Embedding of structures

Builds on Structure Homomorphism Injective function Substructure Isomorphism

Definition

Let $\sigma$ be a signature and let $\mathbf{A}$ and $\mathbf{B}$ be $\sigma$ -structures with domains $A$ and $B$ . An embedding of $\mathbf{A}$ into $\mathbf{B}$ is a map $h \colon A \to B$ such that:

$h$ is a homomorphism from $\mathbf{A}$ to $\mathbf{B}$ ;
$h$ is injective;
for each relation symbol $R \in \sigma$ of arity $n$ and all $a_1, \dots, a_n \in A$ , if $(h(a_1), \dots, h(a_n)) \in R^{\mathbf{B}}$ then $(a_1, \dots, a_n) \in R^{\mathbf{A}}$ .

Remarks

A map $h$ is an embedding if and only if its image is the domain of a substructure $\mathbf{C}$ of $\mathbf{B}$ and $h$ is an isomorphism from $\mathbf{A}$ to $\mathbf{C}$ . For a signature without relation symbols, clause 3 is empty, and an embedding is an injective homomorphism.

Example: for the signature $\set{+, \cdot, 0, 1}$ , the inclusion $\Z \to \Q$ is an embedding, because it is injective and respects sums, products, $0$ and $1$ .

Non-example: view graphs as structures with one binary relation. Let $A$ be the graph with vertices $1, 2$ and no edge, and $B$ the graph with vertices $1, 2$ and one edge between them. The identity map is an injective homomorphism from $A$ to $B$ but not an embedding, because clause 3 fails for the pair $(1, 2)$ .

Reading path

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Homomorphism concept
Injective function concept
Isomorphism concept
Structure concept
Substructure concept
Embedding of structures definition

Tag	`BQGK1FHSKB`
URI	`m2://BQGK1FHSKB`
Kind	definition
Writing	W0 stub
MSC	08A05, 03C07
Defines	Embedding of structures
Uses	Structure, Homomorphism, Injective function, Substructure, Isomorphism

Embedding of structures

Definition

Remarks

Reading path

Further reading

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