On Vaughan Pratt's crossword problem

Abstract

Vaughan Pratt has introduced objects consisting of pairs $(A,W)$ where $A$ is a set and $W$ a set of subsets of $A,$ such that (i) $W$ contains $\emptyset$ and $A,$ (ii) if $C$ is a subset of $A\times A$ such that for every $a\in A,$ both $\{b\mid (a,b)\in C\}$ and $\{b\mid (b,a)\in C\}$ are members of $W$ (a "crossword" with all "rows" and "columns" in $W),$ then $\{b\mid (b,b)\in C\}$ (the "diagonal word") also belongs to $W,$ and (iii) for all distinct $a,b\in A,$ the set $W$ has an element which contains $a$ but not $b.$ He has asked whether for every $A,$ the only such $W$ is the set of all subsets of $A.$ We answer that question in the negative. We also obtain several positive results, in particular, a positive answer to the above question if $W$ is closed under complementation. We obtain partial results on whether there can exist counterexamples to Pratt's question with $W$ countable.