Math gifts

- Art Gallery -

In the mathematics of graph theory, two graphs, G and H, are called homomorphically equivalent if there exists a graph homomorphism \( {\displaystyle f\colon G\to H} \) and a graph homomorphism \( {\displaystyle g\colon H\to G} \). An example usage of this notion is that any two cores of a graph are homomorphically equivalent.

Homomorphic equivalence also comes up in the theory of databases. Given a database schema, two instances I and J on it are called homomorphically equivalent if there exists an instance homomorphism \( {\displaystyle f\colon I\to J} \) and an instance homomorphism \( {\displaystyle g\colon J\to I}. \)

In fact for any category C, one can define homomorphic equivalence. It is used in the theory of accessible categories, where "weak universality" is the best one can hope for in terms of injectivity classes; see [1]
References

Adamek and Rosicky, "Locally Presentable and Accessible Categories".

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License