---
tag: 676H7EGTKJ
kind: concept
name: Substructure
msc: [03C07]
sources:
  - url: https://en.wikipedia.org/wiki/Substructure_(mathematics)
same_as:
  wikidata: Q859254
  wikipedia: Substructure (mathematics)
---

## Summary

A substructure of a [[structure|983GZBQYT4]] is a subset of its domain that is closed under the functions and contains the constants.
Each symbol is interpreted on it by restriction.
It is the common form of subgroups, subrings and induced subgraphs.
Which subsets are substructures depends on the signature, not only on the structure.

## Explanations

### L1 human

Take a structure and keep only some of its elements.
If applying the operations to kept elements always gives kept elements, and the constants are kept, the kept part is again a structure of the same kind.
For example, the even integers with addition, negation and $0$ form a substructure of $(\Z, +, -, 0)$.
In a graph seen as a structure with one edge relation, any nonempty set of vertices with the edges between them is a substructure.