Completely Precontinuous Posets

doi:10.1016/j.entcs.2014.01.014

Electronic Notes in Theoretical Computer Science

Volume 301, 18 February 2014, Pages 169-178

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Abstract

In this paper, concepts of strongly way below relations, completely precontinuous posets, coprimes and Heyting posets are introduced. The main results are: (1) The strongly way below relations of completely precontinuous posets have the interpolation property; (2) A poset P is a completely precontinuous poset iff its normal completion is a completely distributive lattice; (3) An ω-chain complete P is completely precontinuous iff P and $P^{o p}$ are precontinuous and its normal completion is distributive iff P is precontinuous and has enough coprimes; (4) A poset P is completely precontinuous iff the strongly way below relation is the smallest approximating auxiliary relation on P iff P is a Heyting poset and there is a smallest approximating auxiliary relation on P. Finally, given a poset P and an auxiliary relation on P, we characterize those join-dense subsets of P whose strongly way-below relation agrees with the given auxiliary relation.

Keywords

Completely precontinuous poset

completely distributive lattice

normal completion

coprime

auxiliary relation

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¹: Supported by the National Natural Science Foundation of China (Nos. 10861007, 11161023), the Fund for the Author of National Excellent Doctoral Dissertation of China (No. 2007B14), the NFS of Jiangxi Province (Nos. 20114BAB201008), the Fund of Education Department of Jiangxi Province (No. GJJ12657).