# Dualization in Lattices Given by Ordered Sets of Irreducibles

arXiv: https://arxiv.org/abs/1504.01145 · HTML: https://mlsanity.com/paper/1504.01145/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2015.

## Built on by (most cited first, 1 of 1)

- [Avoiding Exponential Blow-Up in Distributive Lattice Submodular Minimization](https://mlsanity.com/paper/2606.14764/) (2026)

## Abstract (arXiv)

Dualization of a monotone Boolean function on a finite lattice can be represented by transforming the set of its minimal 1 to the set of its maximal 0 values. In this paper we consider finite lattices given by ordered sets of their meet and join irreducibles (i.e., as a concept lattice of a formal context). We show that in this case dualization is equivalent to the enumeration of so-called minimal hypotheses. In contrast to usual dualization setting, where a lattice is given by the ordered set of its elements, dualization in this case is shown to be impossible in output polynomial time unless P = NP. However, if the lattice is distributive, dualization is shown to be possible in subexponential time.