English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this article, we study the inconsistency of systems of $\min-\rightarrow$fuzzy relational equations. We give analytical formulas for computing theChebyshev distances $\nabla = \inf_{d \in \mathcal{D}} \Vert \beta - d \Vert$associated to systems of $\min-\rightarrow$ fuzzy relational equations of theform $\Gamma \Box_{\rightarrow}^{\min} x = \beta$, where $\rightarrow$ is aresidual implicator among the G\"odel implication $\rightarrow_G$, the Goguenimplication $\rightarrow_{GG}$ or Lukasiewicz's implication $\rightarrow_L$ and$\mathcal{D}$ is the set of second members of consistent systems defined withthe same matrix $\Gamma$. The main preliminary result that allows us to obtainthese formulas is that the Chebyshev distance $\nabla$ is the lower bound ofthe solutions of a vector inequality, whatever the residual implicator used.Finally, we show that, in the case of the $\min-\rightarrow_{G}$ system, theChebyshev distance $\nabla$ may be an infimum, while it is always a minimum for$\min-\rightarrow_{GG}$ and $\min-\rightarrow_{L}$ systems.

Handling the inconsistency of systems of $\min\rightarrow$ fuzzy relational equations