# Handling the inconsistency of systems of minrightarrow fuzzy relational equations

arXiv: https://arxiv.org/abs/2308.12385 · HTML: https://mlsanity.com/paper/2308.12385/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 15 references that are in the map.
- First posted on arXiv 22 Aug 2023.

## Builds on (most cited first, 2 of 2)

- [Max-min Learning of Approximate Weight Matrices from Fuzzy Data](https://mlsanity.com/paper/2301.06141/) (2023)
- [Chebyshev distances associated to the second members of systems of Max-product/Lukasiewicz Fuzzy relational equations](https://mlsanity.com/paper/2302.08554/) (2023)

## Abstract (arXiv)

In this article, we study the inconsistency of systems of min-rightarrow fuzzy relational equations. We give analytical formulas for computing the Chebyshev distances nabla = infd ∈ D Vert β - d Vert associated to systems of min-rightarrow fuzzy relational equations of the form Gamma Boxrightarrowmin x = β, where rightarrow is a residual implicator among the G"odel implication rightarrowG, the Goguen implication rightarrowGG or Lukasiewicz's implication rightarrowL and D is the set of second members of consistent systems defined with the same matrix Gamma. The main preliminary result that allows us to obtain these formulas is that the Chebyshev distance nabla is the lower bound of the solutions of a vector inequality, whatever the residual implicator used. Finally, we show that, in the case of the min-rightarrowG system, the Chebyshev distance nabla may be an infimum, while it is always a minimum for min-rightarrowGG and min-rightarrowL systems.