Editing Algebraic structure (section)

=== 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.
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* [[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.