Join-closed subsets of an ordered set. Unicity of the join-basis

Achille Achache

We define the notions of cluster points and isolated points of a subset of an arbitrary closure space. We recall the notion of free subset and the notion of basis. We apply all that to the closure space made of the join-closed subsets of an arbitrary ordered set $E$. We establish that a join-closed subset has at most one basis. The set $I(E)$ of the isolated points of $E$ is exactly the set of the completely join-irreducible elements of $E$. When $I(E)$ generates $E$, $I(E)$ is the unique basis of $E$ (we give examples). When $I(E)$ does not generate $E$, $E$ has no basis.