Let $(A,O)$ be an algebraic system. A congruence relation, or simply a congruence $\mathfrak{C}$ on $A$

1. is an equivalence relation on $A$ ; if $(a,b)\in \mathfrak{C}$ we write $a\equiv b\pmod {\mathfrak{C}}$ , and
2. respects every $n$ -ary operator on $A$ : if $\omega_A$ is an $n$ -ary operator on $A$ ($\omega\in O$ ), and for any $a_i,b_i\in A$ , $i=1,\ldots,n$ , we have $$a_i\equiv b_i\pmod {\mathfrak{C}} \qquad\mbox{ implies }\qquad \omega_A(a_1,\ldots,a_n)\equiv \omega_A(b_1,\ldots,b_n) \pmod {\mathfrak{C}}.$$

For example, $A^2$ and $\Delta_A:=\lbrace (a,a)\mid a\in A\rbrace$ are both congruence relations on $A$ . $\Delta_A$ is called the trivial congruence (on $A$ ). A proper congruence relation is one not equal to $A^2$ .

Remarks.

• $\mathfrak{C}$ is a congruence relation on $A$ if and only if $\mathfrak{C}$ is an equivalence relation on $A$ and a subalgebra of the product $A\times A$ .
• The set of congruences of an algebraic system is a complete lattice. The meet is the usual set intersection. The join (of an arbitrary number of congruences) is the join of the underlying equivalence relations. This join corresponds to the subalgebra (of $A\times A$ ) generated by the union of the underlying sets of the congruences. The lattice of congruences on $A$ is denoted by $\operatorname{Con}(A)$ .
• (restriction) If $\mathfrak{C}$ is a congruence on $A$ and $B$ is a subalgebra of $A$ , then $\mathfrak{C}_B$ defined by $\mathfrak{C}\cap (B\times B)$ is a congruence on $B$ . The equivalence of $\mathfrak{C}_B$ is obvious. For any $n$ -ary operator $\omega_B$ inherited from $A$ 's $\omega_A$ , if $a_i \equiv b_i \pmod {\mathfrak{C}_B}$ , then $\omega_B(a_1,\ldots,a_n)=\omega_A(a_1,\ldots, a_n)\equiv \omega_A(b_1,\ldots,b_n)=\omega_B(b_1,\ldots, b_n) \pmod {\mathfrak{C}}$ . Since both $\omega_B(a_1,\ldots,a_n)$ and $\omega_B(b_1,\ldots,b_n)$ are in $B$ , $\omega_B(a_1,\ldots,a_n)\equiv \omega_B(b_1,\ldots, b_n) \pmod {\mathfrak{C}_B}$ as well. $\mathfrak{C}_B$ is the congruence restricted to $B$ .
• (extension) Again, let $\mathfrak{C}$ be a congruence on $A$ and $B$ a subalgebra of $A$ . Define $B^{\mathfrak{C}}$ by $\lbrace a\in A\mid (a,b)\in \mathfrak{C}\mbox{ and }b\in B\rbrace$ . In other words, $a\in B^{\mathfrak{C}}$ iff $a \equiv b \pmod {\mathfrak{C}}$ for some $b\in B$ . We assert that $B^{\mathfrak{C}}$ is a subalgebra of $A$ . If $\omega_A$ is an $n$ -ary operator on $A$ and $a_1,\ldots,a_n\in B^{\mathfrak{C}}$ , then $a_i\equiv b_i \pmod {\mathfrak{C}}$ , so $\omega_A(a_1,\ldots,a_n)\equiv \omega_A(b_1,\ldots,b_n) \pmod {\mathfrak{C}}$ . Since $\omega_A(b_1,\ldots,b_n)\in B$ , $\omega_A(a_1,\ldots,a_n)\in B^{\mathfrak{C}}$ . Therefore, $B^{\mathfrak{C}}$ is a subalgebra. Because $B\subseteq B^{\mathfrak{C}}$ , we call it the extension of $B$ by $\mathfrak{C}$ .
• Let $B$ be a subset of $A\times A$ . The smallest congruence $\mathfrak{C}$ on $A$ such that $a\equiv b\pmod {\mathfrak{C}}$ for all $a,b\in B$ is called the congruence generated by $B$ . $\mathfrak{C}$ is often written $\langle B\rangle$ . When $B$ is a singleton $\lbrace (a,b)\rbrace$ , then we call $\langle B\rangle$ a principal congruence, and denote it by $\langle (a,b)\rangle$ .

Quotient algebra

Given an algebraic structure $(A,O)$ and a congruence relation $\mathfrak{C}$ on $A$ , we can construct a new $O$ -algebra $(A/\mathfrak{C},O)$ , as follows: elements of $A/\mathfrak{C}$ are of the form $[a]$ , where $a\in A$ . We set $$[a]=[b]\mbox{ iff }a\equiv b\pmod {\mathfrak{C}}.$$ Furthermore, for each $n$ -ary operator $\omega_A$ on $A$ , define $\omega_{A/\mathfrak{C}}$ by $$\omega_{A/\mathfrak{C}}\big([a_1],\ldots,[a_n]\big):=\big[\omega_A(a_1,\ldots,a_n)\big].$$ It is easy to see that $\omega_{A/\mathfrak{C}}$ is a well-defined operator on $A/\mathfrak{C}$ . The $O$ -algebra thus constructed is called the quotient algebra of $A$ over $\mathfrak{C}$ .

Remark. The bracket $[\cdot]:A\to A/\mathfrak{C}$ is in fact an epimorphism, with kernel $\ker([\cdot])=\mathfrak{C}$ . This means that every congruence of an algebraic system $A$ is the kernel of some homomorphism from $A$ . $[\cdot]$ is usually written $[\cdot]_{\mathfrak{C}}$ to signify its association with $\mathfrak{C}$ .

Bibliography

1

G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).