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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. V, ‘ 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:

				
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