Lexicographic partial order
Henry H. Crapo · Transactions of the American Mathematical Society · 1978
Given a (partially) ordered set P with the descending chain condition, and an ordered set Q , the set Q P {Q^P} of functions from P to Q has a natural lexicographic order, given by f ⩽ g f \leqslant g if and only if f ( y ) > g ( y ) f(y) > g(y) for all minimal elements of the set { x ; f ( x ) ≠ g ( x ) } \{ x;f(x) e g(x)\} where the functions differ. We show that if Q is a complete lattice, so also is the set Q P {Q^P} , in the lexicographic order. The same holds for the set Hom ( P , Q ) {\operatorname {Hom}}(P,Q) of order-preserving functions, and for the set Op ( P ) {\text {Op}}(P) of increasing order-preserving functions on the set P . However, the set Cl ( P ) {\text {Cl}}(P) of closure operators on P is not necessarily a lattice even if P is a complete lattice.