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}*]⊂

*s _{E}*০

∀

*f*_{|V} = Id_{V}

This naturally
simplifies when reformulating such categories as those of ground
(*L*∪*V*)-drafts: in each draft, variable symbols are replaced (reinterpreted) by
constant symbols added to the language, so Ψ_{E} is extended by
Id_{V}, to form a ground (*L*∪*V*)-draft.

∀*x*∈*O _{D}*,

∃!

The uniqueness comes from a previous proposition.

∀

⋃

(

So, a

By the above interpretation theorem, in any variables-preserving category of

In particular, any ground term

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*.

**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

fofMis both injective (subalgebra of an injective algebra) and minimal, thus a ground termL-algebra, and the morphismfbetween initialL-algebrasKand Imfis 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

Mor_{L}(*E*,*F*) ⊂
Mor_{T}(*E*,*F*) ⊂ Mor_{{t}}(*E*,*F*)

The interpretation ofThis may be seen as a particular case of conservation of logically defined interpreted relations in categories of relational systems, since any term defining an operation can be re-expressed as a formula defining the graph of this operation, using logical symbols ∃ and ∧. The advantage now that it is established for the general case of abstractly conceived terms no matter their size, instead of concretely written terms on which the conservation property must be repeatedly used for each occurrence of symbol it contains.Tas a set ofV-ary algebraic symbols in eachEis defined by ∀v∈E, ∀^{V}t∈T,t(_{E}v) =g(_{v}t) whereg∈Mor_{v}_{L}(T,E) andg_{|T⋂V}=v.

Thus ∀f∈Mor_{L}(E,F), ∀v∈E, (^{V}f০g∈Mor_{v}_{L}(T,F) ∧ (f০g)_{v}_{|T⋂V}=f০v) ∴t(_{F}f০v) = (f০g)(_{v}t) =f(t(_{E}v)). ∎

Back to homepage : 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. Special morphisms

3.3. Algebras

3.4.Algebraic terms and term algebras

3.5. Integers and recursion

3.6. Arithmetic with addition

4.1. Finiteness and countability

4.2. The Completeness Theorem

4.3. Infinity and the axiom of choice

4.4. Non-standard models of Arithmetic

4.5. How theories develop

4.6. The Incompleteness Theorem