FIXED POINTS AND NEGATIVE CIRCUIT FREE IN FINITE LATTICES

Abstract

Let $X$ be a dimensional finite lattice (not necessary distributive) and let $F$ be a mapping from $X$ to $X$. Here we introduce a new notion of neighbours of an element of $X$ and prove that if all the neighbours of each element of $X$ are in $X$ and there is no negative circuit in the interaction graph of $F$, then $F$ has a fixed point.