Definition 11

Let E be a set. A law of composition T and an equivalence relation R on E are said to be compatible if the relations x ≡ x' (mod R) and y ≡ y' (mod R) (for x, x', y, y' ∈ E) imply x T y ≡ x' T y' (mod R); the law of composition on the quotient set E/R which maps the equivalence classes of x and y to the equivalence class of x T y is called the quotient law of the law T with respect to R. The set E/R with the quotient law is called the quotient magma of E with respect to R.

To say that an equivalence relation R on E is compatible with the internal law of composition f: E × E → E on E means that the mapping f is compatible (in the sense of Set Theory, II, § 6, no. 5) with the product equivalence relation R × R on E × E and the equivalence relation R on E. (Set Theory, II, § 6, no. 8). This also means that the graph of R is a submagma of E × E.

If the law T is associative (resp. commutative) so is the quotient law (more briefly we say that associativity, or commutativity, is preserved when passing to the quotient).

The canonical mapping from the magma E to the magma E/R is a homomorphism.

For a mapping g of E/R into a magma F to be a homomorphism it is necessary and sufficient that the composition of g with the canonical mapping of E onto E/R be a homomorphism.

The two following propositions are immediate from the definitions: