The authors investigate partial orders compatible with the operation \(f\) of a given monounary algebra \((A,f)\), i.e. partial orders \(\leq\) such that \(x\leq y\) implies \(f(x)\leq f(y)\) for all \(x,y\in A\). They give a complete description of maximal (with respect to containment) compatible partial orders on \((A,f)\). Those are exactly the so-called \(f\)-quasilinear partial orders. It is shown (Theorem 4.2) that if \((A,f,\leq)\) is a partially ordered monounary algebra, then there exists a compatible partial order on \((A,f)\) that is an extension of \(\leq\) and is \(f\)-quasilinear.
Reviewer’s note. G. Rubanovich described all possible compatible linear orders on a monounary algebra. He gave a method of their construction and obtained their total number [see G. Rubanovich, Uch. Zap. Tartu. Gos. Univ. 281, Trudy Mat. Mekh. 11, 34–48 (1971; Zbl 0331.06011)].