in the plane. A particularly nice result in this field is the characterization of the planar graphs, as those graphs whose vertex-edge incidence relations are circle orders. There are also geometric representations that are not based on inclusion. Indeed, one of the best studied classes among these are the interval orders, which represent the partial order in terms of what might be called disjoint precedence of intervals on the real line: each element x of the poset is represented by an interval [x1, x2], such that for any y and z in the poset, y is below z if and only if y2 < z1. Logic In logic, the representability of algebras as relational structures is often used to prove the equivalence of algebraic and relational semantics. Examples of this include Stone's representation of Boolean algebras as fields of sets, Esakia's representation of Heyting algebras as Heyting algebras of sets, and the study of representable relation algebras and representable cylindric algebras. Polysemy Under certain circumstances, a single function f : X → Y is at once an isomorphism from several mathematical structures on X. Since each of those structures may be thought of, intuitively, as a meaning of the image Y (one of the things that Y is trying to tell us), this phenomenon is called polysemy—a term borrowed from linguistics. Some examples of polysemy include: intersection polysemy—pairs of graphs G1 and G2 on a common vertex set V that can be simultaneously represented by a single collection of sets Sv, such that any distinct vertices u and w in V are adjacent in G1, if and only if their corresponding sets intersect ( Su ∩ Sw ≠ Ø ), and are adjacent in G2 if and only if the complements do ( SuC ∩ SwC ≠ Ø ). competition polysemy—motivated by the study of ecological food webs, in which pairs of species may have prey in common or have predators in common. A pair of graphs G1 and G2 on one vertex set is competition polysemic, if and only if there exists a single directed graph D on the same vertex set, such that any distinct vertices u and v are adjacent in G1, if and only if there is a vertex w such that both uw and vw are arcs in D, and are adjacent in G2, if and only if there is a vertex w such that