In the

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, two graphs, ''G'' and ''H'', are called homomorphically equivalent if there exists a graph homomorphism

f\colon G\to H

and a graph homomorphism

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 In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases spa ...

. Given a

database schema The database schema is the structure of a database described in a formal language supported by the database management system (DBMS). The term "schema" refers to the organization of data as a blueprint of how the database is constructed (divide ...

, two instances I and J on it are called homomorphically equivalent if there exists an instance homomorphism

f\colon I\to J

and an instance homomorphism

g\colon J\to I

. In fact for any

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

''C'', one can define homomorphic equivalence. It is used in the theory of

accessible categories The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects. The theory originates i ...

, where "weak universality" is the best one can hope for in terms of injectivity classes; see Adamek and Rosicky, "Locally Presentable and Accessible Categories".

References

{{math-stub Graph theory Equivalence (mathematics)