---
tag: 983GZBQYT4
kind: concept
name: Structure
msc: [03C07]
sources:
  - url: https://en.wikipedia.org/wiki/Structure_(mathematical_logic)
same_as:
  wikidata: Q1851710
  wikipedia: Structure (mathematical logic)
---

## Summary

A structure for a [[signature|WDVVEYD6AT]] is a set together with a meaning for each symbol of the signature.
Each symbol becomes a function, a relation or an element of the set.
Structures are the objects in which first-order formulas are true or false.
Groups, rings, ordered sets and graphs are all structures for suitable signatures.

## Explanations

### L1 human

A signature is a list of names, such as "$+$", "$0$" and "$<$", with the number of inputs of each name.
A structure picks a set and says what each name means on that set.
For example, the integers with ordinary addition, the number zero and the usual order form a structure for the signature with $+$, $0$ and $<$.
The same signature can have many different structures, such as the rational numbers with the same symbols.