Terms in different drafts *D*, *E* with the same variables, can be so compared
by interpreting ≃ in their coproduct *D* ⊔_{V} *E*.
Equivalently, for any morphisms *f*∈Mor_{L,V}(*D*,*F*),
*g*∈Mor_{L,V}(*E*,*F*) to a functional draft *F*,

∀*x*∈*D*,
∀*y*∈*E*, *x* ≃ *y* ⇔ *f*(*x*) = *g*(*y*).

∀*h*∈Mor_{L,V}(*D*,*E*),
∀*x*,*y*∈*D*, *h*(*x*) ≃ *x* ∧ (*x* ≃ *y* ⇔ *h*(*x*)
≃ *h*(*y*))

Thus ∀

- An algebraic
*L*-draft*D*where*V*=_{D}*V*; -
An injective
*L*-algebra*D*generated by*V*where*V*∩ Im φ_{D}= ∅ (thus*D*\Im φ_{D}=*V*); - A final
*L*-draft in the sense of variables-preserving morphisms Mor_{L,V}; - An
*L*-algebra with basis*V*in the category of*L*-algebras.

Intuitively, a

Proof of equivalence. Already 1. ⇔ 2. ⇒ (3. ∧ 4.).

Proof of 3. ⇒ 1. : any final *L*-draft *D* is

- Functional : ∃
*f*∈Mor_{L,V}(⤓*D*,*D*),*f*⚬ ≃**⃗**_{D}∈ Mor_{L,V}(*D*,*D*) ∴*f*⚬ ≃**⃗**_{D}= Id_{D}∴ Inj ≃**⃗**_{D} - Serial : Mor
_{L,V}(*D*∪(\Dom^{L}D**D**),*D*) ≠∅ *V*=_{D}*V*because*V*is a draft.

Any initial

- minimal : ∃
*f*∈Mor_{L}(*E*, Min_{L}*E*),*f*∈Mor_{L}(*E*,*E*) ∴*f*= Id_{E}∴ Min_{L}*E*=*E* - injective : (
*F*=∧ φ^{L}E_{F}=^{L}φ_{E}) ⇒ φ_{E}⚬^{L}φ_{E}= φ_{E}⚬ φ_{F}⇔ φ_{E}∈Mor_{L}(*F*,*E*)

∴ ∃*f*∈Mor_{L}(*E*,*F*), φ_{E}⚬*f*= Id_{E}∴*f*⚬φ_{E}= φ_{F}⚬=^{L}f^{L}(φ_{E}⚬*f*) = Id_{F}.∎

Thus for any

For any *f*∈Mor_{L}(*K*,*E*) from a ground draft *K*
to an injective system *E*, the system Im *f* is both injective and minimal, i.e. a ground
draft. As with any morphism, if *K* is serial then Im *f* is serial; if *E* is functional
then Im *f* is functional. Thus, if *K* is a ground term algebra and *E* is an injective
partial algebra then Im *f* is a ground term algebra,
and thus *f* is injective (by essential uniqueness of initial or final objects).

- If
*K*is a serial minimal system and*E*is an injective partial algebra then Im*f*is a ground term algebra; - If
*K*is a functional ground draft and*E*is injective then*f*is injective (because ∼_{f}⊂*f*_{⋆}(≃_{Im f}) = ≃_{K}= 𝛿_{K}).

If

If

∀*x*∈*E*,
〈{*x*}〉_{L} = {*f*(*x*)|*f*∈〈*L*〉_{{Id,⚬}}}

∀*X*⊂*E*, 〈*X*〉_{L} =
⋃_{f∈〈L〉{Id,⚬}} *f*[*X*]
= ⋃_{x∈X} 〈{*x*}〉_{L}

Proof of

- Id

∀

∀

Any unary term *L*-algebra *M* (if it exists) is an egg in the category of
*L*-algebras, thus a monoid acting on all *L*-algebras with
Mor_{L} ⊂ Mor_{M}.

If *L* is a language of function symbols and *M* is a unary term
*L*-algebra then *L* is a basis of *M* in the
category of monoids.

Proof:

- Let us confuse every

By the representation theorem, any monoid

Any

Such an

〈

Finally Im

Proof:

- Any

∀

To conclude that (

〈{

This view of free monoids as unary term algebras on languages of function symbols,
can then be re-expressed saying they are made of all finite strings (i.e. tuples) of symbols
from a given set, with concatenation
in the role of composition. This can be easily understood after the definition of ℕ in 4.4,
which is a particular case involved in the general construction.

The unique variables-preserving morphism from any {*e*,•}-draft
to the free monoid with basis its set of variables, is the function of "syntactic simplification" which
removes the occurrences of *e* and "removes all parenthesis", according respectively
to the axioms of identity and associativity which define the category of monoids. Such morphisms from
{*e*,•}-term algebras are surjective.

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory

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

4.7. Countability and Completeness

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. Quotient systems

4.3.

4.4. Integers and recursion

4.5. Presburger Arithmetic

4.6. Finiteness

4.7. Countability and Completeness

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