this paper the author was partially supported by an SERC/EPSRC Advanced Research Fellowship, EPSRC Research grant GR/L54639, and EU Working Group APPSEM

Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, thatis,ifthe components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking. This thesis posits that part of the problem is the usual, concrete and syntaxoriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from

A logic is developed in which function symbols are allowed to represent partial functions. It has the usual rules of logic (in the form of a sequent calculus) except that the substitution rule has to be modi ed. It is developed here in its minimal form, with equality and conjunction, as partial Horn logic. Various kinds of logical theory are equivalent: partial Horn theories, quasi-equational theories (partial Horn theories without predicate symbols), cartesian theories and essentially algebraic theories. The logic is sound and complete with respect to models in Set, and sound with respect to models in any cartesian ( nite limit) category. The simplicity of the quasi-equational form allows an easy predicative constructive proof of the free partial model theorem for cartesian theories: that if a theory morphism is given from one cartesian theory to another, then the forgetful (reduct) functor from one model category to the other has a left adjoint. Various examples of quasi-equational theory are studied, including those of cartesian categories and of other classes of categories. For each quasi-equational theory T another, CartϖT, is constructed, whose models are cartesian categories equipped with models of T. Its initial model, the classifying category for T, has properties similar to those of the syntactic category, but more precise with respect to strict cartesian functors. This is a preprint version of the article published as Annals of Pure and Applied Logic 145 (3) (2007), pp. 314- 353. doi:10.1016/j.apal.2006.10.001

This document is an outline of the theory of sketches with pointers to the literature. An extensive bibliography is given. Some coverage is given to related areas such as algebraic theories, categorial model theory and categorial logic as well. An appendix beginning on page 11 provides definitions of some of the less standard terms used in the paper, but the reader is expected to be familiar with the basic ideas of category theory. A rough machine generated index begins on page 21. I would have liked to explain the main ideas of all the papers referred to herein, but I am not familiar enough with some of them to do that. It seemed more useful to be inclusive, even if many papers were mentioned without comment. One consequence of this is that the discussions in this document often go into more detail about the papers published in North America than about those published elsewhere. The DVI file for this article is available by anonymous FTP from ftp.cwru.edu in the directory

For a set M of graphs the category CatM of all M-complete categories and all strictly M-continuous functors is known to be monadic over Cat. The question of monadicity of CatM over the category of graphs is known to have an affirmative answer when M specifies either (i) all finite limits, or (ii) all finite products, or (iii) equalizers and terminal objects, or (iv) just terminal objects. We prove that, conversely, these four cases are (essentially) the only cases of monadicity of CatM over the category of graphs, provided that M is a set of finite graphs containing the empty graph.

. A graph-based specification language and the corresponding machinery are described as stating a basic framework for semantic modeling and database design. It is shown that a few challenging theoretical questions in the area, and some of hot practical problems as well, can be successfully approached in the framework. The machinery has its origin in the classical sketches invented by Ehresmann and is close to their generalization recently proposed by Makkai. There are two essential distinctions from Makkai's sketches. One consists in a different -- more direct -- formalization of sketches that categorists (and database designers) usually draw. The second distinction is more fundamental and consists in introducing operational sketches specifying complex diagram operations over ordinary (predicate) sketches, correspondingly, models of operational sketches are diagram algebras. Together with the notion of parsing operational sketches, this is the main mathematical contribution of the pape...

We present a class of algebraic theories that are enriched over a novel Symmetrical Monoidal Closed structure on the category of graphs, whose free models are posets that are equipped with an induction principle, which is easily formalized in type theory. We give examples. The development of computer science has given a new impulse to the theory of inductive denitions. It was classically based on set theory a la Zermelo (for a survey see [Acz77]), but the needs of the theory of data types, and those of type theory, has compelled people to look towards universal algebra and category theory for inspiration and paradigms. In particular it has been known for a long time that the notion of free structure is closely related to that of induction principle, at the very least since Lawvere's categorical axiomatization of natural numbers [Law64]. But a lot of mathematical structures, be they algebraic or topological, admit a free model, and it is also known that those that can be said to den...