Weak infinite products of Blaschke products

We study weak infinite products for sequences of Blaschke products. Using properties of these functions, -subsets of zero sets of functions in are studied. An affirmative answer is given to a problem on prime ideals of posed by Gorkin and Mortini.

     

Toric fiber products versus Segre products

The toric fiber product is an operation that combines two ideals that are homogeneous with respect to a grading by an affine monoid. The Segre product is a related construction that combines two multigraded rings. The quotient ring by a toric fiber product of two ideals is a subring of the Segre product, but in general this inclusion is strict. We contrast the two constructions and show that any Segre product can be presented as a toric fiber product without changing the involved quotient rings. This allows to apply previous results about toric fiber products to the study of Segre products. We give criteria for the Segre product of two affine toric varieties to be dense in their toric fiber product, and for the map from the Segre product to the toric fiber product to be finite. We give an example that shows that the quotient ring of a toric fiber product of normal ideals need not be normal. In rings with Veronese type gradings, we find examples of toric fiber products that are always Segre products, and we show that iterated toric fiber products of Veronese ideals over Veronese rings are normal.     

Complete finite semidirect products and wreath products

A complete group is one with a trivial center and with all automorphisms inner. This paper uses group cohomology to give a sufficient condition for a finite semidirect product G = N \rtimes H{G = N \rtimes H} with C _G (N) ≤ N to be complete and proves a partial converse. These results are enough to fully characterize complete finite permutational wreath products and to specialize that characterization in the case of finite standard wreath products.     

Edge-transitive products

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edge-transitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a strong product is edge-transitive if and only if all factors are complete graphs. In addition, a connected, infinite non-trivial Cartesian product graph G is edge-transitive if and only if it is vertex-transitive and if G is a finite weak Cartesian power of a connected, edge- and vertex-transitive graph H, or if G is the weak Cartesian power of a connected, bipartite, edge-transitive graph H that is not vertex-transitive.     

Representing products of polyhedra by products of edge-colored graphs

Given an (m + 1)-colored graph ( Γ γ′) and an (n + 1)-colored graph (, γ″), representing two polyhedra P′, P″ respectively, we present a direct construction of an (m + n+1)-colored graph ( Γ ′? Γ ″, γ′ ? γ″), which represents the product P ′ × P ″. Some examples, applications, and conjectures about the genus of manifold products are also presented. © 1993 John Wiley & Sons, Inc.     

On semidirect products and wreath products of partially ordered groups

The notions of Cartesian and semidirect products for partially ordered groups are considered. A series of results on those products of AO \mathcal{A}\mathcal{O} -groups and interpolation groups is obtained. Some results concerning wreath products of directed groups are obtained.