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

## TL;DR

This paper characterizes when a lattice can be represented as the collection of all up-sets of a poset, providing conditions for such representations and embeddings, especially for finite distributive lattices.

## Contribution

It establishes necessary and sufficient conditions for representing lattices as up-set collections and for embedding lattices while preserving key elements, with a focus on finite distributive lattices.

## Key findings

- A lattice can be represented as all up-sets of a poset under certain conditions.
- Embedding a lattice into another while preserving infima, suprema, top, and bottom is characterized.
- The set of monotonic operators forms a lattice when quotiented by an equivalence relation in finite distributive lattices.

## 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.