The term "unital" has several different definitions used throughout various branches of mathematics.
In geometric combinatorics, a block design of the form (
,
, 1) is said to be a unital. In particular,
then, a unital is a set 
consisting of 
points and arranged into subsets 
so that 
for all 
and every pair of distinct points 
is contained in exactly
one 
.
In abstract algebra, unital means that an algebraic structure has a multiplicative identity.
For example, a unital ring has an element 
with 
for every element 
. This should be distinguished from a unit,
which is an invertible element of a ring
with multiplicative identity. No nonidentity
unit is required for a ring to
be unital.
The adjective unitary is sometimes used with this meaning, but it also occurs in unrelated terminology such as unitary matrices
and unitary divisors. There are also unital
natural transformations in category theory,
characterized by unit compatibility commutative
diagrams (tom Dieck 2004). A unital R-module
over a ring 
with multiplicative
identity satisfies 
for every 
(Dummit and Foote 2004).
The term unital is also commonly used relative to function-theoretic properties in algebra as well. For example, when discussing maps
between some structures 
and 
with multiplicative
identities 
and 
, respectively, the term unital is
often used to describe a map 
for which
 |
In this scenario, it may happen that 
and 
are magmas, algebras, or rings, with 
a homomorphism of
these respective structures.
In functional analysis, algebras of functions often use the constant
function 1 as their multiplicative identity,
while algebras of linear
operators often use the identity operator.
When a subalgebra has a different multiplicative
identity, unitality refers to that identity element.
A map between unital algebras
is unital if it preserves their multiplicative
identities. In particular, a map 
on a unital C*-algebra
with multiplicative identity 
is unital iff 
.
Another unrelated though similarly-named property of Banach algebras is that of being stably unital.
