**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. ∀

If *L* only has symbols with arity 0 or 1 then every *L*-term is condensed.

∀(

Then the interpretation of

If *L* has no constant then ∅ is a ground term *L*-algebra.

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

From any injective *L*-algebra (*E*,φ_{E}) and
*V* ⊂ *E* \ Im φ_{E} one can form the term algebra
〈*V*〉_{L}. In particular the existence of an injective
algebra implies that of a ground term algebra.

Conversely any initial

- minimal : ∃
*f*∈Mor_{L}(*E*,Min_{L}*E*),*f*∈Mor_{L}(*E*,*E*)∴*f*=Id_{E}∴ Min_{L}*E*=*E* - injective : (
*F*=*L*⋆*E*∧ φ_{F}= (φ_{E})_{L}) ⇒ φ_{E}∈Mor_{L}(*F*,*E*) ∴ ∃*f*∈Mor_{L}(*E*,*F*), φ_{E}০*f*= Id_{E}∴*f*০φ_{E}= φ_{F}০*f*= (φ_{L}_{E}০*f*)_{L}= Id_{F}.

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

Proof 2. Im

This

For any subset *A* of an *L*-algebra *E* and any term algebra
whose set of variables is a copy of *A*, the image of its interpretation
in *E* is 〈*A*〉_{L}.

Conversely if

The monoid structure of

For any set

Denoting

*M*is a unary term*X*_{1}-algebra with variable*e*, interpreting the copy*x*'∈*X*_{1}of each*x*∈*X*as ∀*y*∈*M*,*x*'_{M}(*y*) =*j*(*x*)•*y*- For any
*X*_{1}-algebra*E*there is a unique left action ⋅ of*M*on*E*such that ∀*x*∈*X*, ∀*y*∈*E*,*j*(*x*)⋅*y*=*x*_{E}(*y*) *M*is an initial object in the category of*X*-monoids.

The uniqueness of the morphism to other *X*-monoids is expressed by
〈*X*〉_{{e,•}} = *M*.

This equivalence is deducible from seen results, among which the property of trajectories
and the representation theorem.

When writing terms with multiple uses of an associative operation symbol, all parenthesis
may be removed. For monoids, this removal of parenthesis and also of occurrences of *e*
seen as the empty chain of symbols, is operated by the interpretation of any
*V*-ary {*e*,•}-term in the free monoid on *V*.

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3.1. Relational systems and concrete categories4. Model Theory

3.2. Algebras

3.3. Special morphisms

3.4. Monoids

3.5. Actions of monoids

3.6. Invertibility and groups

3.7. Categories

3.8. Algebraic terms

3.9.Term algebras

3.10. Integers and recursion

3.11. Presburger Arithmetic