LIFTING THEOREMS FOR TENSOR FUNCTORS ON MODULE CATEGORIES

Any (co)ring R is an endofunctor with (co)multiplication on the category of abelian groups. These notions were generalized to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate's lifting theorem of functors between categories to related (co)module categories. Curiously enough some of these results were not noticed by researchers in module theory and thus notions like entwining structures and smash products between algebras and coalgebras were introduced (in the nineties) without being aware that these are special cases of the more general theory. The purpose of this survey is to explain several of these notions and recent results from general category theory in the language of elementary module theory focusing on functors between module categories given by tensoring with a bimodule. This provides a simple and systematic approach to smash products, wreath products, corings and rings over corings (C-rings). We also highlight the relevance of the Yang–Baxter equation for the structures on the threefold tensor product of algebras or coalgebras (see 3.6).

Constructing tensegrity structures from one-bar elementary cells

This study aimed to develop a method to construct tensegrity structures from elementary cells, defined as structures consisting of only one bar connected with a few strings. Comparison of various elementary cells leads to the further selection of the so-called ‘Z-shaped’ cell, which contains one bar and three interconnected strings, as the elementary module to assemble the Z-based spatial tensegrity structures. The graph theory is utilized to analyse the topology of strings required to construct this type of tensegrity structures. It is shown that ‘a string net can be used to construct a Z-based tensegrity structure if and only if its topology is a simple and bridgeless cubic graph’. Once the topology of strings has been determined, one can easily design the associated tensegrity structure by adding a deterministic number of bars. Two schemes are suggested for this design strategy. One is to enumerate all possible topologies of Z-based tensegrity for a specified number of bars or cells, and the other is to determine the tensegrity structure from a vertex-truncated polyhedron. The method developed in this paper allows us to construct various types of novel tensegrity structures.