Combinatorial lemmas for oriented complexes

Abstract

A solid combinatorial theory is presented. The
generalized Sperner lemma for chains is derived from the
combinatorial Stokes formula. Many other generalizations follow
from applications of an $n$-index of a labelling defined on chains
with values in primoids. Primoids appear as the most general
structure for which Sperner type theorems can be formulated. Their
properties and various examples are given. New combinatorial
theorems for primoids are proved. Applying them to different
primoids the well-known classic results of Sperner, Fan, Shapley,
Lee and Shih are obtained.