**D**is functional, i.e. Ψ_{D}is injective;*D*has an injective interpretation in some algebra;- For any two distinct elements of
*D*there is an algebra interpreting them differently.

2.⇒3.

3.⇒1. ∀

Any draft *D* can be reduced to a condensed draft as follows.

Then the interpretation of

Let us call

This equivalence relation is included in that of any interpretation of

We can also compare separately given terms by reducing them to this case as any disjoint union of drafts (only keeping variables in common) is a draft.

IfThe condition defined by an equational system is equivalent to : for any equality between terms found true in

Indeed, the morphism

Conversely, any formula made of universal quantifiers over an equality of terms

If the family (

- Each
*K*was only an_{i}*L*-algebra, not necessarily satisfying (*K*,_{j}*B*). Can we replace (_{j}*K*,_{i}*B*) by some (_{i}*K'*,_{i}*B'*) where_{i}*K'*satisfies all (_{i}*K*,_{j}*B*), but that remains equivalent as a predicate on the class of algebras satisfying all (_{j}*K*,_{j}*B*) for_{j}*j*≠*i*? - Can two conditions (
*K*,*B*) and (*K*',*B'*) with the same arity (*B*and*B'*are in bijection) be fused into one ?

These questions will be positively answered below.

**Theorem.** For any *L*-algebra *A* and any
variety *V* of *L*-algebras, the category of all (*M*,*f*)
where *M* is an *L*-algebra in *V* and *f*
∈Mor_{L}(*A*,*M*), has an initial object (*A*',φ), called the quotient of *A* by the
family of conditions defining the variety.

Proof.

LetNow we can answer the previous questions.Hbe the set of all equivalence relationshonAsuch thatA/his anL-algebra inV, with the canonical projectionp_{h}∈Mor_{L}(A,A/h).

Let φ =∏_{h}_{∈}_{H}p_{h}andA' = Im φ ⊂P=∏_{h}_{∈}_{H}A/h.

A' satisfies all (K,_{i}B)_{i}_{i∈I}because it is a subalgebra of a productPofL-algebras that do.

To verify that (A',φ) is an initial object in this category, let (M,f) be another object. We need to show that there is a unique morphismgfrom (A',φ) to (M,f), i.e.g∈Mor_{L}(A',M) such thatg০φ=f.

In the work on quotients (2.9) we saw the existence of a uniquegsuch thatg০φ=f, writteng=f/φ with domain Im φ =A', provided that ~_{φ}⊂~_{f}. The condition of uniqueness here is Im φ =A'. The existence is obtained asg=j০π_{h}, whereh= ~_{f}, π_{h}∈Mor_{L}(A',A/h) is the restriction of the canonical projection ofPon its factorA/h, andj= (f/h) ∈ Mor_{L}(A/h,M). Indeed,০φ =

gj০π_{h}০φ =j০p_{h}=f. ∎

First, to replace (

If the restriction of this quotient φ to

If there is anNow assuming it injective, the arity is preserved; the verification of equivalence of the conditions (L-algebraMwith elementsx≠x' and satisfying all conditions then for allb≠b' inB, ∃u∈M,^{Bi}u(b)=x≠u(b')=x' and ∃f∈Mor_{L}(K,_{i}M),f_{|B}=uthus ∃g∈Mor_{L}(K',_{i}M),g০φ=f, and φ(b)≠φ(b') becausef(b)≠f(b').

Now for fusing several conditions (

Any

In fact, when

In any concrete category,

Section ⇒ Embedding ⇒ Injective morphism

Retraction ⇒ Quotient ⇒ Surjective morphism

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3. Algebra 1

4.1.
Algebraic terms

4.2.**Quotient systems**

4.3. Term algebras

4.4. Integers and recursion

4.5. Presburger Arithmetic

4.6. Finiteness and countability

4.7. The Completeness Theorem

4.8. More recursion tools

4.9. Non-standard models of Arithmetic

4.10. Developing theories : definitions

4.11. Constructions

4.A. The Berry paradox

5. Second-order foundations 4.2.

4.3. Term algebras

4.4. Integers and recursion

4.5. Presburger Arithmetic

4.6. Finiteness and countability

4.7. The Completeness Theorem

4.8. More recursion tools

4.9. Non-standard models of Arithmetic

4.10. Developing theories : definitions

4.11. Constructions

4.A. The Berry paradox

6. Foundations of Geometry