Example

(1) In the set $\mathbf{N}$ of natural numbers 0 is an identity element under addition and 1 is an identity element under multiplication. Each of these two laws gives $\mathbf{N}$ a commutative monoid structure (Set Theory, III, § 3, no. 3).

(2) In the set of subsets of a set $E$, $\varnothing$ is an identity element under the law $\cup$ and $E$ under the law $\cap$. More generally, in a lattice the least element, if it exists, is identity element under the law sup; conversely, if there exists an identity element under this law it is the least element of the set. Similarly for the greatest element and the law inf.

(3) The set $\mathbf{N}$ has no identity element under the law $(x, y) \mapsto x^y$. Under the law $(X, Y) \mapsto X \circ Y$ between subsets of $E \times E$ the diagonal $\Delta$ is the identity element. Under the law $(f, g) \mapsto f \circ g$ between mappings of $E$ into $E$ the identity mapping of $E$ onto $E$ is the identity element.

(4) Let $E$ be a magma and $R$ an equivalence relation on $E$ compatible with the law on $E$ (§ 1, no. 6). If $e$ is an identity element of $E$ the canonical image of $e$ in $E/R$ is an identity element of the magma $E/R$.

The identity element of a unital magma is a unital homomorphism; the composition of two unital homomorphisms is also one. For a mapping to be a unital magma isomorphism it is necessary and sufficient that it be a bijective unital homomorphism and the inverse mapping is then a unital homomorphism. Let $E$ and $E'$ be unital magmas and $e'$ the identity element of $E'$; the constant mapping of $E$ into $E'$ mapping $E$ to $e'$ is a unital homomorphism, called a trivial homomorphism.

The product of a family of unital magmas (resp. monoids) is a unital magma (resp. monoid).

Every quotient magma of a unital magma (resp. monoid) is a unital magma (resp. monoid).

Let $E$ be a unital magma and $e$ its identity element. A submagma $A$ of $E$ such that $e \in A$ is called a unital submagma of $E$. Clearly $e$ is the identity element of the magma $A$. Every intersection of unital submagmas of $E$ is a unital submagma of $E$. If $X$ is a subset of $E$ then there exists a smallest unital submagma of $E$ containing $X$; it is called the unital submagma of $E$ generated by $X$; it is equal to $\{e\}$ if $X$ is empty. If $E$ is a monoid, a unital submagma of $E$ is called a submonoid of $E$.

If $F$ is a magma without identity element a submagma of $F$ may possess an identity element. For example, if $F$ is associative and $h$ is an idempotent element of $F$ (I, § 1, no. 4), the set of $h \uparrow x \uparrow h$, where $x$ runs through $F$, is a submagma of $F$ with $h$ as identity element.

If $E$ is a magma with identity element $e$, a submagma $A$ of $E$ such that $e \notin A$ may still possess an identity element.