Poset extensions, convex sets, and semilattice presentations

The paper is devoted to an algebraic and geometric study of the feasible set of a poset, the set of finite probability distributions on the elements of the poset whose weights satisfy the order relationships specified by the poset. For a general poset, this feasible set is a barycentric algebra. The feasible sets of the order structures on a given finite set are precisely the convex unions of the primary simplices, the facets of the first barycentric subdivision of the simplex spanned by the elements of the set. As another fragment of a potential complete duality theory for barycentric algebras, a duality is established between order-preserving mappings and embeddings of feasible sets. In particular, the primary simplices constituting the feasible set of a given finite poset are the feasible sets of the linear extensions of the poset. A finite poset is connected if and only if its barycentre is an extreme point of its feasible set. The feasible set of a (general) disconnected poset is the join of the feasible sets of its components. The extreme points of the feasible set of a finite poset are specified in terms of the disjointly irreducible elements of the semilattice presented by the poset. Semilattices presented by posets are characterised in terms of various distributivity concepts.