The model companion of the class of pseudo-complemented semilattices is finitely axiomatizable

Transcription

1 The model companion of the class of pseudo-complemented semilattices is finitely axiomatizable (joint work with Regula Rupp and Jürg Schmid) Joel Adler Institut S1 Pädagogische Hochschule Bern AMS Spring Western Sectional Meeting March 3, 2012

2 Contents

3 Basic notions Contents

4 Contents Basic notions Algebraic and existential closedness

5 Contents Basic notions Algebraic and existential closedness Pseudo-complemented semilattices

6 Contents Basic notions Algebraic and existential closedness Pseudo-complemented semilattices Point of departure

7 Contents Basic notions Algebraic and existential closedness Pseudo-complemented semilattices Point of departure Axiomatizing P CS ac

8 Contents Basic notions Algebraic and existential closedness Pseudo-complemented semilattices Point of departure Axiomatizing P CS ac Axiomatizing P CS ec

9 Basic notions Algebraic and existential closedness

10 Basic notions Algebraic and existential closedness Let K be a class of (universal) algebras of the same (finitary) type, and L K a (first-order) language suitable for K. For A K, let L K,A be L extended with an individual constant for each element a A.

11 Basic notions Algebraic and existential closedness Let K be a class of (universal) algebras of the same (finitary) type, and L K a (first-order) language suitable for K. For A K, let L K,A be L extended with an individual constant for each element a A. Definition: (i) An algebra A K is called algebraically closed in K - for short: ac - iff for every finite set φ 1,..., φ n of positive 1 -sentences from L K,A the following holds: Whenever n 1 φ i is satisfied in some extension A A K, then it is satisfied already in A.

12 Basic notions Algebraic and existential closedness Let K be a class of (universal) algebras of the same (finitary) type, and L K a (first-order) language suitable for K. For A K, let L K,A be L extended with an individual constant for each element a A. Definition: (i) An algebra A K is called algebraically closed in K - for short: ac - iff for every finite set φ 1,..., φ n of positive 1 -sentences from L K,A the following holds: Whenever n 1 φ i is satisfied in some extension A A K, then it is satisfied already in A. (ii) A K is called existentially closed - for short: ec - iff (the same as sub (i), with positive removed).

13 Basic notions Algebraic and existential closedness for the working mathematician A is ac in K iff every finite set of equations with parameters from A which is solvable in some extension A A K already has a solution in A, and it is ec in K iff the same holds for every such finite set of equations and negated equations.

14 Basic notions Algebraic and existential closedness for the working mathematician A is ac in K iff every finite set of equations with parameters from A which is solvable in some extension A A K already has a solution in A, and it is ec in K iff the same holds for every such finite set of equations and negated equations. Explicitly, we are dealing with finite sets of formulae of type s(x 1,..., x m, a 1,..., a n ) = t(x 1,..., x m, a 1,..., a n ) resp. s(x 1,..., x m, a 1,..., a n ) t(x 1,..., x m, a 1,..., a n ), where s and t are L-terms and a 1,..., a n A.

15 Basic notions Algebraic versus existential closedness

16 Basic notions Algebraic versus existential closedness The two notions can coincide as in the class of fields: If K is a field and p ( x ) and q ( x ) are polynomials over K, then the satisifiability of the inequality p ( x ) q ( x ) is equivalent to the satisfiability of the equation where x = (x 1,..., x n ). x (p ( x ) q ( x )) = 1,

17 Basic notions Algebraic versus existential closedness The two notions can coincide as in the class of fields: If K is a field and p ( x ) and q ( x ) are polynomials over K, then the satisifiability of the inequality p ( x ) q ( x ) is equivalent to the satisfiability of the equation where x = (x 1,..., x n ). x (p ( x ) q ( x )) = 1, This is not the general situation: In the class of boolean algebras every boolean algebra is algebraically closed whereas boolean algebra is existentially closed if and only if it is atomfree.

18 Basic notions Algebraic versus existential closedness The two notions can coincide as in the class of fields: If K is a field and p ( x ) and q ( x ) are polynomials over K, then the satisifiability of the inequality p ( x ) q ( x ) is equivalent to the satisfiability of the equation where x = (x 1,..., x n ). x (p ( x ) q ( x )) = 1, This is not the general situation: In the class of boolean algebras every boolean algebra is algebraically closed whereas boolean algebra is existentially closed if and only if it is atomfree. An abelian group is algebraically closed if and only if it is divisible, whereas it is existentially closed if and only if it is divisible and contains an infinite direct sum of copies of Q/Z (as a module).

19 The classes K ac and K ec Basic notions

20 The classes K ac and K ec Basic notions For a class K of algebras K ac and K ec denote the subclasses of the algebraically and existentially closed members of K respectively.

21 The classes K ac and K ec Basic notions For a class K of algebras K ac and K ec denote the subclasses of the algebraically and existentially closed members of K respectively. If a class of algebras K is (finitely) axiomatizable (with first-order sentences) the natural question arises whether the subclasses K ac and K ec are (finitely) axiomatizable.

22 The classes K ac and K ec Basic notions For a class K of algebras K ac and K ec denote the subclasses of the algebraically and existentially closed members of K respectively. If a class of algebras K is (finitely) axiomatizable (with first-order sentences) the natural question arises whether the subclasses K ac and K ec are (finitely) axiomatizable. A semantic characterization of finite axiomatizability of a class of L-structures:

23 The classes K ac and K ec Basic notions For a class K of algebras K ac and K ec denote the subclasses of the algebraically and existentially closed members of K respectively. If a class of algebras K is (finitely) axiomatizable (with first-order sentences) the natural question arises whether the subclasses K ac and K ec are (finitely) axiomatizable. A semantic characterization of finite axiomatizability of a class of L-structures: Theoreom: An finitely axiomatizable class of L-structures is finitely axiomatizable iff both the class itself as well as its complementary class are closed under elementary equivalence and ultraproducts.

24 The classes K ac and K ec Basic notions If BA is the class of boolean algebras, then adding the sentence ( x)( y)(0 < x 0 < y < x) to the axioms of boolean algebras yields a finite axiomatization of BA ec.

25 The classes K ac and K ec Basic notions If BA is the class of boolean algebras, then adding the sentence ( x)( y)(0 < x 0 < y < x) to the axioms of boolean algebras yields a finite axiomatization of BA ec. If GA is the class of abelian groups, then adding the infinite set of sentence {( x)( y)(x = ny) n N} to the axioms of abelian groups yields an infinite axiomatization of GA ac. There is no finite axiomatization of GA ac

26 The classes K ac and K ec Basic notions If BA is the class of boolean algebras, then adding the sentence ( x)( y)(0 < x 0 < y < x) to the axioms of boolean algebras yields a finite axiomatization of BA ec. If GA is the class of abelian groups, then adding the infinite set of sentence {( x)( y)(x = ny) n N} to the axioms of abelian groups yields an infinite axiomatization of GA ac. There is no finite axiomatization of GA ac Let L a be the first-order language without relation, function and constant symbols and K be the class of all L-models (of the empty theory). Then K is simply the class of sets. We obtain K ac = K and K ec = {x K x is infinite}. Thus K ac is finitely axiomatizable and K ec is not.

27 Basic notions The model companion of a class of L-structures

28 Basic notions The model companion of a class of L-structures We will not discuss this model theoretic notion here. For our purposes it suffices to notice that under very general assumptions on a class K of algebras the model companion of K coincides with K ec.

29 Basic notions The model companion of a class of L-structures We will not discuss this model theoretic notion here. For our purposes it suffices to notice that under very general assumptions on a class K of algebras the model companion of K coincides with K ec. The class P CS of pseudo-complemented semilattices satisfies these assumptions.

30 Basic notions Pseudo-complemented semilattices: definition

31 Basic notions Pseudo-complemented semilattices: definition Definition: A pseudo-complemented semilattice (for short: p-semilattice) (P ;,, 0, 1) is a meet-semilattice (P ; ) with least element 0 and top element 1, equipped with an unary operation a a such that for all x P, x a = 0 iff x a. With P CS we denote the class of all p-semilattices.

32 Basic notions Pseudo-complemented semilattices: definition Definition: A pseudo-complemented semilattice (for short: p-semilattice) (P ;,, 0, 1) is a meet-semilattice (P ; ) with least element 0 and top element 1, equipped with an unary operation a a such that for all x P, x a = 0 iff x a. With P CS we denote the class of all p-semilattices. Theorem (Frink 1962) P CS is a variety.

33 Basic notions Pseudo-complemented semilattices: definition Definition: A pseudo-complemented semilattice (for short: p-semilattice) (P ;,, 0, 1) is a meet-semilattice (P ; ) with least element 0 and top element 1, equipped with an unary operation a a such that for all x P, x a = 0 iff x a. With P CS we denote the class of all p-semilattices. Theorem (Frink 1962) P CS is a variety. Definition: A distributive pseudo-complemented lattice (for short: p-algebra) (L;,,, 0, 1) is a distributive lattice (L;, ) with least element 0 and top element 1, equipped with an unary operation a a such that for all x L, x a = 0 iff x a. Let PALG be the class of all p-algebras.

34 Basic notions Pseudo-complemented semilattices: examples

35 Basic notions Pseudo-complemented semilattices: examples Every boolean algebra B considered as a (B;,, 0, 1)-reduct is a pcs. 2, F n and A denote the two-element boolean algebra, the n-atom boolean algebra and the countable atomless boolean algebra, respectively.

36 Basic notions Pseudo-complemented semilattices: examples Every boolean algebra B considered as a (B;,, 0, 1)-reduct is a pcs. 2, F n and A denote the two-element boolean algebra, the n-atom boolean algebra and the countable atomless boolean algebra, respectively. For an arbitrary boolean algebra B the ordered structure B obtained from B by adding a new top element is a pcs. 3 denotes three element chain which is obtained from 2 by adding a new top element.

37 Basic notions Pseudo-complemented semilattices: special elements

38 Basic notions Pseudo-complemented semilattices: special elements Let x A P CS. x is dense iff x = 0. D(A) = {x A : x dense} = the dense filter of A.

39 Basic notions Pseudo-complemented semilattices: special elements Let x A P CS. x is dense iff x = 0. D(A) = {x A : x dense} = the dense filter of A. x is skeletal iff x = x. Sk(A) = {x A : x skeletal} = the skeleton of A.

40 Basic notions Pseudo-complemented semilattices: some facts

41 Basic notions Pseudo-complemented semilattices: some facts If P = P ;,, 0, 1 is a pcs, then Sk(P );,,, 0, 1 putting a b := (a b ) is a boolean algebra.

42 Basic notions Pseudo-complemented semilattices: some facts If P = P ;,, 0, 1 is a pcs, then Sk(P );,,, 0, 1 putting a b := (a b ) is a boolean algebra. B, where B is any boolean algebra, are exactly the subdirectly irreducible members of P CS.

43 Basic notions Pseudo-complemented semilattices: some facts If P = P ;,, 0, 1 is a pcs, then Sk(P );,,, 0, 1 putting a b := (a b ) is a boolean algebra. B, where B is any boolean algebra, are exactly the subdirectly irreducible members of P CS. The three-element chain 3 generates the variety P CS, that is P CS = HSP(3). The subclass BA = HSP(2) of boolean algebras is the only non-trivial subvariety.

44 Point of departure In their 1986 paper Finite axiomatizations for existentially closed posets and semilattices, Order 3(1986), , Michael Albert and Stanley Burris asked the following question:

45 Point of departure In their 1986 paper Finite axiomatizations for existentially closed posets and semilattices, Order 3(1986), , Michael Albert and Stanley Burris asked the following question: Does the class of pseudo-complemented semilattices have a finitely axiomatizable model companion?

46 Point of departure In their 1986 paper Finite axiomatizations for existentially closed posets and semilattices, Order 3(1986), , Michael Albert and Stanley Burris asked the following question: Does the class of pseudo-complemented semilattices have a finitely axiomatizable model companion? which is the same as asking

47 Point of departure In their 1986 paper Finite axiomatizations for existentially closed posets and semilattices, Order 3(1986), , Michael Albert and Stanley Burris asked the following question: Does the class of pseudo-complemented semilattices have a finitely axiomatizable model companion? which is the same as asking Is P CS ec finitely axiomatizable?

48 Point of departure In their 1986 paper Finite axiomatizations for existentially closed posets and semilattices, Order 3(1986), , Michael Albert and Stanley Burris asked the following question: Does the class of pseudo-complemented semilattices have a finitely axiomatizable model companion? which is the same as asking Is P CS ec finitely axiomatizable? In the above paper the authors answered the same question for the class of posets and the class of semilattices in the affirmative.

49 Point of departure For the following classes K of algebras with a pseudo-complement the above question was answered in the affirmative:

50 Point of departure For the following classes K of algebras with a pseudo-complement the above question was answered in the affirmative: K the class of pseudo-complemented distributive lattices (J. Schmid, 1982). The same author showed that this also holds for all Lee classes B n which are subvarieties of K.

51 Point of departure For the following classes K of algebras with a pseudo-complement the above question was answered in the affirmative: K the class of pseudo-complemented distributive lattices (J. Schmid, 1982). The same author showed that this also holds for all Lee classes B n which are subvarieties of K. K the class of the so-called Stone semilattices alias the class ISP(3) P CS, which is a quasi-subvariety of P CS (Gerber,Schmid,1991).

52 A finite axiomatization of P CS ac A semantic characterization of P CS ac

53 A finite axiomatization of P CS ac A semantic characterization of P CS ac The following theorem is crucial for finding a finite axiomatization of P CS ac as well as P CS ec. (J. Schmid, 1985) A p-semilattice P is algebraically closed iff for any finite subalgebra F P there exists r, s N and a p-semilattice F isomorphic to 2 r Âs such that F F P.

54 The list of axioms A finite axiomatization of P CS ac

55 The list of axioms A finite axiomatization of P CS ac (AC1) ( a, b, c P )( x, y P )(c a b x a, y b & x y = c)

56 The list of axioms A finite axiomatization of P CS ac (AC1) ( a, b, c P )( x, y P )(c a b x a, y b & x y = c) (AC2) ( a, b, c, t P )( x P )(a = b = c = 0, c < b < a, t c < t b < t a c < x < a, x b = c, t c < t x < t a)

57 The list of axioms A finite axiomatization of P CS ac (AC1) ( a, b, c P )( x, y P )(c a b x a, y b & x y = c) (AC2) ( a, b, c, t P )( x P )(a = b = c = 0, c < b < a, t c < t b < t a c < x < a, x b = c, t c < t x < t a) (AC3) ( d, d m D(P ), f, f m, x P, k Sk(P ))( z x Sk(P ))(d d m, f d m, f m d, f m d m, k d, k f d, x d m k z x d, z x f d, z x f m d m & (z x x) d m ).

58 The list of axioms A finite axiomatization of P CS ac (AC1) ( a, b, c P )( x, y P )(c a b x a, y b & x y = c) (AC2) ( a, b, c, t P )( x P )(a = b = c = 0, c < b < a, t c < t b < t a c < x < a, x b = c, t c < t x < t a) (AC3) ( d, d m D(P ), f, f m, x P, k Sk(P ))( z x Sk(P ))(d d m, f d m, f m d, f m d m, k d, k f d, x d m k z x d, z x f d, z x f m d m & (z x x) d m ). (AC4) ( d D(P ), b 1 Sk(P )( b 2 Sk(P ))(b 1 < d < 1 b 1 < b 2 < d & b 1 b 2 < d)

59 The theorem A finite axiomatization of P CS ac

60 The theorem A finite axiomatization of P CS ac Finally, a finite axiomatization of P CS ac : (,R. Rupp,J. Schmid,2012) Let A be a finite list of first-order sentences axiomatizing P CS. Then the finite set A {(AC1),(AC2),(AC3),(AC4)} axiomatizes P CS ac.

61 Proof sketch A finite axiomatization of P CS ac

62 Proof sketch A finite axiomatization of P CS ac Let P be a p-semilattice satisfying (AC1), (AC2), (AC3) and (AC4) and F 0 P, F 0 < ω. By repeatedly applying axiom (ACi), i = 1,..., 4, we construct a chain F 0 F 1 F 2 F 3 F 4 P s.t.

63 Proof sketch A finite axiomatization of P CS ac Let P be a p-semilattice satisfying (AC1), (AC2), (AC3) and (AC4) and F 0 P, F 0 < ω. By repeatedly applying axiom (ACi), i = 1,..., 4, we construct a chain F 0 F 1 F 2 F 3 F 4 P s.t. F 1 is finite and distributive,

64 Proof sketch A finite axiomatization of P CS ac Let P be a p-semilattice satisfying (AC1), (AC2), (AC3) and (AC4) and F 0 P, F 0 < ω. By repeatedly applying axiom (ACi), i = 1,..., 4, we construct a chain F 0 F 1 F 2 F 3 F 4 P s.t. F 1 is finite and distributive, F 2 is finite, distributive and there is s N s.t. D(F 2 ) = {e, 1} s,

65 Proof sketch A finite axiomatization of P CS ac Let P be a p-semilattice satisfying (AC1), (AC2), (AC3) and (AC4) and F 0 P, F 0 < ω. By repeatedly applying axiom (ACi), i = 1,..., 4, we construct a chain F 0 F 1 F 2 F 3 F 4 P s.t. F 1 is finite and distributive, F 2 is finite, distributive and there is s N s.t. D(F 2 ) = {e, 1} s, there is r N s.t. F 3 = 2 r s i=1 F f(i), f(i) N, i = 1,..., s,

66 Proof sketch A finite axiomatization of P CS ac Let P be a p-semilattice satisfying (AC1), (AC2), (AC3) and (AC4) and F 0 P, F 0 < ω. By repeatedly applying axiom (ACi), i = 1,..., 4, we construct a chain F 0 F 1 F 2 F 3 F 4 P s.t. F 1 is finite and distributive, F 2 is finite, distributive and there is s N s.t. D(F 2 ) = {e, 1} s, there is r N s.t. F 3 = 2 r s i=1 F f(i), f(i) N, i = 1,..., s, F 4 = 2 r (Â) s.

67 A finite axiomatization of P CS ec Existential closedness revisited

68 A finite axiomatization of P CS ec Existential closedness revisited Every finite system of equations and inequalities with coefficients a 1,..., a m P corresponds to a quantifier-free formula ϕ( x, a ). This yields:

69 A finite axiomatization of P CS ec Existential closedness revisited Every finite system of equations and inequalities with coefficients a 1,..., a m P corresponds to a quantifier-free formula ϕ( x, a ). This yields:

70 A finite axiomatization of P CS ec Existential closedness revisited Every finite system of equations and inequalities with coefficients a 1,..., a m P corresponds to a quantifier-free formula ϕ( x, a ). This yields: P is ec iff for any extension Q of P and any quantifier-free formula ϕ from L(P ) the following holds: If Q = ( x 1,..., x n )ϕ(x 1,..., x n, a 1,..., a m ) then there are u 1,..., u n P such that P = ϕ( u, a )

71 A finite axiomatization of P CS ec Existential closedness revisited Every finite system of equations and inequalities with coefficients a 1,..., a m P corresponds to a quantifier-free formula ϕ( x, a ). This yields: P is ec iff for any extension Q of P and any quantifier-free formula ϕ from L(P ) the following holds: If Q = ( x 1,..., x n )ϕ(x 1,..., x n, a 1,..., a m ) then there are u 1,..., u n P such that P = ϕ( u, a ) P is ec iff for any extension Q of P, arbitrary S := {a 1,..., a m } P and V := {v 1,..., v n } Q there is U := {u 1,..., u n } P and an isomorphism f : S U S V with f S = id S.

72 A finite axiomatization of P CS ec Assumptions on the set of coefficients and the solution set

73 A finite axiomatization of P CS ec Assumptions on the set of coefficients and the solution set S P and the set T := S V Q the following can be assumed without loss of generality:

74 A finite axiomatization of P CS ec Assumptions on the set of coefficients and the solution set S P and the set T := S V Q the following can be assumed without loss of generality: with F 0 := 2 and S = F t (t N)

75 A finite axiomatization of P CS ec Assumptions on the set of coefficients and the solution set S P and the set T := S V Q the following can be assumed without loss of generality: with F 0 := 2 and S = F t (t N) T = 2 p F q u (p, q, u N)

76 The list of axioms A finite axiomatization of P CS ec

77 The list of axioms A finite axiomatization of P CS ec

78 The list of axioms A finite axiomatization of P CS ec (EC1) ( b 1, b 2 )( b 3 )((Sk(b 1 ) & Sk(b 2 ) & b 1 < b 2 ) (Sk(b 3 ) & b 1 < b 3 < b 2 ))

79 The list of axioms A finite axiomatization of P CS ec (EC1) ( b 1, b 2 )( b 3 )((Sk(b 1 ) & Sk(b 2 ) & b 1 < b 2 ) (Sk(b 3 ) & b 1 < b 3 < b 2 )) (EC2) ( b 1, d)( b 2 )((Sk(b 1 ) & D(d) & b 1 < d & b 1 d) (Sk(b 2 ) & b 1 < b 2 d & b 2 < 1 & b 1 b 2 < d & b 1 b 2 d))

80 The list of axioms A finite axiomatization of P CS ec (EC1) ( b 1, b 2 )( b 3 )((Sk(b 1 ) & Sk(b 2 ) & b 1 < b 2 ) (Sk(b 3 ) & b 1 < b 3 < b 2 )) (EC2) ( b 1, d)( b 2 )((Sk(b 1 ) & D(d) & b 1 < d & b 1 d) (Sk(b 2 ) & b 1 < b 2 d & b 2 < 1 & b 1 b 2 < d & b 1 b 2 d)) (EC3) ( d)(d(d) & d < 1)

81 The list of axioms A finite axiomatization of P CS ec (EC1) ( b 1, b 2 )( b 3 )((Sk(b 1 ) & Sk(b 2 ) & b 1 < b 2 ) (Sk(b 3 ) & b 1 < b 3 < b 2 )) (EC2) ( b 1, d)( b 2 )((Sk(b 1 ) & D(d) & b 1 < d & b 1 d) (Sk(b 2 ) & b 1 < b 2 d & b 2 < 1 & b 1 b 2 < d & b 1 b 2 d)) (EC3) ( d)(d(d) & d < 1) (EC4) ( d 1, d 2 )( d 3 )((D(d 1 ) & d 1 < d 2 ) (d 1 < d 3 < d 2 ))

82 The list of axioms A finite axiomatization of P CS ec (EC1) ( b 1, b 2 )( b 3 )((Sk(b 1 ) & Sk(b 2 ) & b 1 < b 2 ) (Sk(b 3 ) & b 1 < b 3 < b 2 )) (EC2) ( b 1, d)( b 2 )((Sk(b 1 ) & D(d) & b 1 < d & b 1 d) (Sk(b 2 ) & b 1 < b 2 d & b 2 < 1 & b 1 b 2 < d & b 1 b 2 d)) (EC3) ( d)(d(d) & d < 1) (EC4) ( d 1, d 2 )( d 3 )((D(d 1 ) & d 1 < d 2 ) (d 1 < d 3 < d 2 )) (EC5) ( b, d 1 )( d 2 )((D(d 1 ) & Sk(b) & 0 < b < d 1 ) (D(d 2 ) & d 2 < d 1 & b d 2 & d 1 b = d 2 b ))

83 The theorem A finite axiomatization of P CS ec

84 The theorem A finite axiomatization of P CS ec At last, a finite axiomatization of P CS ec : (,2012) Let A be a finite list of first-order sentences axiomatizing P CS. Then the finite set A {(AC1),...,(AC4),(EC1),...,(EC5)} axiomatizes P CS ec.