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| 1 | Linear Transformation |
| 1 | Vector Space |
| 1 | Binary Operation |
| 1 | Operation |
| 1 | Function |
| 1 | Relation Theory |
| 1 | Identity Element |
| 1 | Groupoid |
| 1 | Group |
| 2 | Lattice |
| 2 | Monoid |
| 3 | Direct Product Of Algebras |
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Idempotency
If 
is a magma, then an element 
is said to be idempotent if 
.
For example, every identity element
is idempotent, and in a group
this is the only idempotent element.
An idempotent element is often just called an idempotent.
If every element of the magma 
is idempotent, then the binary operation
(or the magma itself) is said to be idempotent. For example, the 
and 
operations
in a lattice
are idempotent, because
and
for all 
in the lattice.
A function
is said to be idempotent if
. (This is just a special case of the first definition above, the magma in question being
, the monoid
of all functions from 
to 
with the operation of function composition.) In other words, 
is idempotent if and only if repeated application
of 
has the same effect as a single application:
for all 
. An idempotent linear transformation
from a vector space
to itself is called a projection.