There are several different definitions of the term "unital" 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 collection 
consisting of 
points and arranged into subsets 
so that 
for all 
and every pair of distinct points 
is contained in exactly one 
.
A completely separate notion of unital is used ubiquitously throughout abstract algebra as an adjective to refer to an algebraic structure which contains a unit, e.g., a unitary ring is a ring
which contains at least one unit. Algebraic structures of this kind are sometimes
called unitary, though caution must be exhibited due to numerous unrelated mathematical
notions which are themselves called unitary, e.g., unitary matrices which collectively form the unitary
group, unitary elements, unitary
divisors, etc. One must also exhibit caution when consulting literature on algebraic
topics because even there, some variation will be found with respect to usage of
the term unital. For example, some authors reserve the term unital to refer to a
structure which has non-identity units while other authors apply the term to some
algebraic structures (e.g., magmas and algebras)
solely on the basis that they possess a multiplicative
identity. This confusion is compounded even farther by the fact that some authors
use the term unital in category
theory to apply to a class of natural transformations
which satisfy certain commutative properties
(Dieck 2000), while other authors use the term unital
to apply to R-modules 
satisfying 
for all 
where here, 
is a ring with identity 
(Dummit and Foote 2003).
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 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, rings, modules, etc., with 
a homomorphism of
these respective structures.
In functional analysis, similar notions of unital may be implemented to describe either a function algebra 
which contains an identity operator 
or a map 
between function algebras 
and 
having identity operators 
and 
, respectively, which satisfies the
condition that 
.
Likewise, if 
is a map from a 
-algebra
to itself, then 
is unital precisely if 
.
Another unrelated though similarly-named property of Banach algebras is that of being stably unital.
