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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Lexicographic partial order


Author: Henry Crapo
Journal: Trans. Amer. Math. Soc. 243 (1978), 37-51
MSC: Primary 06A10
MathSciNet review: 0491372
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0491372-3
PII: S 0002-9947(1978)0491372-3
Keywords: Partial order, order, lattice, complete lattice, closure operator, lexicographic order
Article copyright: © Copyright 1978 American Mathematical Society