Mathematical logic and foundations Definition

Substructure

Definition

Let $L$ be a signature and let $\mathcal{M}$ and $\mathcal{N}$ be $L$ -structures with domains $M$ and $N$ . The structure $\mathcal{N}$ is a substructure of $\mathcal{M}$ , written $\mathcal{N} \subseteq \mathcal{M}$ , when the following hold.

$N$ is a subset of $M$ .
For each $n$ -ary function symbol $f$ , the map $f^{\mathcal{N}}$ is the restriction of $f^{\mathcal{M}}$ to $N^n$ .
For each $n$ -ary relation symbol $R$ , $R^{\mathcal{N}} = R^{\mathcal{M}} \cap N^n$ .
For each constant symbol $c$ , $c^{\mathcal{N}} = c^{\mathcal{M}}$ .

Remarks

Clause 2 forces $N$ to be closed under each $f^{\mathcal{M}}$ , and clause 4 forces $N$ to contain each $c^{\mathcal{M}}$ . Conversely, a nonempty subset of $M$ with these two closure properties is the domain of exactly one substructure of $\mathcal{M}$ .

Example: in the signature $\set{+, -, 0}$ , the set $2\Z$ of even integers is closed under $+$ and $-$ and contains $0$ . So it carries a substructure of $(\Z, +, -, 0)$ .

Non-example: the set $\N$ is not the domain of a substructure of $(\Z, +, -, 0)$ , because $-1 \notin \N$ although $1 \in \N$ . In the smaller signature $\set{+, 0}$ , the set $\N$ is the domain of a substructure of $(\Z, +, 0)$ .

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Signature concept
Structure concept
Subset concept
Substructure definition

Tag	`JS7BWX3HW3`
URI	`m2://JS7BWX3HW3`
Kind	definition
Writing	W0 stub
MSC	03C07
Defines	Substructure
Uses	Structure, Signature, Subset
Changed	2 November 2026

Substructure

Definition

Remarks

Reading path

Further reading

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