A semiprime filter-based identity- summand graph of a lattice

Abstract

Let $F$ be a proper filter of a lattice $L$ with the leastelement $0$ and the greatest element $1$. The filter-basedidentity-summand graph of $L$ with respect to $F$, denoted by$\Gamma_{F} (L)$, is the graph with vertices I^*_{F} (L) = \{x\in L \setminus F: x \vee y \in F \, \, \mbox{for some} \, \, y\in L \setminus F \}, and distinct vertices $x$ and $y$ areadjacent if and only if $x \vee y \in F$. We will make anintensive study of the notions of diameter, grith, chromaticnumber, clique number, independence number, domination numberand planar property of this graph. Moreover, Beck$^sconjecture is proved for$\Gamma_{F} (L)$