Reconstruction of Finite Posets of Width Three

Source abstract

We prove that every finite ordered set of width at most three and cardinality at least four is reconstructible from its deck. This settles the width-three part of an open problem posed by Schröder in Ordered Sets; the separate question concerning ordered sets of height one is not addressed here. The new ingredient is a criterion for recovering the marked maximal deck of an ordered set with exactly three maximal elements from its unmarked maximal deck and the isomorphism type of the suborder obtained by deleting all maximal elements. We express the criterion as an orbital balance condition on the attachment ideals in this common base. Necessity follows by double counting; sufficiency uses the structure of automorphism actions in width at most three. The full-deck theorem then follows from reconstruction results of Schröder concerning maximal cards and marked maximal cards. A local reattachment argument correcting one intermediate step in the two-maximal-element case is included in an appendix.