The concept of regular incidence-complexes generalizes the notion of regular polytopes in a combinatorial and group-theoretical sense. An incidence-complex is a special type of partially ordered set with regularity defined by the flag-transitivity of its group of automorphisms. A central problem in the theory of regular polytopes is the construction of d-dimensional polytopes with predescribed facets. In this paper the combinatorial analog for regular incidencecomplexes is considered. It is proved that every d-dimensional regular complex K is a facet of a (d + 1)-dimensional regular complex ℒ, even of a finite and non-degenerate one in case K is finite and non-degenerate. The construction starts from a group-theoretical transformation of the problem into an embedding problem for groups.
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