On starrable lattices

چکیده مقاله

A starrable lattice is one with a cancellative semigroup structure satisfying (x _ y)(x ^ 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;·) is distributive if and only if there is an abelian group (G; +) and an injection f : X ! G such that f(x) + f(y) = f(x _ y) + f(x ^ y) for all x; y 2 X, while it is fully starrable if and only if there is an abelian group (G; +) and a bijection f : X ! G such that f(x) + f(y) = f(x _ y) + f(x ^ y), for all x; y 2 X.