Consequence Operations Based on Hypergraph Satisfiability

Four consequence operators based on hypergraph satisfiability are defined. Their properties are explored and interconnections are displayed. Finally their relation to the case of the Classical Propositional Calculus is shown. 1. Preliminaries. Let us recall some definitions and facts which can also be found in Cowen [1,2] and Kolany [3]. A hypergraph is a structure G = (V, E), where V is a set and E is a family of nonempty subsets of V . The elements of V will be called vertices, and the elements of the set E , edges of the hypergraph G. Sets of vertices will sometimes be called clauses. A hypergraph is compact iff every edge contains a finite one. A hypergraph is locally finite iff every vertex belongs to a finite number of edges only. A hypergraph is edge disjoint iff its edges are pairwise disjoint. Notice that a graph is a hypergraph with at most two-element edges. Here, the vertices of a fixed hypergraph G will be interpreted as some elementary propositions and the edges of G will be inconsistent sets of them. This interpretation leads to the following generalization of satisfiability of families of disjunctions. Definition 1.1. A set of vertices σ satisfies a family of clauses A (wrt. G) iff 1. σ does not contain any edge, 2. σ meets all clauses of A, that is σ ∩ α 6= ∅, for all α in A. If some σ satisfies the family A, we say that A is satisfiable wrt. G, or G-satisfiable, for brevity.