Lexicographic partial order

Abstract:

Given a (partially) ordered set P with the descending chain condition, and an ordered set Q, the set ${Q^P}$ of functions from P to Q has a natural lexicographic order, given by $f \leqslant g$ if and only if $f(y) < g(y)$ for all minimal elements of the set $\{ x;f(x) \ne g(x)\}$ where the functions differ. We show that if Q is a complete lattice, so also is the set ${Q^P}$, in the lexicographic order. The same holds for the set ${\operatorname {Hom}}(P,Q)$ of order-preserving functions, and for the set ${\text {Op}}(P)$ of increasing order-preserving functions on the set P. However, the set ${\text {Cl}}(P)$ of closure operators on P is not necessarily a lattice even if P is a complete lattice.