Quotient algebra

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In mathematics, a quotient algebra, (where algebra is used in the sense of universal algebra), also called a factor algebra, is obtained by partitioning the elements of an algebra into equivalence classes given by a congruence relation, that is an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below.

Compatible relation[edit]

Let A be a set (of the elements of an algebra \mathcal{A}), and let E be an equivalence relation on the set A. The relation E is said to be compatible with (or have the substitution property with respect to) an n-ary operation f if for all a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in A whenever (a_1, b_1) \in E, (a_2, b_2) \in E, \ldots, (a_n, b_n) \in E implies (f (a_1, a_2, \ldots, a_n), f (b_1, b_2, \ldots, b_n)) \in E. An equivalence relation compatible with all the operations of an algebra is called a congruence.

Congruence lattice[edit]

For every algebra \mathcal{A} on the set A, the identity relation on A, and A \times A are trivial congruences. An algebra with no other congruences is called simple.

Let \mathrm{Con}(\mathcal{A}) be the set of congruences on the algebra \mathcal{A}. Because congruences are closed under intersection, we can define a meet operation:  \wedge : \mathrm{Con}(\mathcal{A}) \times \mathrm{Con}(\mathcal{A}) \to \mathrm{Con}(\mathcal{A}) by simply taking the intersection of the congruences E_1 \wedge E_2 = E_1\cap E_2.

On the other hand, congruences are not closed under union. However, we can define the closure of any binary relation E, with respect to a fixed algebra \mathcal{A}, such that it is a congruence, in the following way:  \langle E \rangle_{\mathcal{A}} = \bigcap \{ F \in \mathrm{Con}(\mathcal{A}) | E \subseteq F \}. Note that the (congruence) closure of a binary relation depends on the operations in \mathcal{A}, not just on the carrier set. Now define  \vee: \mathrm{Con}(\mathcal{A}) \times \mathrm{Con}(\mathcal{A}) \to \mathrm{Con}(\mathcal{A}) as E_1 \vee E_2 = \langle E_1\cup E_2 \rangle_{\mathcal{A}} .

For every algebra \mathcal{A}, (\mathcal{A}, \wedge, \vee) with the two operations defined above forms a lattice, called the congruence lattice of \mathcal{A}.

Quotient algebras and homomorphisms[edit]

A set A can be partitioned in equivalence classes given by an equivalence relation E, and usually called a quotient set, and denoted A/E. For an algebra \mathcal{A}, it is straightforward to define the operations induced on A/E if E is a congruence. Specifically, for any operation f^{\mathcal{A}}_i of arity n_i in \mathcal{A} (where the superscript simply denotes that it's an operation in \mathcal{A}) define f^{\mathcal{A}/E}_i : (A/E)^{n_i} \to A/E as f^{\mathcal{A}/E}_i ([a_1]_E, \ldots, [a_{n_i}]_E) = [f^{\mathcal{A}}_i(a_1,\ldots, a_{n_i})]_E, where [a]_E denotes the equivalence class of a modulo E.

For an algebra \mathcal{A} = (A, (f^{\mathcal{A}}_i)_{i \in I}), given a congruence E on \mathcal{A}, the algebra \mathcal{A}/E = (A/E, (f^{\mathcal{A}/E}_i)_{i \in I}) is called the quotient algebra (or factor algebra) of \mathcal{A} modulo E. There is a natural homomorphism from \mathcal{A} to \mathcal{A}/E mapping every element to its equivalence class. In fact, every homomorphism h determines a congruence relation; the kernel of the homomorphism,  \mathop{\mathrm{ker}}\,h = \{(a,a') \in A \times A | h(a) = h(a')\}.

Given an algebra \mathcal{A}, a homomorphism h thus defines two algebras homomorphic to \mathcal{A}, the image h(\mathcal{A}) and \mathcal{A}/\mathop{\mathrm{ker}}\,h The two are isomorphic, a result known as the homomorphic image theorem. Formally, let  h : \mathcal{A} \to \mathcal{B} be a surjective homomorphism. Then, there exists a unique isomorphism g from \mathcal{A}/\mathop{\mathrm{ker}}\,h onto \mathcal{B} such that g composed with the natural homomorphism induced by \mathop{\mathrm{ker}}\,h equals h.

See also[edit]