Lexicographic partial order

Author:
Henry Crapo

Journal:
Trans. Amer. Math. Soc. **243** (1978), 37-51

MSC:
Primary 06A10

DOI:
https://doi.org/10.1090/S0002-9947-1978-0491372-3

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 of functions from *P* to *Q* has a natural lexicographic order, given by if and only if for all minimal elements of the set where the functions differ.

We show that if *Q* is a complete lattice, so also is the set , in the lexicographic order. The same holds for the set of order-preserving functions, and for the set of increasing order-preserving functions on the set *P*.

However, the set of closure operators on *P* is not necessarily a lattice even if *P* is a complete lattice.

**[1]**Garrett Birkhoff,*An extended arithmetic*, Duke Math. J.**3**(1937), 311-316. MR**1545989****[2]**-,*Generalized arithmetic*, Duke Math. J.**9**(1942), 283-302. MR**4**, 74. MR**0007031 (4:74g)****[3]**Mahlon Day,*Arithmetic of ordered systems*, Trans. Amer. Math. Soc.**58**(1945), 1-43. MR**7**, 1. MR**0012262 (7:1a)****[4]**Denis Higgs,*A lattice order for the set of all matroids on a set*, Abstract, Ontario Math. Meeting, Nov. 5, 1966.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0491372-3

Keywords:
Partial order,
order,
lattice,
complete lattice,
closure operator,
lexicographic order

Article copyright:
© Copyright 1978
American Mathematical Society