Quotient (universal algebra)

Quotient algebras and homomorphisms

Any equivalence relation E in a set A partitions this set in equivalence classes. The set of these equivalence classes is usually called the quotient set, and denoted A/E. For an algebra ${\displaystyle \mathcal{A}}$, it is straightforward to define the operations induced on the elements of A/E if E is a congruence. Specifically, for any operation ${\displaystyle f_i^{\mathcal{A}}}$ of arity ${\displaystyle n_i}$ in ${\displaystyle \mathcal{A}}$ (where the superscript simply denotes that it is an operation in ${\displaystyle \mathcal{A}}$, and the subscript ${\displaystyle i\in I}$ enumerates the functions in ${\displaystyle \mathcal{A}}$ and their arities) define ${\displaystyle f_i^{\mathcal{A}/E}:(A/E{)}^{n_i}\to A/E}$ as ${\displaystyle f_i^{\mathcal{A}/E}([a_1{]}_E,\dots ,[a_{n_i}{]}_E)=[f_i^{\mathcal{A}}(a_1,\dots ,a_{n_i}){]}_E}$, where ${\displaystyle [x{]}_E\in A/E}$ denotes the equivalence class of ${\displaystyle x\in A}$ generated by E ("x modulo E").

For an algebra ${\displaystyle \mathcal{A}=(A,(f_i^{\mathcal{A}}{)}_{i\in I})}$, given a congruence E on ${\displaystyle \mathcal{A}}$, the algebra ${\displaystyle \mathcal{A}/E=(A/E,(f_i^{\mathcal{A}/E}{)}_{i\in I})}$ is called the quotient algebra (or factor algebra) of ${\displaystyle \mathcal{A}}$ modulo E. There is a natural homomorphism from ${\displaystyle \mathcal{A}}$ to ${\displaystyle \mathcal{A}/E}$ mapping every element to its equivalence class. In fact, every homomorphism h determines a congruence relation via the kernel of the homomorphism, ${\displaystyle \mathrm{k}\mathrm{e}\mathrm{r}h=\{(a,a^{\prime })\in A^2|h(a)=h(a^{\prime })\}\subseteq A^2}$.

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

Congruence lattice

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

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

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 ${\displaystyle \mathcal{A}}$, such that it is a congruence, in the following way: ${\displaystyle ⟨E{⟩}_{\mathcal{A}}=\bigcap \{F\in \mathrm{C}\mathrm{o}\mathrm{n}(\mathcal{A})\mid E\subseteq F\}}$. Note that the closure of a binary relation is a congruence and thus depends on the operations in ${\displaystyle \mathcal{A}}$, not just on the carrier set. Now define ${\displaystyle \vee :\mathrm{C}\mathrm{o}\mathrm{n}(\mathcal{A})\times \mathrm{C}\mathrm{o}\mathrm{n}(\mathcal{A})\to \mathrm{C}\mathrm{o}\mathrm{n}(\mathcal{A})}$ as ${\displaystyle E_1\vee E_2=⟨E_1\cup E_2{⟩}_{\mathcal{A}}}$.

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

Maltsev conditions

If two congruences permute (commute) with the composition of relations as operation, i.e. ${\displaystyle \alpha \circ \beta =\beta \circ \alpha }$, then their join (in the congruence lattice) is equal to their composition: ${\displaystyle \alpha \circ \beta =\alpha \vee \beta }$. An algebra is called congruence permutable if every pair of its congruences permutes; likewise a variety is said to be congruence-permutable if all its members are congruence-permutable algebras.

In 1954, Anatoly Maltsev established the following characterization of congruence-permutable varieties: a variety is congruence permutable if and only if there exist a ternary term q(x, y, z) such that q(x, y, y) ≈ x ≈ q(y, y, x); this is called a Maltsev term and varieties with this property are called Maltsev varieties. Maltsev's characterization explains a large number of similar results in groups (take q = xy⁻¹z), rings, quasigroups (take q = (x / (y \ y))(y \ z)), complemented lattices, Heyting algebras etc. Furthermore, every congruence-permutable algebra is congruence-modular, i.e. its lattice of congruences is modular lattice as well; the converse is not true however.

After Maltsev's result, other researchers found characterizations based on conditions similar to that found by Maltsev but for other kinds of properties. In 1967 Bjarni Jónsson found the conditions for varieties having congruence lattices that are distributive (thus called congruence-distributive varieties), while in 1969 Alan Day did the same for varieties having congruence lattices that are modular. Generically, such conditions are called Maltsev conditions.

This line of research led to the Pixley–Wille algorithm for generating Maltsev conditions associated with congruence identities.