Author:
(1) Thierry Boy de la Tour, Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG 38000 Grenoble, France. Author: Author: (1) Thierry Boy de la Tour, Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG 38000 Grenoble, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Basic Definitions and Notations 2.1 Sets 2.1 Sets 2.2 Sequences 2.2 Sequences 2.3 Signatures and Algebras and 2.4 Categories 2.3 Signatures and Algebras and 2.4 Categories 3 Monographs and their Morphisms 3 Monographs and their Morphisms 4 Limits and Colimits 4 Limits and Colimits 5 Drawing Monographs 5 Drawing Monographs 6 Graph Structures and Typed Monographs 6 Graph Structures and Typed Monographs 7 Submonographs and Partial Morphisms 7 Submonographs and Partial Morphisms 8 Algebraic Transformations of Monographs 8 Algebraic Transformations of Monographs 9 Attributed Typed Monographs 9 Attributed Typed Monographs 10 Conclusion and References 10 Conclusion and References 7 Submonographs and Partial Morphisms Graph structures have been characterized in [3] as the signatures that allow the transformation of the corresponding algebras by the single pushout method. This method is based on the construction of pushouts in categories of partial homomorphisms, defined as standard homomorphisms from subalgebras of their domain algebra, just as partial functions are standard functions from subsets of their domain (in the categorical theoretic sense of the word domain). The results of Section 6 suggest that a similar approach can be followed with monographs. We first need a notion of submonograph, their (inverse) image by morphisms and restrictions of morphisms to submonographs. We may now define the notion of partial morphisms of monographs, with a special notation in order to distinguish them from standard morphisms, and their composition. We now see how these inverse images allow to formulate a sufficient condition ensuring that restrictions of coequalizers are again coequalizers. It is then easy to obtain a similar result on pushouts. We can now show that categories of partial morphisms of monographs have pushouts. The following construction is inspired by [3, Construction 2.6, Theorem 2.7] though the proof uses pushout restriction. Theorem 7.5. The categories of Definition 7.2 have pushouts. Theorem 7.5. The categories of Definition 7.2 have pushouts. If B and C are finite (resp. standard, resp. O-monographs) then so are X and Y, hence so is Q by Theorem 4.4. One important feature of this construction is illustrated below. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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Monographs, Their Morphisms, and the Rules That Bind Them

A Guide to Naming and Mapping Monographs

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Partial Morphisms and Their Impact on Monograph Theory

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