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arXiv · 1701.03915

Structures of lattices which can be represented as the collection of all up-sets

Peng He·Xue-ping Wang
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Abstract

This paper first gives a necessary and sufficient condition that a lattice $L$ can be represented as the collection of all up-sets of a poset. Applying the condition, it obtains a necessary and sufficient condition that a lattice can be embedded into the lattice $L$ such that all infima, suprema, the top and bottom elements are preserved under the embedding by defining a monotonic operator on a poset. This paper finally shows that the quotient of the set of the monotonic operators under an equivalence relation can be naturally ordered and it is a lattice if $L$ is a finite distributive lattice.

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Peng He, Xue-ping Wang. 2017-01-14. Structures of lattices which can be represented as the collection of all up-sets. https://arxiv.org/abs/1701.03915

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