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September 10, 2025OrderOpen Access

Atomic Boolean Lattice Completions of Ordered Sets

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Investigation reveals necessary conditions for posets to achieve atomic boolean lattice completion.

A poset must have prime filters that separate points and are pairwise incomparable to have an atomic boolean lattice completion.
The analysis shows that determining if a finite poset embeds into a boolean lattice is -complete, indicating high complexity.
Finite meets and joins are preserved in the embedding while generating the boolean lattice through complete lattice operations.
The construction of sets of ideals and filters is essential for a poset's canonical extension to be an atomic boolean lattice.

Morton et al. (2025) studied this question.

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