In this paper, we define a new type of product of association schemes (and of the related objects, permutation groups and orthogonal block structures), which generalizes the direct and wreath products (which are referred to as “crossing” and “nesting” in the statistical literature.) Given two association schemes Qr for r = 1, 2, each having an inherent partition Fr (that is, a partition whose e...

We develop new algebraic tools to reason about concurrent behaviours modelled as languages of Mazurkiewicz traces and asynchronous automata. These reflect the distributed nature underlying causality concurrency between events, can be said support true concurrency. They generalize that have been so efficient in understanding, classifying reasoning word languages. In particular, we introduce an v...

The Peiier product of groups rst arose in work of J.H.C. White-head on the structure of relative homotopy groups, and is closely related to problems of asphericity for two-complexes. We develop algebraic methods for computing the second integral homology of a Peiier product. We s h o w t h a t a Peiier product of superperfect groups is superperfect, and determine when a Peiier product of cyclic...

For a restricted wreath product $G\wr \mathbb{Z}^k$, where $G$ is finite abelian group, we determine (almost in all cases) whether this has the $R_\infty$ property (i.e., each its automorphism infinite Reidemeister number).

We consider the problem of modelling noisy but highly symmetric shapes that can be viewed as hierarchies of whole-part relationships in which higher level objects are composed of transformed collections of lower level objects. To this end, we propose the stochastic wreath process, a fully generative probabilistic model of drawings. Following Leyton’s ”Generative Theory of Shape”, we represent s...

The character table of the fully nonrigid water cluster (cyclic forms), (H_{2}O){_i}, with C{_ih} symmetry derived for the first time, for 2<=i <=6. The group of all feasible permutations is the wreath product of groups S{_i}[S{_2}] which consists of i!2i operations for i = 2, ..., 6 divided into ( w.r.t) 5, 10, 20, 36, 65 conjugacy classes and 5, 10, 20, 36, 65 irreducible representations resp...

In his 1996 work developing the theory of association schemes as a ‘generalized’ group theory, Zieschang introduced the concept of the semidirect product as a possible product operation of certain association schemes. In this paper we extend the semidirect product operation into the entire set of association schemes. We then derive a way to decompose certain association schemes into smaller ass...

In this paper we prove that the wreath product with symmetric group Sn, is cellular for algebra Z2(x). We obtain simple cell modules which satisfy semi-simplicity conditions. make use of method iterated inflations purpose.