Projective Lattices. with applications to isotope maps and databases. Ralph Freese CLA La Rochelle

Transcription

1 Projective Lattices with applications to isotope maps and databases Ralph Freese CLA La Rochelle Ralph Freese () Projective Lattices Oct / 17

2 Projective Lattices A lattice L is projective in a variety V of lattices if whenever f : K L is an epimorphism, there is a homomorphism g : L K such that f (g(a)) = a for all a L. Ralph Freese () Projective Lattices Oct / 17

3 Projective Lattices A lattice L is projective in a variety V of lattices if whenever f : K L is an epimorphism, there is a homomorphism g : L K such that f (g(a)) = a for all a L. Note g is one-to-one and ρ = gf : K K is a retraction of K: ρ is an endomorphism of K and ρ 2 = ρ. Ralph Freese () Projective Lattices Oct / 17

4 Projective Lattices A lattice L is projective in a variety V of lattices if whenever f : K L is an epimorphism, there is a homomorphism g : L K such that f (g(a)) = a for all a L. Note g is one-to-one and ρ = gf : K K is a retraction of K: ρ is an endomorphism of K and ρ 2 = ρ. The sublattice ρ(k) = g(l) of K is isomorphic to L. g(l) is a retract of K. In a slight abuse of terminology, we will also say L is a retract of K. A lattice is projective in V if and only if it a retract of a free lattice F V (X) in V. Ralph Freese () Projective Lattices Oct / 17

5 Projective Lattices Theorem (1978) A lattice L is projective (in the class of all lattices) iff L satisfies Whitman s condition (W), L = D(L) = D d (L), L has the minimal join cover refinement property and its dual, and L is finitely separable. Ralph Freese () Projective Lattices Oct / 17

6 Isotone Sections Let f : K L be an epimorphism. An isotone (order preserving) map g : L K is an isotone section if f (g(a)) = a for a L. Ralph Freese () Projective Lattices Oct / 17

7 Isotone Sections Let f : K L be an epimorphism. An isotone (order preserving) map g : L K is an isotone section if f (g(a)) = a for a L. For which L is there an epimorphism f : FL(X) L with an isotone section? Ralph Freese () Projective Lattices Oct / 17

8 Isotone Sections Let f : K L be an epimorphism. An isotone (order preserving) map g : L K is an isotone section if f (g(a)) = a for a L. For which L is there an epimorphism f : FL(X) L with an isotone section? For which L does every epimorphism f : K L have an isotone section? Ralph Freese () Projective Lattices Oct / 17

9 Isotone Sections Let f : K L be an epimorphism. An isotone (order preserving) map g : L K is an isotone section if f (g(a)) = a for a L. For which L is there an epimorphism f : FL(X) L with an isotone section? For which L does every epimorphism f : K L have an isotone section? Which ordered sets P can be embedded into FL(X) for some X? Ralph Freese () Projective Lattices Oct / 17

10 Isotone Sections Let f : K L be an epimorphism. An isotone (order preserving) map g : L K is an isotone section if f (g(a)) = a for a L. For which L is there an epimorphism f : FL(X) L with an isotone section? For which L does every epimorphism f : K L have an isotone section? Which ordered sets P can be embedded into FL(X) for some X? Yes to all 3 if L (or P) is countable. Ralph Freese () Projective Lattices Oct / 17

11 A Theorem of Galvin and Jónsson Theorem Relatively free lattices are countable unions of antichains. So no free lattice contain an uncountable chain. Ralph Freese () Projective Lattices Oct / 17

12 A Theorem of Galvin and Jónsson Theorem Relatively free lattices are countable unions of antichains. So no free lattice contain an uncountable chain. Proof outline: Let G be the group of automorphisms of F V (X) induced from permutations of X that move only finitely many letters. Ralph Freese () Projective Lattices Oct / 17

13 A Theorem of Galvin and Jónsson Theorem Relatively free lattices are countable unions of antichains. So no free lattice contain an uncountable chain. Proof outline: Let G be the group of automorphisms of F V (X) induced from permutations of X that move only finitely many letters. G has only only countable many orbits. Ralph Freese () Projective Lattices Oct / 17

14 A Theorem of Galvin and Jónsson Theorem Relatively free lattices are countable unions of antichains. So no free lattice contain an uncountable chain. Proof outline: Let G be the group of automorphisms of F V (X) induced from permutations of X that move only finitely many letters. G has only only countable many orbits. Each orbit is an antichain. Ralph Freese () Projective Lattices Oct / 17

15 A Theorem of Galvin and Jónsson Theorem Relatively free lattices are countable unions of antichains. So no free lattice contain an uncountable chain. Proof outline: Let G be the group of automorphisms of F V (X) induced from permutations of X that move only finitely many letters. G has only only countable many orbits. Each orbit is an antichain. If a < σ(a), then applying σ Ralph Freese () Projective Lattices Oct / 17

16 A Theorem of Galvin and Jónsson Theorem Relatively free lattices are countable unions of antichains. So no free lattice contain an uncountable chain. Proof outline: Let G be the group of automorphisms of F V (X) induced from permutations of X that move only finitely many letters. G has only only countable many orbits. Each orbit is an antichain. If a < σ(a), then applying σ a < σ(a) < σ 2 (a) < < σ k (a) = a, Ralph Freese () Projective Lattices Oct / 17

17 A Theorem of Galvin and Jónsson Theorem Relatively free lattices are countable unions of antichains. So no free lattice contain an uncountable chain. Proof outline: Let G be the group of automorphisms of F V (X) induced from permutations of X that move only finitely many letters. G has only only countable many orbits. Each orbit is an antichain. If a < σ(a), then applying σ a < σ(a) < σ 2 (a) < < σ k (a) = a, where k is the order of σ. Ralph Freese () Projective Lattices Oct / 17

18 A Theorem of Galvin and Jónsson Theorem Relatively free lattices are countable unions of antichains. So no free lattice contain an uncountable chain. Proof outline: Let G be the group of automorphisms of F V (X) induced from permutations of X that move only finitely many letters. G has only only countable many orbits. Each orbit is an antichain. If a < σ(a), then applying σ a < σ(a) < σ 2 (a) < < σ k (a) = a, where k is the order of σ. This theorem is also true for free Boolean algebras. Ralph Freese () Projective Lattices Oct / 17

19 Finite Separability L is finitely separable if for each a L there are finite sets A(a) {x L : x a} and B(a) {x L : x a} such that a b = A(a) B(b) Ralph Freese () Projective Lattices Oct / 17

20 Finite Separability L is finitely separable if for each a L there are finite sets A(a) {x L : x a} and B(a) {x L : x a} such that (A for above and B for below.) a b = A(a) B(b) Ralph Freese () Projective Lattices Oct / 17

21 Finite Separability L is finitely separable if for each a L there are finite sets A(a) {x L : x a} and B(a) {x L : x a} such that (A for above and B for below.) a b = A(a) B(b) Every countable lattice is finitely separable. Ralph Freese () Projective Lattices Oct / 17

22 Finite Separability and Isotone Sections Theorem TFAE for a lattice L L is finitely separable. Every epimorphism K L has an isotone section. An epimorphism f : FL(X) L exists with an isotone section. Ralph Freese () Projective Lattices Oct / 17

23 An Interpolation Result w F V (X). There is a smallest subset var(w) of X with w in the sublattice generated by var(w). Ralph Freese () Projective Lattices Oct / 17

24 An Interpolation Result w F V (X). There is a smallest subset var(w) of X with w in the sublattice generated by var(w). Theorem If w u in F V (X), there is a v with w v u such that var(v) var(w) var(u) and r(v) min(r(w), r(u)). Ralph Freese () Projective Lattices Oct / 17

25 An Interpolation Result w F V (X). There is a smallest subset var(w) of X with w in the sublattice generated by var(w). Theorem If w u in F V (X), there is a v with w v u such that var(v) var(w) var(u) and r(v) min(r(w), r(u)). Corollary If L is a projective lattice in V then L is finitely separable. Ralph Freese () Projective Lattices Oct / 17

26 An Interpolation Result Corollary If L is a projective lattice in V then L is finitely separable. Proof. Finite separability is clearly preserved by retracts; thus it suffices to prove this theorem for L = F V (X). For a L, let A(a) = {w F V (X) : w a, var(w) var(a), and r(w) r(a)} and define B(a) dually. The result follows from the interpolation theorem. Ralph Freese () Projective Lattices Oct / 17

27 Projective Ordinal Sums Theorem Let V be a variety of lattices and let L = L 0 L 1, where L i V for i = 0, 1. Then L is projective in V if and only if both L 0 and L 1 are and one of the following hold: 1 L 0 has a greatest element. 2 L 1 has a least element. 3 L 0 has a countable cofinal chain and L 1 has a countable coinitial chain. Ralph Freese () Projective Lattices Oct / 17

28 Projective Ordinal Sums Corollary F V (X) F V (Y ) is projective in V iff X is finite, Y is finite, or both X and Y are countable. Ralph Freese () Projective Lattices Oct / 17

29 F V (P) for P a partially ordered set Theorem If P is a partially ordered set, then F V (P) (the V lattice freely generated by P subject to the order relations of P) is projective in V if and only if P is finitely separable. Ralph Freese () Projective Lattices Oct / 17

30 F V (P) for P a partially ordered set Theorem If P is a partially ordered set, then F V (P) (the V lattice freely generated by P subject to the order relations of P) is projective in V if and only if P is finitely separable. Example: P consists of two uncountable antichains A 0 and A 1 with x 0 < x 1 for x 0 A 0 and x 1 A 1. Ralph Freese () Projective Lattices Oct / 17

31 F V (P) for P a partially ordered set Theorem If P is a partially ordered set, then F V (P) (the V lattice freely generated by P subject to the order relations of P) is projective in V if and only if P is finitely separable. Example: P consists of two uncountable antichains A 0 and A 1 with x 0 < x 1 for x 0 A 0 and x 1 A 1. F V (P) = F V (A 0 ) F V (A 1 ) is not projective. Ralph Freese () Projective Lattices Oct / 17

32 F V (P) for P a partially ordered set Theorem If P is a partially ordered set, then F V (P) (the V lattice freely generated by P subject to the order relations of P) is projective in V if and only if P is finitely separable. Example: P consists of two uncountable antichains A 0 and A 1 with x 0 < x 1 for x 0 A 0 and x 1 A 1. F V (P) = F V (A 0 ) F V (A 1 ) is not projective. Let Q = P {r} with x 0 < r < x 1. Ralph Freese () Projective Lattices Oct / 17

33 F V (P) for P a partially ordered set Theorem If P is a partially ordered set, then F V (P) (the V lattice freely generated by P subject to the order relations of P) is projective in V if and only if P is finitely separable. Example: P consists of two uncountable antichains A 0 and A 1 with x 0 < x 1 for x 0 A 0 and x 1 A 1. F V (P) = F V (A 0 ) F V (A 1 ) is not projective. Let Q = P {r} with x 0 < r < x 1. Then F V (Q) = F V (A 0 ) 1 F V (A 1 ) is projective. Ralph Freese () Projective Lattices Oct / 17

34 F V (P) for P a partially ordered set Theorem If P is a partially ordered set, then F V (P) (the V lattice freely generated by P subject to the order relations of P) is projective in V if and only if P is finitely separable. Example: P consists of two uncountable antichains A 0 and A 1 with x 0 < x 1 for x 0 A 0 and x 1 A 1. F V (P) = F V (A 0 ) F V (A 1 ) is not projective. Let Q = P {r} with x 0 < r < x 1. Then F V (Q) = F V (A 0 ) 1 F V (A 1 ) is projective. So Q, and hence P, is embeddable into a free V lattice. Ralph Freese () Projective Lattices Oct / 17

35 Another Example P be the induced ordered set of atoms and coatrooms of the lattice of subsets of a uncountable set. Ralph Freese () Projective Lattices Oct / 17

36 Another Example P be the induced ordered set of atoms and coatrooms of the lattice of subsets of a uncountable set. P cannot be embedded into any (relatively) free lattice. Ralph Freese () Projective Lattices Oct / 17

37 Another Example P be the induced ordered set of atoms and coatrooms of the lattice of subsets of a uncountable set. P cannot be embedded into any (relatively) free lattice. Proof: like Galvin-Jónsson, Ralph Freese () Projective Lattices Oct / 17

38 Another Example P be the induced ordered set of atoms and coatrooms of the lattice of subsets of a uncountable set. P cannot be embedded into any (relatively) free lattice. Proof: like Galvin-Jónsson, but harder. Ralph Freese () Projective Lattices Oct / 17

39 The (join) dependency relation D For join irreducibles a b, a depends on b a D b p with a b p and a c p for c < b. Ralph Freese () Projective Lattices Oct / 17

40 The (join) dependency relation D For join irreducibles a b, a depends on b a D b p with a b p and a c p for c < b. Assume L is finite. Ralph Freese () Projective Lattices Oct / 17

41 The (join) dependency relation D For join irreducibles a b, a depends on b a D b p with a b p and a c p for c < b. Assume L is finite. (T refines S) For S, T L: T S if for all t T, there exists s S with t s. Ralph Freese () Projective Lattices Oct / 17

42 The (join) dependency relation D For join irreducibles a b, a depends on b a D b p with a b p and a c p for c < b. Assume L is finite. (T refines S) For S, T L: T S if for all t T, there exists s S with t s. S is a minimal join cover of a if a S and if a T and T S, then S T. Ralph Freese () Projective Lattices Oct / 17

43 The (join) dependency relation D For join irreducibles a b, a depends on b a D b p with a b p and a c p for c < b. Assume L is finite. (T refines S) For S, T L: T S if for all t T, there exists s S with t s. S is a minimal join cover of a if a S and if a T and T S, then S T. a D b iff b belongs to a minimal nontrivial join cover of a. Ralph Freese () Projective Lattices Oct / 17

44 The (join) dependency relation D For join irreducibles a b, a depends on b a D b p with a b p and a c p for c < b. Assume L is finite. (T refines S) For S, T L: T S if for all t T, there exists s S with t s. S is a minimal join cover of a if a S and if a T and T S, then S T. a D b iff b belongs to a minimal nontrivial join cover of a. D(L) = L if the D relation is acyclic. Ralph Freese () Projective Lattices Oct / 17

45 Applications of the dependency relation D These notions played a role in the study of Free lattices Projective lattices Finitely presented lattices Bounded homormorphisms Transferable lattices Congruence lattices of lattices Representation of finite lattices as congruence lattices of finite algebras (Pudlák and Tůma). Ordered direct bases in database theory. Ralph Freese () Projective Lattices Oct / 17

46 The dependency relation in databases This is the work of K. Adaricheva, J. B. Nation, R. Rand, "Ordered direct implicational basis of a finite closure system." Ralph Freese () Projective Lattices Oct / 17

47 The dependency relation in databases This is the work of K. Adaricheva, J. B. Nation, R. Rand, "Ordered direct implicational basis of a finite closure system." Let S, φ be a reduced closure system. (φ(x) = φ(y) = x = y) Ralph Freese () Projective Lattices Oct / 17

48 The dependency relation in databases This is the work of K. Adaricheva, J. B. Nation, R. Rand, "Ordered direct implicational basis of a finite closure system." Let S, φ be a reduced closure system. (φ(x) = φ(y) = x = y) X Y if x X implies there is y Y with x φ(y). Ralph Freese () Projective Lattices Oct / 17

49 The dependency relation in databases This is the work of K. Adaricheva, J. B. Nation, R. Rand, "Ordered direct implicational basis of a finite closure system." Let S, φ be a reduced closure system. (φ(x) = φ(y) = x = y) X Y if x X implies there is y Y with x φ(y). x Y (Y covers x): x φ(y ) and x φ(y) for all y Y. Ralph Freese () Projective Lattices Oct / 17

50 The dependency relation in databases This is the work of K. Adaricheva, J. B. Nation, R. Rand, "Ordered direct implicational basis of a finite closure system." Let S, φ be a reduced closure system. (φ(x) = φ(y) = x = y) X Y if x X implies there is y Y with x φ(y). x Y (Y covers x): x φ(y ) and x φ(y) for all y Y. Y a minimal cover of x: if x Z, Z Y then Y Z. Ralph Freese () Projective Lattices Oct / 17

51 The dependency relation in databases This is the work of K. Adaricheva, J. B. Nation, R. Rand, "Ordered direct implicational basis of a finite closure system." Let S, φ be a reduced closure system. (φ(x) = φ(y) = x = y) X Y if x X implies there is y Y with x φ(y). x Y (Y covers x): x φ(y ) and x φ(y) for all y Y. Y a minimal cover of x: if x Z, Z Y then Y Z. The D-basis of S, φ, Σ D, is the union of {y x : y x φ(y)} and (1) {X x : X a miminal cover of x}. (2) Ralph Freese () Projective Lattices Oct / 17

52 The dependency relation in databases This is the work of K. Adaricheva, J. B. Nation, R. Rand, "Ordered direct implicational basis of a finite closure system." Let S, φ be a reduced closure system. (φ(x) = φ(y) = x = y) X Y if x X implies there is y Y with x φ(y). x Y (Y covers x): x φ(y ) and x φ(y) for all y Y. Y a minimal cover of x: if x Z, Z Y then Y Z. The D-basis of S, φ, Σ D, is the union of {y x : y x φ(y)} and (1) {X x : X a miminal cover of x}. (2) Ordered D-basis is an ordering of Σ D with (1) before (2). Ralph Freese () Projective Lattices Oct / 17

53 The dependency relation in databases The D-basis of S, φ, Σ D, is the union of {y x : y x φ(y)} and (1) {X x : X a miminal cover of x}. (2) Ordered D-basis is an ordering of Σ D with (1) before (2). Ralph Freese () Projective Lattices Oct / 17

54 The dependency relation in databases The D-basis of S, φ, Σ D, is the union of {y x : y x φ(y)} and (1) {X x : X a miminal cover of x}. (2) Ordered D-basis is an ordering of Σ D with (1) before (2). If Σ D is ordered {s 1, s 2,..., s n }, then φ(x) = X n where X 0 = X and s k+1 = A b Ralph Freese () Projective Lattices Oct / 17

55 The dependency relation in databases The D-basis of S, φ, Σ D, is the union of {y x : y x φ(y)} and (1) {X x : X a miminal cover of x}. (2) Ordered D-basis is an ordering of Σ D with (1) before (2). If Σ D is ordered {s 1, s 2,..., s n }, then φ(x) = X n where X 0 = X and s k+1 = A b { X k {b} if A X k X k+1 = otherwise X k Ralph Freese () Projective Lattices Oct / 17

56 The dependency relation in databases The D-basis of S, φ, Σ D, is the union of {y x : y x φ(y)} and (1) {X x : X a miminal cover of x}. (2) Ordered D-basis is an ordering of Σ D with (1) before (2). If Σ D is ordered {s 1, s 2,..., s n }, then φ(x) = X n where X 0 = X and s k+1 = A b { X k {b} if A X k X k+1 = otherwise X k So only one pass through Σ D is needed to fine the closure. Ralph Freese () Projective Lattices Oct / 17

57 K. Adaricheva, J.B. Nation, and R. Rand. Ordered direct implicational basis of a finite closure system. Discrete Applied Math., 161: , K. Bertet and B. Monjardet. The multiple facets of the canonical direct unit implicational basis. Theoret. Comput. Sci., 411(22-24): , Ralph Freese and J. B. Nation. Projective lattices. Pacific J. Math., 75:93 106, P. Pudlák and J. Tuma. Yeast graphs and fermentation of algebraic lattices. In Lattice theory, pages North-Holland Publishing Co., Amsterdam, Proceedings of the Colloquium held in Szeged,1974. Colloquia Mathematica Societatis János Bolyai, vol. 14. Ralph Freese () Projective Lattices Oct / 17