On the complexity of posets

AbstractThe purpose of this paper is to discuss several invariants each of which provides a measure of the intuitive notion of complexity for a finite partially ordered set. For a poset X the invariants discussed include cardinality, width, length, breadth, dimension, weak dimension, interval dimension and semiorder dimension denoted respectively X, W(X), L(X), B(X), dim(X). Wdim(X), Idim(X) and Sdim(X). Among these invariants the following inequalities hold. B(X)⩽Idim(X)⩽Sdim(X)⩽Wdim(X)⩽dim(X)⩽W(X). We prove that every poset X with three of more points contains a partly with Idim(X) Idim(X) {x,v}). If M denotes the set of maximal elements and A an arbitrary anticham of X we show that Idim(X)⩽W(X-M) and Idim(X)⩽2W(X-A). We also show that there exist functions f(n,t) and (gt) such that I(X)⩽n and Idim(X)⩽tsimply dim(X)⩽f(n,t and Sdim(X)⩽t implies dim(X)⩽g(t)

Core Rationalizability in Two-Agent Exchange Economies

We provide a characterization of selection correspondences in two-person exchange economies that can be core rationalized in the sense that there exists a preference profile with some standard properties that generates the observed choices as the set of core elements of the economy for any given initial endowment vector. The approach followed in this paper deviates from the standard rational choice model in that a rationalization in terms of a profile of individual orderings rather than in terms of a single individual or social preference relation is analyzed.