There are two ways to present this work; the most efficient is of course to start with the main syntactical definitions, and to end with semantics: this is the presentation that we follow in the body of the text: section 1, syntex; section 2, semantics. Another possibility is to follow the order of discovery of the concepts, which (as expected) starts with the semantics and ends with the syntex; we adopt this second way for our introduction, hoping that this orthogonal look at the same object will help to apprehend the concepts.

We present a result of C.C. Squier relating two topics:

— the canonical rewriting systems, which have been widely studied by computer scientists as a tool for solving word problems,

— the homology of groups (more generally of monoids), which belongs to the core of pure mathematics, on the borders of algebra and topology.

In the literature ther are two main notins of model for the second order λ-calculus: one by Bruce, Meyer and Mitchell (the BMM-model, for short) in set-theoretical formulation and one category-theoretical by Seely. Here we generalise Seely's notion, using semifunctors and semi-adjunctions from Hayashi, and introduce λ2-algebras, λη2-algebras, λ2-models and λη-models, similarly to the untyped λ-calculus. Non-extensional abstraction of both term and type variables is described by semi-adjunctions (essentially as in Martini's thesis). We show that also for second order sume, the β-(and commutation!) conversions correspond to semi-functoriality and that the (additional) η-conversion corresponds to ordinary fuctioriality.

In the above framework various examples – well known ones and variations – are described. Also, we determine the place of the BMM-models; an earlier version of the latter has been reported by Jacobs.

High-level replacement systems are formulated in an axiomatic algebraic framework based on categories pushouts. This approach generalizes the well-known algebraic approach to graph grammars and several other types of replacement systems, especially the replacement of algebraic specifications which was recently introduced for a rule-based approach to modular system design.

in this paper basic notions like productions, derivations, parellel and sequential independence are introduced for high-level replacement syetms leading to Church-Rosser, Parallelism and concurrency Theorems previously shown in the literature for special cases only. In the general case of high-level replacement systems specific conditions, called HLR1- and HLR2-conditions, are formulated in order to obtain these results.

Several examples of high-level replacement systems are discussed and classified w.r.t. HLR1- and HLR2-conditions showing which of the results are valid in each case.