Description of the (≤ 3)-hypomorphic multiposets; application to the (≤ 3)-reconstruction

Abstract

Consider a multiposet R := (V, (Pi)i∈I ) made of a family (Pi)i∈I (the components of R) of strict orders on a possibly infinite set V (the vertex set of R). R is linear if at least one of its components is a chain and its other components which are not anti-chains are equal up to duality to this chain. For i ∈ I, a subset M of V is a module of Pi if for every x ∈ V \ M, all the elements of M share the same comparability with x in Pi. A module of R is a common module of its components. A linear-module of R is a module M of R such that the restriction R ↾ M of R to M is linear. R is prime if |V | ≥ 3 and its only modules are the empty set, the singletons of its vertex set, and its own vertex set. Let k be a positive integer. Two multiposets R and R′ on a same vertex set are (≤ k)-hypomorphic if for every set K of at most k vertices, the two restrictions R ↾ K and R′ ↾ K are isomorphic. A multiposet R is (≤ k)-reconstructible if every multiposet R′ that is (≤ k)-hypomorphic to R is isomorphic to R. In this paper, we begin by obtaining a morphological description of the difference classes, introduced by Lopez in 1972, of the pairs of (≤ 3)-hypomorphic multiposets. Then we use this result to describe the pairs of (≤ 3)-hypomorphic multiposets. As a first corollary, we obtain that a multiposet is (≤ 3)-reconstructible if and only if its linear-modules are finite. As a second corollary, we obtain that given two (≤ 3)-hypomorphic multiposets R and R′ with at least four vertices, if R is prime, then R′ = R.