Editing Homomorphism (section)

===Monomorphism===
For algebraic structures, [[monomorphism]]s are commonly defined as [[injective]] homomorphisms.<ref name="Birkhoff.1967"/>{{rp|134}} <ref name="Burris.Sankappanavar.2012"/>{{rp|29}}

In the more general context of [[category theory]], a monomorphism is defined as a [[morphism]] that is '''[[Cancellation property|left cancelable]]'''.<ref name=workmath>{{cite book | at=Exercise 4 in section I.5 | first=Saunders | last=Mac Lane| author-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | volume=5 | series=[[Graduate Texts in Mathematics]] | publisher=Springer | isbn=0-387-90036-5 | year=1971 | zbl=0232.18001 }}</ref> This means that a (homo)morphism <math>f:A \to B</math> is a monomorphism if, for any pair <math>g</math>, <math>h</math> of morphisms from any other object <math>C</math> to <math>A</math>, then <math>f \circ g = f \circ h</math> implies <math>g = h</math>.

These two definitions of ''monomorphism'' are equivalent for all common algebraic structures. More precisely, they are equivalent for [[field (mathematics)|fields]], for which every homomorphism is a monomorphism, and for [[variety (universal algebra)|varieties]] of [[universal algebra]], that is algebraic structures for which operations and axioms (identities) are defined without any restriction (the fields do not form a variety, as the [[multiplicative inverse]] is defined either as a [[unary operation]] or as a property of the multiplication, which are, in both cases, defined only for nonzero elements).

In particular, the two definitions of a monomorphism are equivalent for [[set (mathematics)|sets]], [[magma (algebra)|magmas]], [[semigroup]]s, [[monoid]]s, [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]], [[vector space]]s and [[module (mathematics)|modules]].

A '''[[split monomorphism]]''' is a homomorphism that has a [[inverse function#Left and right inverses|left inverse]] and thus it is itself a right inverse of that other homomorphism. That is, a homomorphism <math>f\colon A \to B</math> is a split monomorphism if there exists a homomorphism <math>g\colon B \to A</math> such that <math>g \circ f = \operatorname{Id}_A.</math> A split monomorphism is always a monomorphism, for both meanings of ''monomorphism''. For sets and vector spaces, every monomorphism is a split monomorphism, but this property does not hold for most common algebraic structures.

{{collapse top|Proof of the equivalence of the two definitions of monomorphisms}}
''An injective homomorphism is left cancelable'': If <math>f\circ g = f\circ h,</math> one has <math>f(g(x))=f(h(x))</math> for every <math>x</math> in <math>C</math>, the common source of <math>g</math> and <math>h</math>. If <math>f</math> is injective, then <math>g(x) = h(x)</math>, and thus <math>g = h</math>. This proof works not only for algebraic structures, but also for any [[category (mathematics)|category]] whose objects are sets and arrows are maps between these sets. For example, an injective continuous map is a monomorphism in the category of [[topological space]]s.

For proving that, conversely, a left cancelable homomorphism is injective, it is useful to consider a ''[[free object]] on <math>x</math>''. Given a [[variety (universal algebra)|variety]] of algebraic structures a free object on <math>x</math> is a pair consisting of an algebraic structure <math>L</math> of this variety and an element <math>x</math> of <math>L</math> satisfying the following [[universal property]]: for every structure <math>S</math> of the variety, and every element <math>s</math> of <math>S</math>, there is a unique homomorphism <math>f: L\to S</math> such that <math>f(x) = s</math>. For example, for sets, the free object on <math>x</math> is simply <math>\{x\}</math>; for [[semigroup]]s, the free object on <math>x</math> is <math>\{x, x^2, \ldots, x^n, \ldots\},</math> which, as, a semigroup, is isomorphic to the additive semigroup of the positive integers; for [[monoid]]s, the free object on <math>x</math> is <math>\{1, x, x^2, \ldots, x^n, \ldots\},</math> which, as, a monoid, is isomorphic to the additive monoid of the nonnegative integers; for [[group (mathematics)|group]]s, the free object on <math>x</math> is the [[infinite cyclic group]] <math>\{\ldots, x^{-n}, \ldots, x^{-1}, 1, x, x^2, \ldots, x^n, \ldots\},</math> which, as, a group, is isomorphic to the additive group of the integers; for [[ring (mathematics)|rings]], the free object on <math>x</math> is the [[polynomial ring]] <math>\mathbb{Z}[x];</math> for [[vector space]]s or [[module (mathematics)|modules]], the free object on <math>x</math> is the vector space or free module that has <math>x</math> as a basis.

''If a free object over <math>x</math> exists, then every left cancelable homomorphism is injective'': let <math>f\colon A \to B</math> be a left cancelable homomorphism, and <math>a</math> and <math>b</math> be two elements of <math>A</math> such <math>f(a) = f(b)</math>. By definition of the free object <math>F</math>, there exist homomorphisms <math>g</math> and <math>h</math> from <math>F</math> to <math>A</math> such that <math>g(x) = a</math> and <math>h(x) = b</math>. As <math>f(g(x)) = f(h(x))</math>, one has <math>f \circ g = f \circ h, </math> by the uniqueness in the definition of a universal property. As <math>f</math> is left cancelable, one has <math>g = h</math>, and thus <math>a = b</math>. Therefore, <math>f</math> is injective.

''Existence of a free object on <math>x</math> for a [[variety (universal algebra)|variety]]'' (see also {{slink|Free object|Existence}}): For building a free object over <math>x</math>, consider the set <math>W</math> of the [[well-formed formula]]s built up from <math>x</math> and the operations of the structure. Two such formulas are said equivalent if one may pass from one to the other by applying the axioms ([[identity (mathematics)|identities]] of the structure). This defines an [[equivalence relation]], if the identities are not subject to conditions, that is if one works with a variety. Then the operations of the variety are well defined on the set of [[equivalence class]]es of <math>W</math> for this relation. It is straightforward to show that the resulting object is a free object on <math>x</math>.
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