Ideals in atomic posets

Abstract

The “bottom” of a partially ordered set (poset) Q is the set Qℓ of its lower bounds (hence, Qℓ is empty or a singleton). The poset Q is said to be atomic if each element of Q∖Qℓ dominates an atom, that is, a minimal element of Q∖Qℓ. Thus, all finite posets are atomic. We study general closure systems of down-sets (referred to as ideals) in posets. In particular, we investigate so-called m-ideals for arbitrary cardinals m, providing common generalizations of ideals in lattices and of cuts in posets. Various properties of posets and their atoms are described by means of ideals, polars (annihilators) and residuals, defined parallel to ring theory. We deduce diverse characterizations of atomic posets satisfying certain distributive laws, e.g. by the representation of specific ideals as intersections of prime ideals, or by maximality and minimality properties. We investigate non-dense ideals (down-sets having nontrivial polars) and semiprime ideals (down-sets all of whose residuals are ideals). Our results are constructive in that they do not require any set-theoretical choice principles.