A Guide to Naming and Mapping Monographs

Table of Links

Abstract and 1 Introduction

2 Basic Definitions and Notations

2.1 Sets

2.2 Sequences

2.3 Signatures and Algebras and 2.4 Categories

3 Monographs and their Morphisms

4 Limits and Colimits

5 Drawing Monographs

6 Graph Structures and Typed Monographs

7 Submonographs and Partial Morphisms

8 Algebraic Transformations of Monographs

9 Attributed Typed Monographs

10 Conclusion and References

5 Drawing Monographs

It is sometimes necessary to name the edges in a drawing. We may then adopt the convention sometimes used for drawing diagrams in a category: the bullets are replaced by the names of the corresponding nodes, and arrows are interrupted to write their name at a place free from crossing, as in

Note that no confusion is possible between the names of nodes and those of other edges, e.g., in

it is clear that x and z are nodes since arrow tips point to them, and that y is the name of an edge of length 3.

One particularity of monographs is that edges can be adjacent to themselves, as in

Of course, knowing that a is a morphism sometimes allows to deduce the type of an edge, possibly from the types of adjacent edges. In the present case, indicating a single type would have been enough to deduce all the others.

In particular applications it may be convenient to adopt completely different ways of drawing (typed) monographs.

6 Graph Structures and Typed Monographs

The next lemma is central as it shows that no graph structure is omitted by the functor S if we allow sort-preserving isomorphisms of graph structures. We assume the Axiom of Choice through its equivalent formulation known as the Numeration Theorem [5].

It is therefore clear that if S were full it would be an equivalence of categories, but this is not the case as we now illustrate on graphs.

The type indicated by the syntax (and consistent with the drawings of E-graphs in [2]) is of course T1.