3.1. Morphisms of relational systems and concrete
categories
For simplicity, let us focus the study on systems with only one type.
Languages. A language is a set L of "symbols", with
the data of the intended arity n_{s}∈ℕ of each
symbol s∈L. It may be
 A relational language if its symbols aim to represent relations
 An algebraic language if its symbols aim to represent operations.
For any language L and any set E, let
L⋆E = ∐_{s∈L}
E^{ns}
For a relational language L, an Lsystem
is any pair (E,E) made of a set E with an Lstructure
E⊂L⋆E.
Most often, we shall only use one Lstructure on each set, so that
E can be treated as implicit, determined by E. Precisely, let us
assume a fixed choice of a class of Lsystems where the predicate
(r,x)∈E with arguments r,x is independent of
the (E,E) in this class such that (r,x)∈L⋆E
(i.e. x∈E^{nr}).
This way, each r∈L defines an n_{r}ary
predicate (or class of n_{r}tuples) r(x)
⇔ (r,x)∈E for any (E,E) in this class
such that x∈E^{nr}, so that the
determination of E by E can be written
The interpretation of each symbol in each system will be written
r_{E} =
{x∈E^{nr}  r(x)}
= ⃗E(r)
E=∐_{r∈L} r_{E}.
Morphism. Between any 2 Lsystems E,F,
we define the set of Lmorphisms from E to F
as
Mor_{L}(E,F) = {f
∈F^{E}∀r∈L,∀x∈E^{nr},
r(x)⇒ r(f०x)}
= {f ∈F^{E}∀(r,x)∈E,
(r,f०x)∈F}.
Introducing for any function f the function f_{L}
= (L⋆Domf ∋(r,x) ↦ (r,f०x)),
we get the shorter definitions for sets of morphisms,
Mor_{L}(E,F) = {f
∈F^{E} f_{L}[E]⊂F} =
{f ∈F^{E} E⊂f_{L}*F}.
Let us introduce the general concept of quotient of a relational
system, before using it in a particular case.
For any relational language L, any Lsystem
(E,E) and any equivalence relation
R on E, the quotient set E/R has a natural Lstructure
defined as ⃗R_{L}[E].
It is the smallest Lstructure on E/R such that
⃗R∈ Mor(E, E/R).
Extending the language by defined structures
We can see that any Lmorphism f ∈Mor_{L}(E,F)
preserves (= remains a morphism for) any further structure
defined by a formula (without parameters) using symbols in L
and logical symbols ∧,∨,0,1,=,∃. Indeed,
 Substituting arguments of a r∈L by a map σ to n' other variables
(∀E,∀x∈E^{n'}, r'(x)⇔r(x०σ)),
works :
r'(x) ⇒ r(x०σ) ⇒
r(f०x०σ) ⇒ r'(f०x).
 ∀r,r'∈L,n_{r}=n_{r'} ⇒
∀x∈E^{nr}, (r(x)∧r'(x)) ⇒
(r(f०x)∧r'(f०x))
 ∀r,r'∈L,n_{r}=n_{r'}⇒∀x∈E^{nr}
, (r(x)∨r'(x)) ⇒ (r(f०x)∨r'(f०x))
 For 0 and 1 it is trivial
 ∀x,y∈E, x=y ⇒ f(x)=f(y)
 ∀x∈E^{nr},(∃y∈E,
r(x,y)) ⇒ (∃z=f(y)∈F,
r(f०x, z))
Moreover, the use of unions and intersections can be generalized
from the binary cases (∨ and ∧) to arbitrary families (indexed by
possibly infinite subsets of L; the empty one corresponds to
0 and 1).
Thus, if there exists an f ∈Mor_{L}(E,F),
then any ground formula with language L using the only
logical symbols (=,∧,∨,0,1,∃), that is true in E, is also
true in F.
However morphisms may no more preserve structures defined with other
symbols (¬,⇒,∀).
Concrete categories
The concept of concrete category is what remains of a kind of
systems with their morphisms, when we forget which are the
structures that the morphisms are preserving (as we saw that this
structures list can be extended without affecting the sets of
morphisms). Let us introduce a slightly different (more concrete)
version of this concept than the one usually found elsewhere: here,
a concrete category will be the data of
 A class of sets called objects
 A class of functions called morphisms; for any objects
E,F we have a set Mor(E,F)⊂F^{E}
of morphisms from E to F, that is the set of all
functions from E to F which are morphisms
satisfying the following axioms
 Every morphism belongs to some Mor(E,F), i.e.
its domain is an object and its image is included in an object
(in practice, images of morphisms will be objects too);
 For any object E, Id_{E} ∈ Mor(E,E)
;
 Any composite of morphisms is a morphism: for any 3 objects E,F,G
, ∀f ∈ Mor(E,F), ∀g∈Mor(F,G),
g०f ∈Mor(E,G).
The last condition is easily verified for Lmorphisms :
∀(r,x)∈E, (r,f०x)∈F ∴
(r,g०f०x)∈G.Other concepts of category
(Abstract) categories differ from concrete categories, by forgetting that objects
are sets (ordered by inclusion) and that morphisms are functions; what remains is the
sets Mor(E,F) treated as pairwise disjoint, and the composition operations,
Mor(F,G)×Mor(E,F)→Mor(E,G).
A category (either concrete or abstract) is small if its class of objects is a set.
Morphisms between systems with several types
While we introduced the notion of morphism in the case of systems
with a single type, it may be extended to systems with several types as
well. Between systems E,F with a common list τ of types (and
interpretations of a common list of structure symbols), morphisms
can equivalently be conceived in the following 2 ways,
apart from having to preserve all structures:
 A tuple (or family) of functions (f_{t})_{t}_{∈}_{τ},
where ∀t∈τ, f_{t}:E_{t}→ F_{t}
where E_{t}⊂E, F_{t} ⊂F
are the interpretations of type t in E anf F
 A function f:E→ F that is a morphism
when regarding τ as a list of unary relation symbols (by the
same idea as the use of classes
instead of types in set theory); or equivalently, such
that h_{F}०f=h_{E}
where h_{E}:E→τ, h_{F}
:F→τ are the functions giving the type of each object.
3.2. Special morphisms
Let us introduce diverse possible qualifications for morphisms of
relational systems, and show them related by the following
dependencies (without converses in general):
Automorphism ⇔ (Endomorphism ∧ Isomorphism)
Isomorphism ⇔ (Retraction ∧ Monomorphism) ⇔ (Section ∧ Epimorphism)
Retraction ⇒ Surjective morphism ⇒ Epimorphism
Section ⇒ Embedding ⇒ Injective morphism ⇒ Monomorphism
Isomorphism ⇒ Elementary embedding ⇒ Embedding
Functions defined by composition. In any category (whether
concrete or not), any f ∈ Mor(E,F) defines
functions by currying the operation of composition with other
morphisms to or from another object X: let us denote (following wikipedia)
 Hom(X,f) = (Mor(X, Dom
f)∋g↦ f০g),
with target Mor(X,F) for any target F of f.
 Hom_{F}(f,X) = (Mor(F,
X)∋g↦ g০f), with target
Mor(E,X). In abstract categories where f determines
F, the notation simplifies as Hom(f,X).
The first one respects composition, while the second one reverses
it: for any 4 objects E,F,G,X , ∀f
∈Mor(E,F), ∀g∈Mor(F,G),
Hom(X,g) ০ Hom(X,f) =
Hom(X,g০f)
Hom_{F}(f,X) ০
Hom_{G}(g,X) =
Hom_{G}(g০f,X)
Monomorphism. In a category, a morphism
f∈Mor(E,F)
is called monic, or a monomorphism, if Hom(X,f)
is injective for all objects X.
Epimorphism. In an abstract category, a morphism f∈Mor(E,F)
is called epic, or an epimorphism, if Hom(f,X)
is injective for all objects X:
∀g,h∈Mor(F,X),
g০f=h০f ⇒ g=h.
In our concept of concrete category, we must specify F: we say that
f∈Mor(E,F) is Fepic, or an Fepimorphism,
if all Hom_{F}(f,X) are injective.
In any concrete category, all injective morphisms are monic, and
any morphism with image F is Fepic.
However, the converses may not hold, and exceptions may be uneasy
to classify, especially as the condition depends on the whole
category and not just the morphism at hand.
The following 2 concepts may be considered cleaner as they admit
a local characterization:
Sections. A morphism f∈Mor(E,F) is
called a section (or section in F if the category
is concrete), if Id_{E}∈Im(Hom_{F}(f,E)),
i.e. ∃g∈Mor(F,E),g০f=Id_{E}.
Then f is monic and for all objects X we have
Im(Hom_{F}(f,X)) = Mor(E,X).
Retraction. A morphism g∈Mor(F,E)
is called a retraction (or retraction on E if the
category is concrete), if Id_{E}∈Im(Hom(E,g)),
i.e. ∃f∈Mor(E,F),g০f=Id_{E}.
Then g is epic and for all objects X we have
Im(Hom(X,g)) = Mor(X,F).
When g০f=Id_{E} we also say that f
is a section of g, and that g is a retraction of f.
Proof: if g০f=Id_{E} then for all objects X,
Hom_{F}(f,X) ০ Hom_{E}(g,X)
= Hom_{E}(Id_{E},X)
= Id_{Mor(E,X)}, thus
 Hom_{E}(g,X) is injective (g is epic)
 Im(Hom_{F}(f,X)) = Mor(E,X). Namely,
∀h∈Mor(E,X), h =
h০g০f
= Hom_{F}(f,X)(h০g).
Similarly, Hom(X,g) ০ Hom(X,f) =
Hom(X,Id_{E}) = Id_{Mor(X,}_{E)} thus
 Hom(X,f) is injective (f is monic)
 Im(Hom(X,g)) = Mor(X,F).∎
Isomorphism. In a concrete category, an isomorphism
between objects E,F , is a bijection f
: E ↔ F such that f ∈Mor(E,F)
and f^{ 1}∈Mor(F,E).
In any category, an isomorphism is an f ∈Mor(E,F) such
that ∃g∈Mor(F,E),
g০f= Id_{E}
∧ f০g= Id_{F} (this inverse g of f
is unique).
In this case, Hom(X,f) and Hom(X,g) are
bijections, inverse of each other, between Mor(X,E)
and Mor(X,F).
Any epic section f∈Mor(E,F) is an isomorphism
: g০f=Id_{E} ⇒ f০g০f
= Id_{F}০f ⇒ f০g=Id_{F}
Similarly, any monic retraction is an isomorphism.
Two objects E, F are said to be isomorphic
if there exists an isomorphism between them.
Endomorphisms. An endomorphism of an object E
in a category, is a morphism from E to itself. Their set
is written End(E)=Mor(E,E).
Automorphisms. An automorphism of an object E is an isomorphism
of E to itself.
Thus it is both an endomorphism and an
isomorphism. However an endomorphism of E which is an isomorphism to a strict
subset of E, is not an automorphism.
Strong preservation. A function f ∈ F^{E} is said to
strongly preserve a relation symbol or formula r interpreted in each of E
and F, if it preserves both r and ¬r : ∀x∈
E^{nr}, x∈r_{E}
⇔ f০x∈r_{F}.
Embeddings. An f ∈Mor_{L}(E,F)
is called an Lembedding if it strongly preserves all structures :
∀r∈L,∀x∈
E^{nr}, x∈r_{E}
⇔ f০x∈r_{F}.
Embeddings will usually be supposed injective, as it means strongly preserving the equality
relation. Things can come down to this case by replacing equality in the concept of injectivity by a
properly defined equivalence relation, or replacing systems by their quotient by this relation,
where the canonical surjections would be noninjective embeddings.
Injective embeddings are isomorphisms to their images.
Embeddings still strongly preserve structures defined using the
symbols in L and the logical symbols ∧,∨,0,1,¬, and also = in
the case of injective embeddings.
Thus, they also preserve invariant structures defined
using symbols of L and ∧,∨,¬,0,1,∃ where any
occurrence of ¬ comes after (inside) any occurrence of ∃.
Now removing this order restriction on the use of logical symbols provides the full use
of firstorder logic:
Elementary embedding. An f ∈
Mor_{L}(E,F) is called an elementary embedding
(or elementary Lembedding) if it (strongly) preserves all
invariant structures (defined by firstorder formulas with language L).
Every isomorphism is an elementary embedding.
If f∈ End(E) is an invariant elementary embedding then it is an automorphism:
Im f is also invariant (as a unary relation
∃y, f(y)=x)
∴ ∀x∈ E,
x∈Im f ⇔ f(x)∈Im f
∴ Im f = E. ∎
The existence
of an elementary embedding f ∈Mor_{L}(E,F)
implies that E and F are elementarily equivalent:
Elementary equivalence. 2 systems are said to be
elementarily equivalent, if every ground firstorder formula true
in the one is true in the other.
The most usual practice of mathematics focuses on systems where all elementarily
equivalent ones are connected by isomorphisms, which are the only elementary embeddings.
However, nonsurjective elementary
embeddings exist and play a special role in foundational issues,
such as Skolem's
paradox and nonstandard
models of arithmetic.
In any category, an object X is called an initial
object (resp. a final object) if for all objects Y
the set Mor(X,Y) (resp. Mor(Y,X)) is a
singleton.
Such objects have this remarkable property: when they exist, all such objects are
isomorphic, by a unique isomorphism between any two of them.
Proof: For any initial objects X and Y, ∃f∈Mor(X,Y),
∃g∈Mor(Y,X), g০f ∈Mor(X,X)
∧ f০g ∈Mor(Y,Y).
But Id_{X} ∈ Mor(X,X) which is a
singleton, thus g০f= Id_{X}.
Similarly, f০g=Id_{Y}.
Thus f is an isomorphism, unique because Mor(X,Y)
is a singleton.∎
By this isomorphism X and Y may be treated as identical to
each other. We may say that an initial object is essentially unique.
Such objects exist in many categories, but are not always interesting. For
example, in any category of relational systems containing representatives
(copies) of all possible ones with a given language:
 Final objects are the singletons (where all relations are
constantly true),
 The only initial object is the empty set (where any nullary
relation, i.e. boolean constant, is false).
Exercise. Given two fixed sets X and Y, consider the
category whose objects are all (E,φ) where E is a set and
φ: E×X→Y, and the morphisms from (E,φ) to (E',φ')
are all f : E→E' such that ∀a∈E,∀x∈X,
φ(a,x) = φ'(f(a),x).
Does it have an initial object ? a final object ?
3.3. Notion of algebra
Algebras. Given an algebraic language L, an Lalgebra
is a set E with an interpretation of each s∈L as an
n_{s}ary operation in E.
Again, let us assume a fixed class of Lalgebras E where each s
is interpreted as the restriction s_{E}
of an n_{s}ary operator s independent of E,
s_{E} = (E^{ns}
∋ x↦ s(x)).
These can be packed as one function
φ_{E} = ∐_{s∈L}
s_{E} = ((s,x) ↦
s(x)) : L⋆E → E.
Such a class of algebras
forms a concrete category with the following concept of morphism.
Morphisms of algebras. For any Lalgebras E,
F,
Mor_{L}(E,F) = {f∈F^{E} 
∀(s,x)∈L⋆E, s_{F}(f০x) =
f(s_{E}(x))} = {f∈F^{E}
φ_{F}০f_{L} = f০φ_{E}}.
When c∈L is a constant (i.e. n_{c}
=0), this condition on f says f(c_{E})=c_{F}.
Such categories can be seen as particular categories
of relational systems, as follows.
Let the relational language L' be a copy of L where to
each s∈L corresponds s'∈L' with increased arity
n_{s'} = n_{s}+1, so that
L'⋆E ≡ ∐_{s∈L}
E^{ns}×E ≡
(L⋆E)×E also expressible as the set of triples
(s,x,y) such that s∈L, x∈
E^{ns} and y∈E.
Each n_{s}ary
operation s_{E} defines an n_{s'}ary
relation s'_{E} ≡ Gr s_{E}.
These are packed as an L'structure
E = Gr φ_{E} ≡ ∐_{s∈L}
s'_{E}.
The conditions for an f ∈ F^{E} to be a
morphism are equivalent : (∀(x,y)∈E,
(f_{L}(x),f(y))∈F) ⇔
(∀x∈L⋆E, φ_{F}(f_{L}(x))=
f(φ_{E}(x))).
Every injective morphism f between algebras is an embedding
:
∀(s,x,y)∈L'⋆E, f(y)
= s_{F}(f০x) = f(s_{E}(x))
⇒ y=s_{E}(x).
Any embedding f ∈ Mor_{L}(E,F),
is injective as soon as Im φ_{E} = E or some
s_{E} is injective for one of its arguments.
Bijective morphisms of algebras are isomorphisms. This can be deduced
from the fact they are embeddings, or by
φ_{E}০f_{L}^{1} = f
^{1}০f০φ_{E}০f_{L}^{1} = f
^{1}০φ_{F}০f_{L}০f_{L}^{1}
= f ^{1}০φ_{F}.
Singletons are the final objects in any category of algebras
with a fixed language where they are admitted as objects.
Subalgebras. A subset A⊂E of an Lalgebra
E will be called an Lsubalgebra of E, if
φ_{E}[L⋆A]⊂A.
Then the restriction φ_{A} of φ_{E}
to L⋆A gives it a structure of Lalgebra.
The set of all Lsubalgebras of E
will be denoted Sub_{L} E. It is nonempty
as E ∈ Sub_{L} E.
For any formula of the form (∀(variables), some formula without
any binder), its truth in E implies its truth in each A∈Sub_{L}
E.
Images of algebras. f ∈Mor_{L}(E,F)
⇒ Im f ∈ Sub_{L}F.
The proof uses the finite choice theorem
with (AC 1)⇒(6):
∀(s,y)∈ L⋆Im f,
∃x∈E^{ns},
f০x = y ∴ s_{F}(y)
= f(s_{E}(x))
∈ Im f ∎
Thus trying to exend this result to algebras with infinitary operations, would require the axiom of choice,
otherwise it anyway still holds for injective morphisms.
Let us generalize the concept of algebra, to any L'systems
(E,E), where E ⊂ (L⋆E)×E
needs not be functional. They form the same kind of categories previously
defined, with a different notation (through the canonical bijection depending on the choice of
distinguished argument) by which more concepts can be introduced.
Stable subsets. The concept of subalgebra is generalized as that of stability
of a subset A of E by L :
A ∈ Sub_{L} E ⇔ (E_{*}(L⋆A)
⊂A) ⇔ (∀(s,x,y)∈E,
Im x⊂A ⇒ y∈A).
Stability is no more preserved by direct images by morphisms, but is still preserved by inverse images:
Preimages of stable subsets. ∀f ∈
Mor_{L}(E,F),
∀B ∈ Sub_{L}F, f *(B)∈Sub_{L}
E.
Let A=f *B. Proof for Lalgebras:
∀(s,x)∈L⋆A, f০x
∈ B^{ns}∴ f(s_{E}(x))
= s_{F}(f০x) ∈B ∴
s_{E}(x)∈A.
Proof for L'systems:
∀(x,y)∈E, (f_{L}(x),f(y))∈F∴
(x∈L⋆A ⇒ f_{L}(x)∈L⋆B
⇒ f(y)∈B ⇒ y∈A).∎
Proposition. For any L'system E and any
Lalgebra F,
∀f,g∈Mor_{L}(E,F),
{x∈Ef(x)=g(x)}∈ Sub_{L}E.
Proof : ∀(s,x,y)∈E, f০x=g০x
⇒ f(y) = s(f০x) = g(y). ∎
Intersections of stable subsets. ∀X ⊂ Sub_{L}E,
∩X ∈ Sub_{L} E where ∩X ≝
{x∈E∀B∈X, x∈B}.
Proof:
∀(x,y)∈E, x∈L⋆∩X ⇒ (∀B∈X,
x∈L⋆B ∴ y∈B) ⇒ y∈∩X. ∎
Other way:
E_{*}(L⋆∩X) =
E_{*}(∩_{B∈X} L⋆B)
⊂∩_{B∈X}
E_{*}(L⋆B) ⊂∩X.
Subalgebra generated by a subset. ∀A ⊂ E, the
Lsubalgebra of E generated by A, written
〈A〉_{L,E} or more simply 〈A〉_{L},
is the smallest Lsubalgebra of E including A:
〈A〉_{L}=
∩{B∈Sub_{L}EA⊂B}=
{x∈E∀B∈Sub_{L}E, A⊂B
⇒ x∈B}∈ Sub_{L}E.
For fixed E and L, this function of A is a
closure with
image Sub_{L}E.
We say that A generates E if 〈A〉_{L}=E.
Minimal subalgebra. The minimal subalgebra of an algebra E,
is its subalgebra Min_{L}E =〈∅〉_{L,E}
= ∩Sub_{L}E. (This can also be used on stable subsets
of algebraic systems which are not algebras)
An Lalgebra E is minimal when E = Min_{L}
E, or equivalently Sub_{L}E
= {E}. Then Min_{L}E is the only subalgebra
of E that is a minimal algebra.
We can redefine generated subalgebras in terms of minimal
subalgebra with a different language: 〈A〉_{L,E}=
Min_{L∪A} E where A is seen as
a set of constants.
Proposition. For any Lalgebra E,
Proofs: φ_{E}[L⋆A] ⊂
Imφ_{E}⊂A
Imφ_{E}⊂ A∪Imφ_{E} ∴
A∪Imφ_{E}∈Sub_{L}E
∀B ∈ Sub_{L}F, f
*(B)∈Sub_{L} E ∴ Min_{L}E
⊂ f*(B) ∴ f [Min_{L}E]⊂B.∎
Injective, surjective algebras. An Lalgebra (E,φ_{E})
will be called injective if φ_{E} is injective, and surjective if
Im φ_{E} = E.
Any minimal Lalgebra is surjective. Thus, the minimal
subalgebra of any algebra is also a surjective algebra.
Proposition. If E is a surjective algebra and
F is an injective one then
∀f ∈Mor_{L}(E,F),
A= {x∈E  ∀y∈E, f(x) =
f(y) ⇒ x=y}
∈ Sub_{L}E.
Proof: ∀(s,x)∈L⋆A, ∀y∈E,
f(s_{E}(x)) = f(y) ⇒
(∃(t,z)∈φ_{E}^{•}(y),
s_{F}(f০x)
= f(s_{E}(x)) = f(y) = f(t_{E}(z))
= t_{F}(f০z) ∴ (s=t
∧f০x=f০z) ∴ x=z)
⇒ s_{E}(x)=y.
Other view : under the same assumptions, for each uniqueness quantifier Q (either ∃! or !),
B = {y∈F  Qx∈E, y =
f(x)} ∈ Sub_{L}F
Proof: as φ_{F} is injective,
∀y∈φ_{F}[L⋆B], ∃!: φ_{F}^{•}(y) ⊂ L⋆B ∴
Qz∈ L⋆E, φ_{F}(f_{L}(z)) = y.
As φ_{F}০f_{L} = f০φ_{E} and
φ_{E} is surjective, we conclude Qx∈E,
y = f(x). ∎
3.4. Algebraic terms and term algebras
Algebraic drafts
As the concept of term was only intuitively
introduced in 1.5, let us now formalize the case of terms using
a purely algebraic language (without logical symbols), called
algebraic terms, as mathematical systems in
a set theoretical framework.
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 drafts.
Variables will have a special treatment, without adding them as constants
in the language.
Given an algebraic language L, an Ldraft 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_{D} = Im D
⊂D is called the set of occurrences in D, and
its complement V_{D}=D\O_{D}
is called the set of variables of D;
 〈V_{D}〉_{L}= D
(wellfoundedness condition).
Let us denote ∀x∈O_{D},
Ψ_{D}(x)=(s(x), l_{x})
∈ L⋆D where s(x)∈L and
l_{x}∈D^{ns}^{(x)}.
We can also denote s_{D}(x)=s(x)
to let s_{D} be a function with domain O_{D}.
Wellfoundedness can be equivalently written in any of these forms
∀A⊂O_{D}, (∀x∈O_{D},
Im l_{x} ⊂ A∪V_{D} ⇒ x∈A)
⇒ A=O_{D}
∀A⊂D, (V_{D}⊂A ∧
D_{*}(L⋆A) ⊂A) ⇒
A = D
∀A⊂D, V_{D}⊂A≠D
⇒∃x∈O_{D}\A, Im l_{x}⊂A
∀A⊂O_{D}, A≠∅ ⇒ ∃x∈A,
A⋂ Im l_{x} = ∅
The set of used variables of (D,D), those which effectively occur,
is V_{D}⋂⋃_{x∈OD}
Im l_{x}. Unused variables can be added or removed
in D while keeping D fixed (by changing
D⊃O_{D}∪(⋃_{x∈OD}
Im l_{x})), so that their presence may be irrelevant.
A ground draft is a draft with no variable, i.e. V_{D}=∅. Thus,
Ψ_{D}: D→ L⋆D and Sub_{L}D
= {D}.
More generally a draft is groundlike if it has no used variable (Dom D⊂ L⋆O_{D}).
Subdrafts and terms
Drafts do not have interesting stable subsets (by wellfoundedness), but let us introduce
another stability concept for them.
A subset A⊂D is a subdraft of D
(or a costable subset of D) if, denoting O_{A}
= A⋂O_{D} and Ψ_{A}= Ψ_{DOA},
we have (Im Ψ_{A}⊂ L⋆A), i.e. ∀x∈O_{A},
Im l_{x}⊂A.
Indeed, it remains wellfounded, as can be seen on the last formulation of wellfoundedness.
Like with stable subsets, any intersection of subdrafts is also a subdraft; the subdraft
cogenerated by a subset is the intersection of all subdrafts that include it.
A term is a draft cogenerated by a single element
which is its root.
Moreover, any union of subdrafts is also a subdraft (which was
not the case for subalgebras because an operation with arity
>1 whose arguments take values in different subalgebras may
give a result outside their union).
There is a natural order relation on each draft D defined by x
≤ y ⇔ x∈ (the term with root y). It is obviously
a preorder. Its antisymmetry is less obvious; a proof for integers will be given
in 3.6, while the general case will come from properties of wellfounded relations
in the study of Galois connections.
Categories of drafts
As particular relational systems, classes of Ldrafts form concrete categories.
Between two Ldrafts D,E,
f ∈Mor_{L}(D,E) ⇔
(f[O_{D}]⊂O_{E} ∧
Ψ_{E} ০ f_{OD}=
f_{L}০Ψ_{D})
where the equality condition can be split as
s_{E}০f_{OD}
= s_{D}
∀x∈O_{D},
l_{f(x)}=f০l_{x}
Another kind of category of drafts can be considered, with objects also Ldrafts
but with a common set of variables (V_{D}=V_{E}=V)
and taking smaller sets of morphisms: the variablespreserving morphisms,
i.e. moreover satisfying
f_{V} = Id_{V}
But for any element t in any draft, the term T cogenerated
by {t} has as set of variables T⋂V (which is the set of used variables of
T unless T={t}⊂V) generally smaller than V,
so the admission of terms defined as subsets cogenerated by singletons in
such a category requires this loosening of the condition. 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.
Intepretations of drafts in algebras
For any Ldraft D and any Lalgebra E, an interpretation of
D in E is a morphism f∈Mor_{L}(D,E),
i.e. f_{OD}=
φ_{E}০f_{L}০Ψ_{D},
which can also be written
∀x∈O_{D}, f(x) =
s(x)_{E}(f০l_{x})
Theorem. For any Ldraft D with set of variables V and
any Lalgebra E, any v∈E^{V}
is uniquely extensible to an interpretation of D:
∃!h∈Mor_{L}(D,E), h_{V}
= v, equivalently ∃!h∈E^{OD}, v∪h
∈Mor_{L}(D,E).
The uniqueness comes from a previous proposition.
Proof of existence.
S = {A⊂D  V⊂A∧ Im Ψ_{A}⊂
L⋆A}
v ∈ K = ⋃_{A∈}_{S}
{f∈Mor_{L}(A,E) 
f_{V} =v}
∀f,g ∈K, B = Dom f ⋂ Dom g
⇒ (f_{B}∈K ∧
g_{B}∈K)
⇒ f_{B}=g_{B}
⋃_{f∈K} Gr f = Gr h
C= Dom h = ⋃_{f∈K} Dom f
∈ S
h ∈ K
(C∪D_{*}(L⋆C) ∋ x↦ (x∈C
? h(x) : φ_{E}(h_{L}(Ψ_{D}(x)))))
∈ K (see conditional operator)
D_{*}(L⋆C) ⊂ C
C=D ∎
Term algebras
An Lalgebra (E,φ_{E}) is called a term algebra
if it is injective and 〈E\Im φ_{E}〉_{L}
= E. Thus it is also an Ldraft with Ψ_{E}
= φ_{E}^{1}. As such, it is ground if
moreover E=Im φ_{E}.
So, a ground term algebra is an algebra both minimal and injective,
and thus also bijective.
By the above interpretation theorem, in any variablespreserving
category of Ldrafts with a fixed set V of variables
(category of ground (L∪V)drafts), any term Lalgebra
F, when present, is a final object. Thus any 2 of them are isomorphic,
by a unique variablespreserving isomorphism.
In particular, any ground term Lalgebra is
a final object in any category of ground Ldrafts,
and an initial object in any category of Lalgebras.
Proposition. For any ground term Lalgebra K
and any injective Lalgebra 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 (subalgebra
of an injective algebra) and minimal,
thus a ground term Lalgebra, and the morphism f between initial Lalgebras
K and Im f is an isomorphism.
Any term algebra F plays the role of the "set of all terms"
with its list V of variable symbols, for the following reason:
Each element of F bijectively defines a term in F as the subdraft of
F it cogenerates, thus where it is the root.
For any
Lterm T with root t and variables ⊂V,
the unique f∈Mor(T,F) such that
f_{T⋂V} = Id_{T⋂V}
represents it in F as the term Imf with root f(t).
Then its interpretation in any Lalgebra E extending any
v∈E^{V}, is determined by the unique
g∈Mor_{L}(F,E) extending v,
as g০f∈Mor(T,E), with result g(f(t)).
The same for terms whose set of variables V' is interpreted in E by the
composite of a function from V' to V, with one from
V to E (instead of having V'⊂V).
For any subset A of an Lalgebra E, any term algebra
F_{A} whose set of variables is a copy of A,
represents the set of all Lterms with variables interpreted in A.
Then, the Lsubalgebra 〈A〉_{L} of E is the
image of the interpretation of F_{A} in E, i.e. the set of
all values of these terms.
Sets of morphisms between Lalgebras E,F
remain unchanged when adding to L any operation symbol
defined by an Lterm T with root t and
variables among V :
Mor_{L}(E,F) ⊂
Mor_{T}(E,F) ⊂ Mor_{{t}}(E,F)
Proof:The interpretation of T as a set of Vary algebraic
symbols in each E is defined by ∀v∈E^{V},
∀t∈T, t_{E}(v) = g_{v}(t)
where g_{v}∈Mor_{L}(T,E)
and g_{T⋂V}=v.
Thus ∀f∈Mor_{L}(E,F), ∀v∈E^{V},
(f০g_{v}∈Mor_{L}(T,F)
∧ (f০g_{v})_{T⋂V} =
f০v) ∴ t_{F}(f০v) =
(f০g_{v})(t) = f(t_{E}(v)). ∎
If L does not contain any constant then ∅ is a ground
term Lalgebra.
If L only contains constants, then ground term Lalgebras
are the copies of L.
3.5. Integers and recursion
The set ℕ
Any theory that aims to describe the system ℕ of natural numbers is
called an Arithmetic. In details, there are several such formal theories,
depending on the language (list of basic structures), and the
logical framework (affecting the expressibility of axioms).
Our first "definition" of ℕ will characterize it in a set theoretical framework.
This way of starting to formalize ℕ now may look circular, as we already used
natural numbers as arities of operation symbols of algebras, of which arithmetic
is a particular case. But it only uses operation symbols with arity
0, 1 or 2, for which previous definitions might as well be
specially rewritten without any general reference to integers.
Definition. The set ℕ of natural numbers is a ground term algebra with
a language of two symbols: one constant symbol 0 ("zero"), and
one unary symbol S.
The interpretation of S there is called the successor, understood as adding one
unit (Sn=n+1).
This concept of ground term algebra can be expanded as the following 3 axioms on this
{0,S}algebra :
∀n∈ℕ, Sn ≠ 0

(H0), i.e. 0 ∉ Im S

∀n,p∈ℕ, Sn =
Sp ⇒ n = p 
(Inj), i.e. S is injective 
∀A⊂ℕ, (0∈A ∧ ∀n∈A,Sn∈A)
⇒ A=ℕ 
(Ind) : induction axiom (ℕ is a minimal (0,S)algebra). 
We can define 1=S0, 2=SS0...
This insertion of
arithmetic into set theory, adding to set theory the constant symbol ℕ and whatever
chosen language for arithmetic interpreted in this ℕ, is the natural way
set theory is completed to form the standard foundation of mathematics as
practiced by most mathematicians. Its implicit choice of a fixed ℕ in the class of ground term
{0,S}algebras, does not introduce any arbritrariness, since, as an initial {0,S}algebra, it
is essentially unique (any two systems in this class are identifiable
by an isomorphism which is unique inside the same universe).
A set E is called countable if there exists an
injection from E to ℕ.
Recursively defined sequences
A sequence of elements of a set E, is a function from ℕ to E
(a family of elements of E indexed by ℕ).
In particular, a recursive sequence in E is a sequence defined as the
only u ∈ E^{ℕ} such that u ∈ Mor(ℕ,(E,a,f)),
where (E,a,f) is the {0,S}algebra E
interpreting 0 as a∈E and S as f∈ E^{E} :
u_{0}=a
∀n∈ℕ, u_{Sn} = f(u_{n}).
As u_{n} also depends on a and f, let us write it as f^{ n}(a).
This notation is motivated as follows.
As an element of a ground term {0,S}algebra, each integer n represents a term with symbols 0 and
S respectively interpreted in E as a and f. So, f^{ n}(a) abbreviates
the term with shape n using symbols a and f:
f^{ 0}(a) = a
f^{ 1}(a) = f(a)
f^{ 2}(a) = f(f(a)) 
In another curried view of this map from E×E^{E}×ℕ
to E we can reinterpret 0 as a variable instead of a constant symbol.
Then each integer n becomes a term
S^{n} with n occurrences of the function symbol S
and one variable, interpreted in each {S}algebra (E,f) as the
function f^{ n}∈ E^{E}
recursively defined by
f^{ 0} = Id_{E}
∀n∈ℕ, f^{ Sn} = f০f^{ n}
In particular, f^{1}=f.
More generally, for any functions f∈ E^{E},
g∈E^{X}, the sequence of functions recursively
defined by
h_{0}=g
∀n∈ℕ, h_{Sn} = f০h_{n}
is h_{n}=f^{ n}০g.
Addition
The operation of addition in ℕ can be defined as n+p =
S^{p}(n), i.e. by the recursive definition
n + 0 = n
∀p∈ℕ, n+S(p) = S(n+p).
Thus,
n+1 = n+S0 =
S(n+0) = Sn
As ∀a∈E ,∀f∈ E^{E}, (p↦f^{
n+p}(a))∈Mor(ℕ,(E,f^{n}(a),f)),
i.e. f^{n+0}=f^{n} and ∀p∈ℕ,
f^{n+Sp} = f^{S(n+p)
} = f০f^{n+p}
we have
∀n,p∈ℕ , f^{ n+p} = f^{
p}০f^{ n}.
Addition is associative: (a+b)+n = S^{n}(S^{b}(a)) =
S^{b}^{+}^{n}(a) =
a+(b+n).
Multiplication
Multiplication in ℕ can be defined as x⋅y =
(S^{x})^{y}(0), so that
∀x∈ℕ, x⋅0 = 0
∀x,y∈ℕ, x⋅(Sy) = (x⋅y)+x
More generally, for any a∈E and f∈ E^{E},
we have f^{x}^{⋅}^{y}(a)
= (f^{x})^{y}(a).
A more general form of recursion
Some useful sequences need recursive definitions where the term defining
u_{Sn} uses not only u_{n} but also n
itself. Somehow it would work all the same, but trying to directly adapt to this case
the proof we gave would require to define some special generalizations of previous
concepts, and specify their resulting properties. To simplify, let us proceed another
way, similar to the
argument in Halmos's Naive Set Theory, but generalized.
For any algebraic language L, let us introduce a general concept of "recursive condition"
for functions u : E → F, where, instead of a draft, E is first
assumed to be an Lalgebra (then a ground term algebra to conclude).
The version we saw was formalized by giving the term in the recursive definition as an
Lalgebra structure on F, φ_{F}: L⋆F → F,
then expressing the request for u to satisfy this condition as u∈Mor(E,F),
namely ∀(s,x)∈L⋆E,
u(s_{E}(x)) = φ_{F}(s,u০x).
Let us generalize this as u(s_{E}(x))
= φ(s,x,u০x) which by the canonical bijection
Dom φ ≡ ∐_{s∈L} E^{ns
}×F^{ns} ≡ ∐_{s∈L}
(E×F)^{ns} =
L⋆(E×F) can be written using h : L⋆(E×F)
→ F such that ∀(s,x,y)∈ Dom φ,
φ(s,x,y) = h(s,x×y), asu(s_{E}(x))
= h(s,x×(u০x)).
As ∀u∈F^{E}, x×(u০x)
= (Id_{E}×u)০x, this becomes
the second component of the formula Id_{E}×u ∈ Mor(E, E×F)
when giving E×F the structure φ_{E×F} =
(φ_{E}০π_{L})×h.
The first component (φ_{E}০π_{L}) we give to φ_{E×F},
makes π∈ Mor(E×F, E) and makes tautological the first component
of the formula Id_{E}×u
∈ Mor(E, E×F), namely
Id_{E}(s_{E}(x)) = φ_{E}(s,x)
= (φ_{E}০π_{L})(s,x×(u০x)).
It is then possible to conclude by reusing the previous result of existence of interpretations:
If E is a closed term Lalgebra then
∃!f ∈ Mor(E, E×F), which is
of the form Id_{E}×u because
π০f ∈ Mor(E, E) ∴ π০f = Id_{E}.
But one can do without it, based on the following property of this Lalgebra E×F:
∀u∈F^{E}, Id_{E}×u
∈ Mor_{L}(E, E×F) ⇔ Gr u ∈ Sub_{L}(E×F)
The ⇒ is a case of image of an algebra by a morphism, Gr u = Im (Id_{E}×u).
For the converse, the inverse of the bijective morphism π_{Gr u}
∈ Mor_{L}(Gr u, E)
is a morphism Id_{E}×u ∈ Mor_{L}(E, Gr u)
⊂ Mor_{L}(E, E×F).
This reduces the issue to the search of subalgebras of E×F which are graphs of functions from E to F.
Now if E is a ground term Lalgebra then M =
Min_{L}(E×F) is one of them because
π_{M}∈ Mor_{L}(M, E) from a surjective algebra to a ground term algebra
must be bijective.
Any other subalgebra of E×F must include M, thus to stay functional it must equal M. ∎
Commutativity, associativity and commutants
A binary operation # in a set E, is called
 Commutative when ∀x,y∈E, x#y
= y#x.
 Associative when ∀x,y,z ∈E,
x#(y#z) = (x#y)#z, so that we
can write x#y#z to mean either of these terms.
The commutant of any subset A⊂E for a binary operation # in E,
is defined as
C(A) = {x∈E∀y∈A,
x#y = y#x}.
This is a Galois connection: ∀A,B⊂E, B⊂C(A)
⇔ A⊂C(B).
C(E)= E expresses the commutativity of #.
Proposition. For any associative
operation # on a set E, ∀A⊂E,
 C(A) ∈ Sub_{#}F
 If A⊂C(A) then # is commutative in 〈A〉_{#}
Proof:
 ∀x,y∈C(A), (∀z∈A, (x#y)#z
= x#z#y = z#(x#y)) ∴ x#y∈C(A)
 A⊂C(A)∈ Sub_{#}F
⇒〈A〉_{#}⊂C(A)
⇒ A⊂C(〈A〉_{#})∈ Sub_{#}F ⇒
〈A〉_{#}⊂C(〈A〉_{#}).
This concept will first be used for arithmetic, then generalized to
monoids and
clones.
3.6. Arithmetic with addition
Firstorder theories of arithmetic
Our first formalization of ℕ
was based on the framework of set theory, where it used the powerset to characterize ℕ in an
"essentially unique" way (specify its isomorphism class). It allowed recursion,
from which we could define addition and multiplication.
Let us now consider formalizations in the framework of firstorder logic, thus
called firstorder arithmetic. As firstorder logic cannot fully express the powerset,
the (∀A⊂ℕ) in the
induction axiom must be replaced by a weaker version : it can only be expressed with
A ranging over all classes
of the theory, that is, the only subsets of ℕ defined by firstorder formulas in the given
language, with bound variables and parameters in ℕ. For this, it must take the form of
a schema of axioms, one for each formula that can define a class.
But without the set theoretical framework we used to express and justify the definiteness
of recursive definitions, the successor function no more suffices to define addition and
multiplication. This leaves us with several nonequivalent versions of firstorder arithmetic
depending on the choice of the primitive language, thus nonequivalent versions of the
axiom schema of induction whose range of expressible classes depends on this language:

Bare arithmetic, with the mere symbols 0 and S, is very poor.

Presburger
arithmetic, with addition, starts to be interesting, but still cannot define multiplication.
 Full arithmetic, with addition and multiplication, is finally
able to express all recursive definitions.
Presburger arithmetic
Presburger arithmetic has been proven by experts to be a decidable theory, i.e.
all its ground formulas are either provable or refutable from its axioms. Let us
present its shortest equivalent formalization, describing the set ℕ* of nonzero natural
numbers, with 2 primitive symbols: the constant 1 and the operation +. Axioms will be
∀x,y∈ℕ*, x + (y+1)
= (x+y)+1 
(A1) : + is associative on 1 
∀A⊂ℕ*,(1∈A ∧ ∀x,y∈A,
x+y∈A) ⇒A=ℕ* 
(Min) 
∀x,y∈ℕ*, x + y
≠ y 
(F) 
In set theory, (Min) would make ℕ* a minimal {1,+}algebra. But let us regard our present
use of set theoretical notations as mere abusive abbreviations of works in firstorder logic,
as long as we only consider subsets of ℕ* defined by firstorder formulas in this arithmetical
language. In particular, (Min) will be meant as abbreviating a schema of axioms with A
ranging over all classes in this theory.
(A1) is a particular case of
∀x,y,z∈ℕ*, x + (y+z)
= (x+y)+z 
(As) : + is associative 
Conversely, (A1 ∧ Min) ⇒ (As) :
Let A={a∈ℕ* ∀x,y ∈ℕ*,
x+(y+a) = (x+y)+a}. ∀a,b∈A,
∀x,y ∈ℕ*, x + (y+(a+b))
= x + ((y+a)+b) = (x + (y+a))+b
= ((x + y)+a)+b = (x+y)+(a+b)
∴ a+b ∈ A.
(A1) ⇔ 1∈A.
(A1 ∧ Min) ⇒ A=ℕ* ∎
(As ∧ Min) ⇒ (+ is commutative), as deduced from 1∈C({1}) by the above proposition.
Now take ℕ = ℕ*∪{0} where 0∉ℕ*, to which + is extended as ∀n∈ℕ,
0+n = n+0 = n. This extension preserves its properties
of commutativity and associativity.
Define S as ℕ∋x↦ x+1.
These definitions directly imply (H0).
(Ind) ⇒ (Min) :
∀A⊂ℕ*, the set A_{0}= A∪{0}
satisfies 0∈A_{0} and
(1∈A ∧ (∀x,y∈A, x+y ∈A))
⇒ (S0∈A_{0} ∧ (∀x∈A, Sx=x+1
∈A⊂A_{0})) ⇒ A_{0}=ℕ∎
(As ∧ Min) ⇒ (Ind)
Let A∈Sub_{{0,S}}ℕ, and B
= {y∈ℕ* ∀x∈A, x+y∈A}.
∀y,z∈B, (∀x∈A, x+y∈A ∴
x+y+z∈A) ∴ y+z ∈B.
(∀x∈A, x+1 ∈A) ⇔ 1∈B ⇒ ((Min)⇒
B=ℕ*).
0∈A ⇒ (∀y∈B, 0+y∈A) ⇒ B⊂A∎
(F) ⇔ (∀x∈ℕ*, ∀y∈ℕ, x+y ≠ y)
because x+0 = x ≠ 0.
(Inj ∧ Ind ∧ A1) ⇒ (F) : ∀x∈ℕ*,
x+0 ≠ 0
∀y∈ℕ, x+y ≠ y ⇒ x+y+1
≠ y+1∎
For the converse, we need to use the order relation.
The order relation
From the operation of addition, let us define binary relations < and ≤ on ℕ and
show that they form an order and its strict order (even in first order arithmetic);
its equivalence with the definition of the order
between terms in set theory in the common particular case of ℕ with full
set theoretical induction, will be left here as an intuitive fact.x<y ⇔ ∃z∈ℕ*, y =
x+z
x≤y ⇔ ∃z∈ℕ, y = x+z
These are consequences of (Ind ∧ A1) :
 < is transitive
 x≤y ⇔ (x<y ∨ x=y)
 x<y ⇔ x+1≤y
 ∀A⊂ℕ, A≠∅ ⇒ ∃x∈A, ∀y∈A,
x≤y (this may also be interpreted either set theoretically or as a schema in first order logic)
 ∀x,y∈ℕ, x≤y ∨ y≤x
 x<y ⇒ x+z < y+z
Proofs :
 using (As), x < y < z
⇒ (∃n,p∈ℕ*, z = y+p = x+n+p)
⇒ x <z.
 obvious from definitions;
 thanks to (Ind), ℕ is a bijective
{0,S}algebra;
 x≤y ⇒ (x+1≤y ∨ x=y)
0∈{x∈ℕ ∀y∈A, x≤y}=B
∀x∈B, x+1∈B ∨ x∈A
A⋂B=∅ ⇒ (B=ℕ ∴ A=∅)
 from 4. with A={x,y}
 y = x+n ⇒ y+z = x+z+n
∎
(for 5. it is also possible to more directly prove for A={x∈ℕ
∀y∈ℕ, x<y ∨ x=y ∨ y<x}
that 0∈A and ∀x∈A, x+1∈A)
Now, (F) means that < is irreflexive, thus a strict total order
thanks to 1. and 5.
Moreover it implies ∀x,y,z∈ℕ, (x<y
⇔ x+z < y+z) and (x =
y ⇔ x+z = y+z), which
gives (Inj) as a particular case.
Proof: the direct implications were shown above; the converses are deduced from there as
< is a strict total order : one of (x<y, x
= y, y<x) must be true while only one of
(x+z<y+z,
x+z=y+z, y+z<x+z)
can.∎
Finally, ≤ is a total order with strict order < and
every nonempty subset of ℕ has a smallest element, which is unique by antisymmetry.
Arithmetic with order
It is possible to express a firstorder arithmetic with language {0,S, ≤}, stronger than {0,S}
but weaker than Presburger arithmetic, in the sense that addition cannot be defined from ≤.
Namely, it can be based on the characteristion of the order by the property:
For all n ∈ℕ, the set {x∈ℕ  n ≤ x}
is the unique A⊂ℕ such that
∀x∈ℕ, x∈A ⇔ (x = n
∨∃y∈A, Sy=x).
Its existence in ℘(ℕ) can be
deduced by induction on n; its uniqueness for a fixed n
is deduced by induction on x.