For convenience, let us work with only one type (the generalization to many types is easy) and introduce a class of systems more general than terms, that we shall call

Given an algebraic language *L*, an *L*-*draft* will be an
*L'*-system
(*D*,**D**) where **D**⊂ (*L*⋆*D*)×*D*,
such that:

- The transpose
^{t}**D**of**D**is the graph of a function Ψ_{D}:*O*→_{D}*L*⋆*D*, whose domain*O*= Im_{D}**D**⊂*D*is called the set of*occurrences*in*D*, and its complement*V*=_{D}*D*\*O*is called the set of_{D}*variables*of*D*; - 〈
*V*〉_{D}_{L}=*D*(well-foundedness condition).

Well-foundedness can be equivalently written in any of these forms

∀*A*⊂*O _{D}*, (∀

∀

∀

A

More generally a draft is ground-like if it has no used variable (Dom

A subset

Indeed, it remains well-founded, as can be seen on the last formulation of well-foundedness.

Like with stable subsets, any intersection of sub-drafts is also a sub-draft; the sub-draft
co-generated by a subset is the intersection of all sub-drafts that include it.

A *term* is a draft co-generated by a single element
which is its root.

Moreover, any union of sub-drafts is also a sub-draft (which was
not the case for sub-algebras because an operation with arity
>1 whose arguments take values in different sub-algebras may
give a result outside their union).

*f* ∈Mor_{L}(*D*,*E*) ⇔
(*f*[*O _{D}*]⊂

σ* _{E}*০

∀

But for any element

∀*x*∈*O _{D}*,

∃!

The uniqueness comes from a previous proposition.

∀

⋃

(

As a particular case of a relation defined by a tuple, here (Id

If *L* does not contain any constant then ⌀ is a ground
term *L*-algebra.

If *L* only contains constants, then ground term *L*-algebras
are the copies of *L*.

The existence of term algebras in other cases will be discussed in
the next section; let us
admit it for now.

**Proposition.** For any ground term *L*-algebra *K*
and any injective *L*-algebra *M*, the unique
*f*∈Mor_{L}(*K*,*M*) is injective.

Proof 1. By a previous result,
{*x*∈*K* | ∀*y*∈*K*,
*f*(*x*) = *f*(*y*) ⇒ *x*=*y*}
∈ Sub_{L}*K*, thus = *K*.

Proof 2. The subalgebra Im *f* of *M* is both injective and
minimal,
thus a ground term *L*-algebra, so the morphism *f* between initial *L*-algebras
*K* and Im *f* is an isomorphism.

Each element ofThe same for terms whose set of variablesFbijectively defines a term inFas the sub-draft ofFit co-generates, thus where it is the root.

For anyL-termTwith roottand variables ⊂V, the uniquef∈Mor(T,F) such thatf_{|T⋂V}= Id_{T⋂V}represents it inFas the term Imfwith rootf(t).

Then its interpretation in anyL-algebraEextending anyv∈E, is determined by the unique^{V}g∈Mor_{L}(F,E) extendingv, asg০f∈Mor(T,E), with resultg(f(t)).

For any subset

If

Binding all variables modifies the view on the above concept of interpretation of a term

We might not need the full sets of these, but at least, an algebra of these (a subset stable by all needed logical operations).

We took ℕ for the case we would need to see "all possible formulas" as terms interpreted in one same algebra. By the way, what is ℕ ? This is the object of the next section.

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3.1. Morphisms of relational systems and concrete categories4. Model Theory

3.2. Algebras

3.3. Special morphisms

3.4. Monoids

3.5. Actions of monoids

3.6. Categories

3.7.Algebraic terms and term algebras

3.8. Integers and recursion

3.9. Arithmetic with addition