Lattice Polynomials and Polytopes

dc.creatorJacob Farinholt
dc.description.abstractA polyhedron, expressed in its canonical form as the intersection of half-spaces, is implicitly defined with respect to a particular ``component-wise'' partial order. This partial order on $\mbb{R}^n$ is a distributive lattice. While a polyhedron with this partial order may not be a sublattice of $\mbb{R}^n$, it may still nevertheless retain some of its lattice structure. This thesis characterizes and classifies polyhedra in $\mbb{R}^n$ according to how much lattice structure is retained. This is done by investigating their closure under convex clones of lattice polynomials. In addition, we investigate the join irreducibles of join semilattice polytopes, and show that they necessarily form faces of the polytope. We then characterize various attributes of these ``join irreducible faces.''
dc.titleLattice Polynomials and Polytopes Mason University


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