On Vaughan Pratt's crossword problem

TL;DR

This paper investigates Vaughan Pratt's crossword problem, providing a negative answer to his question about the structure of certain sets W, and offers positive results under specific conditions, including closure under complementation.

Contribution

The paper disproves Pratt's conjecture that only the power set of A satisfies the conditions, and establishes positive results when W is closed under complementation.

Findings

Counterexamples exist to Pratt's question.

W closed under complementation satisfies the conditions.

Partial results on countable counterexamples.

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.