. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the well-known characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are one-to-one with the arrows of the free gs-monoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gs-monoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...

Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.

. We address the problem of defining an algebraic framework for modularization supporting separate type-checking. In order to do that we introduce the notions of abstract type system and logic of constraints and we present a canonical construction of a model part, on top of a logic of constraints. This canonical construction works under standard assumptions on the underlying type system. We show that the framework is suitable for defining the static and dynamic semantics of module languages, by giving a concrete example of construction on top of the type system of a simple typed module language. As a result, the subtyping relation between module interfaces is captured in a natural way by the notion of signature morphism. Introduction Modularization has been considered since the early 70s an essential principle for managing the complex task of software development [29]. Nowadays there exist many modular programming languages offering rather advanced features for modularization. Neverth...

: The composition of modular specifications can be modeled, in a category theoretic framework, by colimits of diagrams. Pushouts in particular describe the combination of two specifications sharing a common part. This work extends this classic idea along three lines. First, we define a term language to represent modular specifications built with colimit constructions over a category of base specifications. This language is formally characterized by a finitely cocomplete category. Then, we propose to associate with each term a diagram. This interpretation provides us with a more abstract representation of modular specifications because irrelevant steps of the construction are eliminated. We define a category of diagrams, which is a completion of the base category with finite colimits. We prove that the interpretation of terms as diagrams defines an equivalence between the corresponding categories, which shows the correctness of this interpretation. At last, we propose an algorithm to no...