Definition:Canonical Injection (Abstract Algebra)

Definition

Let $\left({S_1, \circ_1}\right)$ and $\left({S_2, \circ_2}\right)$ be algebraic structures with identities $e_1, e_2$ respectively.

The following mappings:

$\operatorname{in}_1: \left({S_1, \circ_1}\right) \to \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right): \forall x \in S_1: \operatorname{in}_1 \left({x}\right) = \left({x, e_2}\right)$
$\operatorname{in}_2: \left({S_2, \circ_2}\right) \to \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right): \forall x \in S_2: \operatorname{in}_2 \left({x}\right) = \left({e_1, x}\right)$

are called the canonical injections.

General Definition

Let $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_j, \circ_j}\right), \ldots, \left({S_n, \circ_n}\right)$ be algebraic structures with identities $e_1, e_2, \ldots, e_j, \ldots, e_n$ respectively.

Then the canonical injection $\displaystyle \operatorname{in}_j: \left({S_j, \circ_j}\right) \to \prod_{i \mathop = 1}^n \left({S_i, \circ_i}\right)$ is defined as:

$\operatorname{in}_j \left({x}\right) = \left({e_1, e_2, \ldots, e_{j-1}, x, e_{j+1}, \ldots, e_n}\right)$

Also known as

The canonical injections are also referred to by some sources as natural monomorphisms.