We consider a uni ed setting for studying local valuated groups and coset-valuated groups, emphasizing the associated ltrations rather than the values of elements. Stable exact sequences, projectives and injectives are identi ed in the encompassing category, and in the category corresponding to coset-valuated groups.

The dualisation and the first-order formulation of the D = 7 abelian Yang-Mills supergravity which is the low energy effective limit of the D = 7 fully Higssed heterotic string is discussed. The nonlinear coset formulation of the scalars is enlarged to include the entire bosonic sector by introducing dual fields and by constructing the Lie superalgebra which generates the dualized coset element.

We present formulae for the characters of coset conformal field theories and apply these to specific examples to determine the integer shift of the conformal weights of primary fields. We also present an example of a coset conformal field theory which cannot be described by the identification current method.

In this paper, we introduce a class of double coset cayley digraphs induced by right solvable ward groupoids. These class of graphs can be viewed as a generalization of double coset cayley graphs induced by groups. Further, many graph properties are expressed in terms of algebraic properties.