Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Algebraic structure
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{Short description|Set with operations obeying given axioms}} {{more footnotes|date=August 2024}} {{Algebraic structures}} In [[mathematics]], an '''algebraic structure''' or '''algebraic system'''<ref>{{Cite book |last=F.-V. Kuhlmann (originator) |title=Encyclopedia of Mathematics |title-link=Encyclopedia of Mathematics |publisher=[[Springer-Verlag]] |isbn=1402006098 |chapter=Structure |chapter-url=https://encyclopediaofmath.org/index.php?title=Structure&oldid=14175}}</ref> consists of a nonempty [[Set (mathematics)|set]] ''A'' (called the '''underlying set''', '''carrier set''' or '''domain'''), a collection of [[operation (mathematics)|operation]]s on ''A'' (typically [[binary operation]]s such as addition and multiplication), and a finite set of [[identity (mathematics)|identities]] (known as [[axiom#Non-logical axioms|''axioms'']]) that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a [[vector space]] involves a second structure called a [[field (mathematics)|field]], and an operation called ''scalar multiplication'' between elements of the field (called ''[[scalar (mathematics)|scalars]]''), and elements of the vector space (called ''[[vector (mathematics and physics)|vectors]]''). [[Abstract algebra]] is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in [[universal algebra]]. [[Category theory]] is another formalization that includes also other [[mathematical structure]]s and [[function (mathematics)|functions]] between structures of the same type ([[homomorphism]]s). In universal algebra, an algebraic structure is called an ''algebra'';<ref>P.M. Cohn. (1981) ''Universal Algebra'', Springer, p. 41.</ref> this term may be ambiguous, since, in other contexts, [[an algebra]] is an algebraic structure that is a vector space over a [[field (mathematics)|field]] or a [[module (ring theory)|module]] over a [[commutative ring]]. The collection of all structures of a given type (same operations and same laws) is called a [[variety (universal algebra)|variety]] in universal algebra; this term is also used with a completely different meaning in [[algebraic geometry]], as an abbreviation of [[algebraic variety]]. In category theory, the collection of all structures of a given type and homomorphisms between them form a [[concrete category]].
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)