NP-completeness of Partial Chirotope Extendibility

In the monograph “Axioms and Hulls” (1992) Donald Knuth studies some axiomatizations of geometric situations. The structures described by one of the axiom systems are called CC-systems. Knuth proves that it is NP-complete to decide, whether a partially defined CC-system can be extended to a complete CC-system. The aim of this note is to show that Knuth’s proof of this result also implies that it is NP-complete to decide the extendability of partially defined chirotopes.