Editing Algebraic structure (section)

== Common algebraic structures ==
{{Main|Outline of algebraic structures#Types of algebraic structures}}

=== One set with operations ===

'''Simple structures''': '''no''' [[binary operation]]:

* [[Set (mathematics)|Set]]: a degenerate algebraic structure ''S''   having no operations.

'''Group-like structures''': '''one''' binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers.

* [[Group (mathematics)|Group]]: a [[monoid]] with a unary operation (inverse), giving rise to [[inverse element]]s.
* [[Abelian group]]: a group whose binary operation is [[commutative]].

'''Ring-like structures''' or '''Ringoids''': '''two''' binary operations, often called [[addition]] and [[multiplication]], with multiplication [[distributivity|distributing]] over addition.

* [[Ring (mathematics)|Ring]]: a semiring whose additive monoid is an abelian group.
* [[Division ring]]: a [[zero ring|nontrivial]] ring in which [[division (mathematics)|division]] by nonzero elements is defined.
* [[Commutative ring]]: a ring in which the multiplication operation is commutative.
* [[field (mathematics)|Field]]: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element).

'''Lattice structures''': '''two''' or more binary operations, including operations called [[meet and join]], connected by the [[absorption law]].<ref>Ringoids and [[Lattice (order)|lattice]]s can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the [[distributive law]]; in the case of lattices, they are linked by the [[absorption law]]. Ringoids also tend to have numerical [[model theory|model]]s, while lattices tend to have [[set theory|set-theoretic]] models.
</ref>

* [[Complete lattice]]: a lattice in which arbitrary [[meet and join]]s exist.
* [[Bounded lattice]]: a lattice with a [[greatest element]] and least element.
* [[Distributive lattice]]: a lattice in which each of meet and join [[distributive lattice|distributes]] over the other. A [[power set]] under union and intersection forms a distributive lattice.
* [[Boolean algebra (structure)|Boolean algebra]]: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.

=== Two sets with operations ===

* [[module (mathematics)|Module]]: an abelian group ''M'' and a ring ''R'' acting as operators on ''M''. The members of ''R'' are sometimes called [[scalar (mathematics)|scalar]]s, and the binary operation of ''scalar multiplication'' is a function ''R''&nbsp;×&nbsp;''M'' → ''M'', which satisfies several axioms. Counting the ring operations these systems have at least three operations.
* [[Vector space]]: a module where the ring ''R'' is a [[field (mathematics)|field]] or, in some contexts, a [[division ring]].

* [[Algebra over a field]]: a module over a field, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and [[Bilinear map|linearity]] with respect to multiplication.
* [[Inner product space]]: a field ''F'' and vector space ''V'' with a [[definite bilinear form]] {{nowrap|''V'' × ''V'' → ''F''}}.