Fixed points and products: Width 3 (Q1810804)

We say that an ordered set \(P\) has the fixed point property (FPP), if for each endomorphism \(f\) of \(P\) there exists \(p\in P\) with \(f(p)= p\). This paper is a contribution to the general problem: ``If ordered sets \(P\) and \(X\) have FPP, does \(P\times X\) have FPP?'' In Order 11, No. 1, 11--14 (1994; Zbl 0814.06003), \textit{M. S. Roddy} solved this problem affirmatively provided that \(P\) is finite. \textit{B. Li} and \textit{E. C. Milner} [Order 12, No. 2, 159--171 (1995; Zbl 0835.06001)] and \textit{M. S. Roddy}, \textit{A. Rutkowski} and \textit{B. S. Schröder} [unpublished manuscript (1994)] (independently) extended this result to the case when \(P\) is chain complete with no infinite anti-chain. In the present paper the author solves the problem positively for \(P\) having width at most three.