Pre-fixed points of polynomial functions of lattices (Q1319061)

An element \(c\) of an ordered set \(P\) is called lower (upper) pre-fixed point of a function \(f:P\to P\) if \(c\leq f(c)\) \((f(c)\leq c)\). Lemma 1. If every order-preserving selfmap of an ordered set \(P\) has an upper pre-fixed point, then each well-ordered subset has an upper bound (\(P\) is inductive). Corollary 1. A lattice \(L\) has a greatest (least) element iff each order- preserving selfmap of \(L\) has a lower (upper) pre-fixed point. A lattice is order-polynomially complete iff each order-preserving selfmap of \(L\) is a polynomial function. A lattice \(L\) is \(\omega\)-chain continuous if each ascending sequence has a supremum and \(\bigvee \{a_ n\): \(n\in\omega\}\wedge b= \bigvee\{a_ n \wedge b\): \(n\in\omega\}\) for all \(b\in L\). Using these notions the author proves: Corollary 2. Any order-polynomially complete lattice is bounded. Corollary 3. Any order-polynomially complete \(\omega\)-chain continuous lattice is complete. Corollary 4. An infinite order-polynomially complete lattice cannot contain any forest of the same cardinality. Here, by a forest the authors mean an ordered set whose nonempty chains are precisely those subsets which have a maximum.