Quotient algebra

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 in equivalence classes given by a congruence, that is an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below.

Compatible operation

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

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

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 defined 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, $\mathop{\mathrm{ker}}\,h = \{(a,a') \in A \times A | f(a) = f(a')\}$.

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

• quotient ring

References

• Klaus Denecke; Shelly L. Wismath (2009). Universal algebra and coalgebra. World Scientific. pp. 14–17. ISBN 9789812837455. http://books.google.com/books?id=NgTAzhC8jVAC&pg=PA14. 
• Purna Chandra Biswal (2005). Discrete mathematics and graph theory. PHI Learning Pvt. Ltd.. p. 215. ISBN 9788120327214. http://books.google.com/books?id=hLX6OG1U5W8C&pg=PA215.