A characterization of ordered loops by monotone mappings (Q1203519)

Let \((L,+)\) be a loop endowed with an order relation \(<\) on \(L\). The author shows that \(L\) is an ordered loop, if and only if all right and left translations of \(L\) and at least one of its inverse mappings are monotone with respect to \(<\). The proof fills a small (and only local) gap in the exposition of ordered loops and ordered webs in his celebrated book on projective planes [the author, Projective Ebenen. 2. Aufl. (1975; Zbl 0307.50001); (1. ed. 1955; Zbl 0066.387)].