The Range of a Steiner Operation

Abstract

This paper answers a fundamental question in the theory of Steiner operations (StOps) as defined and studied in the monograph, "Global Methods for Combinatorial Isoperimetric Problems" (GMCIP). StOps are morphisms for combinatorial isoperimetric problems, analogous to Steiner symmetrization for continuous isoperimetric problems. The usefulness of a StOp, Phi, a function from the power set of V (a finite set) to the power set of V, depends on having an efficient representation of its range. In GMCIP the problem was treated case-by-case. In each case the StOp induced a partial order, P, on V so that Range(Phi)=I(P), the set of all order ideals of P. Here we show (directly from the axioms for a StOp) that every idempotent StOp admits such a representation of its range (P is then called the StOp-order of Phi). That result leads to another question: What additional structure does Range(Phi) have? The answer is none. We show that every finite poset is the StOp-order of some idempotent Steiner operation.