Abstract

A starrable lattice is one with a cancellative semigroup structure satisfying $(x\vee y)(x\wedge y)=xy$. If the cancellative semigroup is a group, then we say that the lattice is fully starrable. In this paper, it is proved that distributivity is a strict generalization of starrability. We also show that a lattice $(X,\le)$ is distributive if and only if there is an abelian 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 $x,y\in X$, while it is fully starrable if and only if there is an abelian group $(G,+)$ and a bijection $f:X\to G$ such that $f(x)+f(y)=f(x\vee y)+f(x\wedge y)$, for all $x,y\in X$.

Keywords: Lattice; distributive lattice; starrable lattice

MSC: 06B99

Pages:  1$-$11     

Volume  68 ,  Issue  1 ,  2016