Quotient algebra

Builds on Algebraic structure Congruence relation Quotient set

Definition

Let $\mathbf{A}$ be an algebra with universe $A$ and let $\theta$ be a congruence on $\mathbf{A}$ . For $a \in A$ , let $a/\theta$ be the class of $a$ , and let $A/\theta$ be the quotient set of all classes. The quotient algebra $\mathbf{A}/\theta$ has universe $A/\theta$ and the same signature as $\mathbf{A}$ . Each $n$ -ary fundamental operation $f$ is interpreted by $f^{\mathbf{A}/\theta}(a_1/\theta, \dots, a_n/\theta) = f^{\mathbf{A}}(a_1, \dots, a_n)/\theta.$

Remarks

The operations are well defined because $\theta$ is compatible with each $f$ . The map $\nu_\theta \colon a \mapsto a/\theta$ is a surjective homomorphism $\mathbf{A} \to \mathbf{A}/\theta$ , the natural map.

Example: for the ring $\Z$ and congruence modulo $n$ , the quotient $\Z/\theta$ is the ring $\Z/n\Z$ . For instance $(3/\theta) \cdot (4/\theta) = 12/\theta = 2/\theta$ when $n = 5$ .

Non-example: for an equivalence relation that is not compatible, the formula gives no operation. Take $(\Z, +)$ with the classes "negative", "zero" and "positive". Computing with $1$ and $-1$ gives the class of $0$ , but computing with $2$ and $-1$ gives the class of $1$ .

For a group $G$ and the congruence of a normal subgroup $N$ , the quotient algebra is the quotient group $G/N$ .

Reading path

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Tag	`10TJ890QK0`
URI	`m2://10TJ890QK0`
Kind	definition
Writing	W0 stub
MSC	08A30
Defines	Quotient algebra
Uses	Algebraic structure, Congruence relation, Quotient set

Quotient algebra

Definition

Remarks

Reading path

Further reading

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