (A B) l Z Z-right left G-ideals Gal(A, B) A B G-ideals June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 12

41 RELATION QUANTALES A, B complete lattices Z = { } 0-Ant(A, B) (A B) l 0 F 1 Ant(A, B) D(A B) G 1 [Erné, Picado, preprint] 0-right left G-ideals f( Ï S) = Î f(s) S Z Z-Ant(A, B) (A B) l Z Z-right left G-ideals Gal(A, B) A B G-ideals June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 12

42 Ï Î RELATION QUANTALES [Erné, Picado, preprint] A, B complete lattices Ant(A, B) F 1 D(A B) G 1 Z = { } 0-Ant(A, B) (A B) l 0 0-right left G-ideals if Z Z-Ant(A, B) f( S) = f(s) (A B) l Z Z-right left G-ideals S Z Gal(A, B) A B G-ideals June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 12

43 RELATION QUANTALES A, B complete lattices [Erné, Picado, preprint] A 0 = A {0}, B 0 = B {0} L(A 0 B 0 ): (truncated) left G-ideals T (A 0 B 0 ): (truncated) G-ideals Z-T (A 0 B 0 ): (truncated) Z-right left G-ideals June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 12

44 RELATION QUANTALES A, B complete lattices F 1 Ant(A, B) D(A B) G 1 [Erné, Picado, preprint] 0-Ant(A, B) (A B) l 0 L(A 0 B 0 ) if Z Z-Ant(A, B) (A B) l Z Z-T (A 0 B 0 ) F 2 G 2 D(A 0 B 0 ) Gal(A, B) A B T (A 0 B 0 ) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 12

45 RELATION QUANTALES [Erné, Picado, preprint] R, S A 0 A 0 R S:= {(a, c) A 0 A 0 b A 0 : (a, b) R and (b, c) S} (the usual product relation) (D(A 0 A 0 ), ) is a quantale June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 13

46 RELATION QUANTALES [Erné, Picado, preprint] R, S A 0 A 0 R S:= {(a, c) A 0 A 0 b A 0 : (a, b) R and (b, c) S} (the usual product relation) (D(A 0 A 0 ), ) is a quantale L(R) := {(a, U 2 ) {a} U 2 R} closure op. T(R) := {(a, U 2 ) {a} U 2 R} {( U 1, b) U 1 {b} R} preclosure op. T(R) := {S T (A 0 B 0 ) R S} closure op. June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 13

47 RELATION QUANTALES [Erné, Picado, preprint] THEOREM: The following conditions on a complete lattice A are equivalent: (1) A is pseudocomplemented. (2) U A (0 a U b 0 : b a U).. (8) L is a quantic nucleus on the quantale (D(A 0 A 0 ), ). (9) T is a quantic nucleus on the quantale (D(A 0 A 0 ), ). (10) L(A 0 A 0 ) is a quantic quotient of the quantale (D(A 0 A 0 ), ). (11) T (A 0 A 0 ) is a quantic quotient of the quantale (D(A 0 A 0 ), ). June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 14

48 RELATION QUANTALES [Erné, Picado, preprint] THEOREM: The following conditions on a complete lattice A are equivalent: (1) A is Z-pseudocomplemented. (2) U Z (0 a U b 0 : b a U).. (6) T Z is a quantic nucleus on the quantale (D(A 0 A 0 ), ). (7) Z-T (A 0 A 0 ) is a quantic quotient of the quantale (D(A 0 A 0 ), ). June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 14

49 RELATION QUANTALES F 1 Ant(A, A) D(A A) G 1 0-Ant(A, A) (A A) l 0 Z-Ant(A, A) (A A) l Z F 2 G 2 [Erné, Picado, preprint] D(A 0 A 0 ) L(A 0 A 0 ) Z-T (A 0 A 0 ) Gal(A, A) A A T (A 0 A 0 ) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 14

50 RELATION QUANTALES F 1 Ant(A, A) D(A A) G 1 0-Ant(A, A) (A A) l 0 Z-Ant(A, A) (A A) l Z F 2 G 2 [Erné, Picado, preprint] (D(A 0 A 0 ), ) (D(A 0 A 0 ), L) (Z-T (A 0 A 0 ), Z) T Gal(A, A) A A (T (A 0 A 0 ), T) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 14

51 RELATION QUANTALES Ant(A, A) D(A A) G 1 (0-Ant(A, A), ) (A A) l 0 (Z-Ant(A, A), ) (A A) l Z F 1 F 2 G 2 [Erné, Picado, preprint] (D(A 0 A 0 ), ) (D(A 0 A 0 ), L) (Z-T (A 0 A 0 ), Z) T (Gal(A, A), ) A A (T (A 0 A 0 ), T) June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 14

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