---
tag: QJGPT2W0FT
kind: concept
name: Algebraic structure
msc: [08A05, 08A62]
specializes: [983GZBQYT4]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_structure
same_as:
  wikidata: Q205464
  wikipedia: Algebraic structure
---

## Summary

An algebraic structure, or algebra, is a set together with a family of [[finitary operations|EJ8YJVM9BW]] on it.
The names and arities of the operations form a [[signature|WDVVEYD6AT]] with no relation symbols, so an algebraic structure is a [[structure|983GZBQYT4]] of a special kind.
Groups, rings, fields, lattices and vector spaces are the main examples.
A kind of algebraic structure is usually fixed by its signature and by a list of axioms, often identities, that the operations satisfy.

## Explanations

### L1 human

An algebraic structure is a set with ways to combine its elements.
The integers with addition, negation and the number $0$ form one: $+$ combines two integers, negation changes one integer, and $0$ is a fixed element.
The axioms say how these operations behave, for example $a + 0 = a$.
A [[group|G5S1M9]] is an algebraic structure with one binary operation, one unary operation and one constant that satisfy the group axioms.