This paper provides an algorithm that, given two finite algebras A and B each of arbitrary finite similarity type, determines whether or not A and B are categorically equivalent. Although the algorithm is not practical in general, we consider some conditions on the algebras that improve its performance.

Two operations are presented for a modular approach to the definition of frameworks for rigorous development of software, formally represented as institutions.

The first generalizes models, allowing them to have more structure than the minimum required by their declared signatures, as happens for software modules, which may have local routines that do not appear in their interface.

The second extends sentences, and their interpretation in models, allowing sentences on richer signatures to be used as formulae for poorer ones.

Combining the application of these operations, powerful institutions can be defined, like those for very abstract entities, or for hyper-loose algebraic specifications.

The compatibility of different sequential applications of these operations and properties of the resulting institutions are also studied.

In recent years, the problem of incorporating a set-building type-constructor into a domain theoretic data model has been addressed by different authors. In Jung and Puhlmann (1995) and Puhlmann (1995) we have shown why the so-called snack powerdomain is particularly suitable for modelling a set constructor. We obtain a generalized database model that covers the nested relational model.

While, with the snack powerconstruction, the data structure of domain theoretic databases seems clear, suitable operations for the data model are still to be defined.

In this paper we start this task by defining the operations nest and unnest for the passage between different nesting-levels of the snack powerconstruction. These functions are shown to form an embedding-projection pair, a property that the corresponding functions of nested relational databases do not have. This demonstrates the usefulness of the domain-theoretic approach for modelling databases: for the first time we have operators for re-grouping nested data that respect the idea of an information ordering.

The use of the snack powerdomain leads to fairly complex formulas. To help the reader, illustrations and pictorial interpretations of formulas are given throughout the paper.