The polarity between approximation and distribution
by
Marcel Erné
Institut für Mathematik, Universität Hannover

The words ``approximation'' and ``distribution'' have here a rather wide meaning that is perhaps a bit vague from the perspective of applied mathematics. However, in theoretical computer science and other mathematically founded disciplines involving ordered structures, it is a custom to call certain transitive relations (modelling hierarchies of information, representation, computation etc.) approximating if each object is the least upper bound of all elements preceding it. Surprisingly, such an approximation property often entails the validity of certain (infinite) distributive laws. To record two typical examples, let us recall that a complete lattice is completely distributive iff the relation \tl, defined by x \tl y iff y <= \Ve W \implies x <= w \some w in W, has the approximation property. Similarly, arbitrary meets distribute over directed joins iff the celebrated way-below relation has the approximation property. Perhaps even more interesting is the fact that in an up-complete poset (DCPO) each element is the join of a directed set of way-below elements iff the Scott topology is completely distributive, and a similar characterization holds for lattices with completely distributive ideal lattices. In fact, a much more general theory has been developed for so-called \Z-distributive and \Z-continuous lattices and posets, encompassing the previous (and other) examples.

Often, one has to work with ordered structures having weaker local approximation properties. For example, the ideal lattice of a join-semilattice S is known to be distributive (and even a frame) iff for all finite Z\inc S and all x <= \/ Z, there is a finite Z'\inc \da Z with x= \/ Z'. Or, the Scott topology of a complete lattice is a dual frame iff the respective property holds for directed instead of finite subsets. Again, these are special instances of a general phenomenon observed for join-ideals with respect to general subset systems. An important recent observation is that all T₀ closure systems that are frames arise in that way as \Z-join ideal systems from so-called locally approximating systems \Z. Similarly, all completely distributive T₀ closure systems come from strongly approximating systems in that manner.

The appropriate framework for these investigations is that of a polarity in the sense of Birkhoff, arising from a ``canonical'' relation associated with a fixed closure operation. In the aforementioned cases, this relation is defined in terms of the cut operator of the given poset.

More generally, every closure operation \de on a frame (locale) L with join-dense range X gives rise to a polarity induced by the relation
\rho = { (y, z): z <= y \imp \de z <= y}.
Using that polarity, one finds that the sublocales of L containing X are precisely the systems of Z-\de-closed elements y (satisfying \de z <= y whenever z in Z and z <= y) for locally approximating subsets of L, i.e. subsets Z contains or equal X such that for all x in X and z in Z with x <= \de z, it follows that x\we z in Z and x=\de (x\we z). Suitable restriction yields a dual isomorphism between the collection of all locally approximating sets that are saturated with respect to the above polarity and the complete lattice of all sublocales containing X. Similar results hold for complete distributivity. These polarity theorems have manifold applications in order theory and related fields, for example in domain theory and pointfree topology.

Date received: February 1, 2001