Lexicographic partial order
Trans. Amer. Math. Soc. 243 (1978), 37-51
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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.
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Denis Higgs, A lattice order for the set of all matroids on a set, Abstract, Ontario Math. Meeting, Nov. 5, 1966.
- Garrett Birkhoff, An extended arithmetic, Duke Math. J. 3 (1937), 311-316. MR 1545989
- -, Generalized arithmetic, Duke Math. J. 9 (1942), 283-302. MR 4, 74. MR 0007031 (4:74g)
- Mahlon Day, Arithmetic of ordered systems, Trans. Amer. Math. Soc. 58 (1945), 1-43. MR 7, 1. MR 0012262 (7:1a)
- 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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