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

Consider a multiposet R := (V, (P_i)i ∈ I) made of a family (P_i)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 P_i if for every x ∈ V M , all the elements of M share the same comparability with x in P_i . 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 .