On starrable lattices

Abstract

A starrable lattice is one with a cancellativesemigroup structure satisfying $(x\vee y)(x\wedge y)=xy$. If thecancellative semigroup is a group, then we say that the lattice isfully starrable. In this paper, it is proved that distributivityis a strict generalization of starrability. We also show that alattice $(X,\le )$ is distributive if and only if there is anabelian group $(G,+)$ and an injection $f:X\to G$ such that$f(x)+f(y)=f(x\vee y)+f(x\wedge y)$ for all $x,yınX$, while itis fully starrable if and only if there is an abelian group$(G,+)$ and a bijection $f:X\to G$ such that $f(x)+f(y)=f(x\text{veey})+f(x\wedge y)$, for all $x,yınX$.