An Isomorphism between Abstract Polyhedral Cones and Definite Boolean Functions

Polyhedral cones can be represented by sets of linear inequalities that express inter-variable relationships. These inequalities express inter-variable relationships that are quantified by the ratios between the variable coefficients. However, linear inequalities over a non-negative variable domain with only unit variable coefficients and no constants other than zero can represent relationships that can be valid in non-numeric domains. For instance, if variables are either non-negative or zero itself, that is, a strictly two-point domain, then 0 <= x, 0 <= y, x <= y, expresses a dependency between x and y, since if y is known to be zero, then so is x. By defining an abstraction operator that effectively puts aside the scaling coefficients whilst retaining the inter-variable aspect of these relationships polyhedral cones can express the same dependency information as Def, a class of Boolean function. Boolean functions are considered over a fixed finite set of variables and Def is a subset of the positive Boolean functions, which return the value true when every variable returns true. Def is a complete lattice ordered by logical consequence and it will be shown that the abstract cones also form a complete lattice, ordered by set inclusion, that is isomorphic to Def.