Powers of ordered sets (Q2266033)

For ordered sets P and Q, \(P^ Q\) denotes the set of all order- preserving maps of Q to P with the ordering \(f\leq g\) in \(P^ Q\) if f(x)\(\leq g(x)\) for all \(x\in Q\). This survey article is concerned with efforts to answer two of Birkhoff's questions [see \textit{G. Birkhoff}, Duke Math. J. 9, 283-302 (1942; Zbl 0060.126)]: (1) Does \(P^ Q\cong P^ R\) imply \(Q\cong R?\) (2) Does \(P^ R\cong Q^ R\) imply \(P\cong Q?\) In the paper under review, two results are presented. The first is a theorem of the author and \textit{I. Rival} [Can. J. Math. 30, 797-807 (1978; Zbl 0497.06004)], settling (1) in the finite case. The second, regarding (2), is due to \textit{B. Jónsson} and \textit{R. McKenzie} [Math. Scand. 51, 87-120 (1982; Zbl 0501.06001)]. The author discusses central ideas of these papers. Also, a series of remarks concerning related topics is presented. The paper ends with an extensive list of references.