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Product a-frames and a related (almost) pre- apartness Douglas S. Bridges University of Canterbury, Christchurch, New Zealand The setting: Bishop-style constructive mathematics (BISH)— that is, math- ematics with intuitionistic logic and an appropriate set-theoretic foundation such as the Aczel-Rathjen-Myhill CST. The setting: Bishop-style constructive mathematics (BISH)— that is, math- ematics with intuitionistic logic and an appropriate set-theoretic foundation such as the Aczel-Rathjen-Myhill CST. The object of study: a habitive a-frame— that is, a complete lattice L in which ^ is in…nitely distributive over _ and there are a unary operation of complement, a unary habitation predicate hab, and a binary pre-apartness relation ./. Axioms for the complement: C1 x ^ x=0 C2 x ^ x=x C3 (x _ y ) = x^ y: Example: The complement of S R, given by S fx 2 R : 8s2S (x 6= s)g ; where x 6= s means jx sj > 0. Axioms for the habitation relation: H1 hab(x) ) : (x = 0) H2 (hab(x) & x 6 y ) ) hab(y ) H3 The join-existential property: For any family (xi)i2I of elements of L, if hab i2I i x , then there exists i 2 I such that hab(xi) _ H4 hab(1) Example: For a subset S of R, hab (S ) 9x (x 2 S ) : Axioms for the pre-apartness: A1 1 ./ 0 A2 x ./ y ) x 6 y A3 (x1 _ x2) ./ (y1 _ y2) , 8i;j xi ./ yj ; Example: For elements S; T of the lattice of subsets of a metric space X , S ./ T 9r>08x2S 8y2T ( (x; y ) > r) : The relation 6= is de…ned on L by x 6= y , (hab (x ^ y ) or hab ( x ^ y )) and has the standard properties of an inequality relation. Then x= fy 2 L : y 6= xg : _ The complement x should be distinguished from the apartness comple- ment, x fy 2 L : y ./ xg ; _ and the logical complement, :x fy 2 L : y ^ x = 0g : _ A mapping f : L ! M between a-frames is called a frame map if for each family (xi)i2I of elements of L; f ( xi ) ; 0 1 i2I i2I _ _ f@ xi A = and for all x 2 L; x 6= 0 if and only if f (x) 6= 0. The map is then order- preserving, so 6 f ( xi ) 0 1 i2I i2I ^ ^ f@ xi A for all families (xi)i2I of elements of L. Starting with habitive a-frames L and M; each carrying a pre-apartness, we de…ne a product frame for L and M (in that order) to be a frame P together with a mapping (x; y ) x y of L M into P and frame maps p1 : P ! L and p2 : P ! M that satisfy a number of axioms. We denote the least and greatest elements of the “factors” L and M by 0; 1 respectively, and their counterparts in P by 0; 1. Example: For the lattices L,M of subsets of metric spaces X; Y the product frame will be the lattice of all subsets of X Y . We take S T to be S T , and de…ne the projection maps p1,p2 as usual. Note that every element of that product is a union (join) of a family of elements of the form S T . In the abstract case, one of the axioms for the product of two a-frames is EP3 For each x 2 P there exists a family ((ui; vi))i2I of elements of L M such that x = ( ui v i ) : i2I W In the rest of this talk, L is an a-frame and P is a product of L and itself. For nonzero elements a; b of L we write (a b) 9c (b _ c = 1 & a ./ c) : In the rest of this talk, L is an a-frame and P is a product of L and itself. For nonzero elements a; b of L we write (a b) 9c (b _ c = 1 & a ./ c) : We de…ne Bw to be the set of all elements of P of the form m bi bi i=1 _ where there exist elements a1; : : : ; am of L such that ai bi for each i; and m ai = 1. i=1 _ In the rest of this talk, L is an a-frame and P is a product of L and itself. For nonzero elements a; b of L we write (a b) 9c (b _ c = 1 & a ./ c) : We de…ne Bw to be the set of all elements of P of the form m bi bi i=1 _ where there exist elements a1; : : : ; am of L such that ai bi for each i; and m ai = 1. i=1 _ Note that since 1 = 1 _ 0 = 1 _ 1 and 1 ./ 1; 1=1 1 2 Bw : Our aim is to explore Bw further, with particular attention to an associated binary relation that has almost all the properties of a pre-apartness on L, and that under certain circumstances can be shown to coincide with the original pre-apartness on L. An inhabited subset F of a habitive lattice is called a lattice …lter base if for all x; y 2 F; x 6= 0 and x ^ y 2 F: Recall that an atom of a lattice is an element a such that if 0 6= x 6 a; then x = a. Proposition 1. Bw is lattice …lter base in P; and x x 6 b for each atom x of L and each b 2 Bw : In the metric-space model, this proposition says that each set in Bw contains the diagonal f(x; x) : x 2 Xg of X X: We say that the inequality on L is zero-tight if 8x2L (: (x 6= 0) ) x = 0) : Note that this does not imply that the inequality is tight in the sense that 8x;y2L (: (x 6= y ) ) x = y ) : Lemma 2. Suppose that the inequality on L is zero-tight, and let x; y; b; c be elements of L such that b _ c = 1; 0 6= x 6 b, and x y 6 (b b) : Then y 6 c: Now de…ne a binary relation ./w on L by 8x;y2L x ./w y , 9b2Bw (x y6 b) : Anticipating that, under certain conditions, this relation will behave somewhat like a pre-apartness on L; for each x 2 L we de…ne wx = fz 2 L : z ./w xg : _ A pre-apartness ./ on L is said to be symmetric if 8x;y2L (x ./ y ) y ./ x) : Symmetry comes into play when, as we now do, we seek connections between ./ and ./w . Proposition 3. Suppose that 8x;y2L ((x 6= 0 ) x 6 y ) ) x 6 y ) ; Then (i) the binary relation ./w between subsets of L satis…es axioms A1–A3, and w y 6 y for each y 2 L; (ii) the statement 8x;y2L (x ./w y ) x ./ y ) (1) implies the following condition: 8x;y2L (x y = 0 ) x ./ y ) ; and (iii) if ./ is symmetric, then A1s implies (1). Note that the eccentric condition 8x;y2L ((x 6= 0 ) x 6 y ) ) x 6 y ) ; holds in the classical set model and implies the zero-tightness of the inequality on L: Corollary 4. Under the hypotheses of Proposition 3, for each atom x 2 L and each s 2 L; if x ./w s; then x ./ s: Lemma 5. If the pre-apartness on L is symmetric and if x ./ y; then x y6 ( x x) ^ ( y y) : Proof. By axiom A2 for a lattice pre-apartness, y6 y . It follows from Proposition 2.1 of Bridges (2008) that x y6 y y6 y y6 ( y y) : Likewise, by the symmetry of the pre-apartness, x y 6 ( x x) : Putting these two together, we obtain the desired conclusion. c The Efremoviµ condition EF 8x;y2L x ./ y ) 9e2L (x ./ :e ^ y ./ e) is the frame analogue of the strongest separation condition on a set-set apart- ness. Here are some of its useful consequences: 8x;y;z2L ((x ./ y & y 6 x) ) x ./ z ) 8x;y2L (x ./ y ) x ./ ::y ) 8x;y2L (x ./ y ) x ./ y) We say that our a-frame L is locally decomposable if 8x2L x= f y2L: x _ y = 1g : _ Local decomposability is an extremely important property in the set-set model, and has some important applications in the context of an a-frame. Proposition 6. Suppose that L is locally decomposable, that its apartness is symmetric and satis…es EF, and that if x 2 L is an atom and x 6 u; then x _ u = 1: Then for each atom x 2 L and each y 2 L; x ./ y ) x ./w y: Proof. Let x ./ y ; then x 6 y: By Proposition 4.7 of Bridges (2008), b=( x x) _ ( y y) belongs to Bw : Lemma 5 shows that x y6 b; whence x ./w y: Corollary 7. Under the hypotheses of Proposition 6, for each atom x 2 L and each y 2 L; x ./ y , x ./w y: Proof. The result follows from Proposition 6 and Corollary 4. It might seem that in several of the foregoing results the requirement of sym- metry is excessive. However, ./w has every appearance of being symmetric, and actually is so if we impose on ./ the following very natural additional condition: 8x;y;b2L (x y6 (b b) ) y x6 (b b)) : The hypotheses of Proposition 6 hold in the set model. They deserve further investigation in the a-frame context. In the classical set-set proximity theory, proximity is the negation of ./. More- c over, if the Efremoviµ condition holds, then ./w and ./ coincide. Since in our constructive a-frame theory, 8x;y2L x ./w y , 9b2Bw (x y6 b) ; it makes some sense to de…ne the proximity relation on L by (x; y ) 8b2Bw ((x y ) ^ b 6= 0) : The closure of x in L is x t 2 L : 8u2L (t ^ u 6= 0 ) x ^ u 6= 0) : _ The closure of x in L is x t 2 L : 8u2L (t ^ u 6= 0 ) x ^ u 6= 0) : _ Proposition 8. For each x 2 L, x6 ft 2 L : (t; x)g : _ As a partial converse we have Proposition 9. Suppose that L is locally decomposable, that ./ is symmetric and satis…es EF, and that if x 2 L is an atom and x 6 u, then x _ u = 1. Then for each x 2 L, ft 2 L : t is an atom and (t; x)g 6 x: _ Propositions 8 and 9 together give a weak analogue of the classical result that, in the set-set theory, the closure of A in the apartness topology is the set of points that are near A. Concluding remark: The constructive theory of apartness on lattices seems to ‡ow naturally, once the “right” axioms are chosen. The foregoing gives a taste of what has been done, and perhaps a hint that a lot more needs to, and can, be done. Here are the three extant papers on the subject: D.S. Bridges, ‘ , Product a-frames and proximity’ Math. Logic Quarterly 54(1), 12–25, 2008. ta A D.S. Bridges and L.S. Vî¸¼, ‘ constructive theory of apartness on lattices’, Scientiae Math. Japon. 69(2), 187–206, 2009. D.S. Bridges, ‘ , Product a-frames and a related (almost) pre-apartness’ sub- mitted to CCC 2009. Acknowledgement. This work was supported by Marsden Award UOC0502 from the Royal Society of New Zealand. dsb, CCA 2009, Köln, 15 July 2009 Appendix: Axioms for a product of a-frames: Let L; M be habitive a-frames. A product of L and M (n that order) comprises a habitive a-frame P, a mapping (x; y ) x y of L M into P, and frame maps p1 : P ! L; and p2 : P ! M, such that the following axioms hold. I Complement axioms: PC1 (x 1) = x 1 PC2 (1 x) = 1 x I Habitation axiom: PI 8s2P hab(s) , 9a2L9b2M (a b 6 s & hab(a) & hab(b)) : I Product element axioms: EP1 1 1=1 EP2 x 0=0=0 x EP3 For each x 2 P there exists a family ((ui; vi))i2I of elements of L M such that x = ( ui v i ) : i2I W I Join and meet axioms: 2 PJ1 (a1 _ a2) (b1 _ b2) = ai bj i;j=1 W PJ2 (a1 ^ a2) (b1 ^ b2) = (a1 b1) ^ (a2 b2) I Projection axioms: PP1 p1(x y ) 6 x and p2(x y ) 6 y ; if x 6= 0 and y 6= 0; then p1(x y ) = x and p2(x y ) = y PP2 p1(0) = 0 2 L and p2(0) = 0 2 M PP3 For each s 2 P; s 6 p1(s) p2(s) PP4 If p1(x) 6 p1(y ) or p2(x) 6 p2(y ); then x 6 y: