Shuffling lattices.

Abstract

By assigning a distinct positive integer to each join-irreducible of a lattice, with each element of the lattice can be associated to word corresponding to the join-irreducibles less than or equal to that element. If this is done to two lattices, then we can construct a new poset by partially ordering all of the possible words obtained by shuffling the words in the two lattices. This thesis examines in depth this shuffling operation. The resulting poset inherits a good deal of its combinatorial structure from its parent lattices. The algebraic properties are particularly nice. The shuffling operation preserves Cohen-Macaulayness, shellability, edge labelings, and supersolvability. We give a complete description of the Mobius function and homology of the resulting poset. Descriptions of the rank-generating function, characteristic polynomial, and zeta polynomial are also gotten. Finally, we obtain bounds on the dimension of the shuffle of two lattices.