Part 3 : Algebra 1
3.1. Morphisms of relational systems and concrete
categories
Let us formalize further the concept of systems, focusing for simplicity
on those with only one type. For
any number n∈ℕ and any set E, let us denote
 E^{n} = E×...×E = E^{Vn},
set of all ntuples of
elements of E (nary product
or exponentiation).
 Rel_{E}^{(n)} = ℘(E^{n}),
set of all nary relations in E
 Op_{E}^{(n)}
= E^{En}, set of all nary operations in E.
A language L is a set of symbols, with the data of the intended arity
n_{s}∈ℕ of each s∈L. For any set E, let
L⋆E = ∐_{s∈L}
E^{ns}
A relational language is a language L of relation symbols, where each
s∈L aims to be interpreted in any set E as an n_{s}ary relation.
These form a family called an Lstructure on E, element of
∏_{s∈L} ℘(E^{ns})
≅ ℘(L⋆E)
A relational system with language L, or Lsystem, is the data
(E,E) of a set E with an Lstructure E⊂L⋆E.
The case of an algebraic language, whose symbols aim to represent operations, will be studied in 3.2.
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 imagine given a class of
Lsystems where each E is the intersection of L⋆E
with a fixed class of (s,x), denoted as a predicate s(x) for how it is naturally
curried: each symbol s∈L is interpreted in each system E as the
n_{s}ary relation s_{E} somehow independent of E,
s_{E} =
{x∈E^{ns}  s(x)}
= ⃗E(s)
E = {(s,x)∈L⋆E  s(x)} =
∐_{s∈L} s_{E}.
For any function f : E → F, let f_{L} :
L⋆E → L⋆F defined by (s,x) ↦ (s,f০x).
Morphism. Between any Lsystems E,F,
we define the set Mor_{L}(E,F) ⊂ F^{E}
of Lmorphisms from E to F
by ∀f∈F^{E},
f ∈ Mor_{L}(E,F) ⇔ ∀(s,x)∈E,
(r,f০x)∈F
⇔ (∀s∈L,∀x∈E^{ns},
s(x) ⇒ s(f০x))
⇔
f_{L}[E]⊂F ⇔ E⊂f_{L}*F.
Concrete categories
The concept of concrete category is what remains of a class of systems with their
morphisms, when we forget which are the structures that the morphisms are preserving
(as we will see this list of structures can be extended without affecting the sets of morphisms).
Let us formalize concrete categories as made of the following data (making this
slightly "more concrete" than the official concept of concrete category from other authors)
 a class of sets called objects ;
 a class of functions called morphisms; then for any objects
E,F, we define
Mor(E,F) = {f∈F^{E}  f is a morphism}
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),
g০f ∈ Mor(E,G).
The last condition is easily verified for Lmorphisms : ∀(s,x)∈E, (s,f০x)∈F ∴
(s,g০f০x)∈G.
A relational symbol s with the data of an interpretation
s_{E}⊂E^{ns}
in every object E of a given concrete category, is said to be
preserved if all morphisms of the category are also morphisms for this symbol,
i.e. ∀f∈Mor(E,F), ∀x∈s_{E}, f০x∈s_{F}.
From definitions, each symbol in a language L is preserved in any
category of Lsystems.
A category is small if its class of objects
is a set.
Preservation of some defined structures
In any given category of Lsystems, or any concrete category with a given list L
of preserved interpreted symbols, any further invariant structure
whose defining formula only uses symbols in L and logical symbols
∧,∨,0,1,=,∃ is preserved.
Indeed, for any Lmorphism
f∈Mor_{L}(E,F),
 Substituting arguments of a s∈L by a map σ to n' other variables
(∀E,∀x∈E^{n'}, s'(x)⇔s(x০σ)),
works :
s'(x) ⇒ s(x০σ) ⇒
s(f০x০σ) ⇒ s'(f০x).
 ∀s,s'∈L,n_{s}=n_{s'} ⇒
∀x∈E^{ns}, (s(x)∧s'(x)) ⇒
(s(f০x)∧s'(f০x))
 ∀s,s'∈L,n_{s}=n_{s'}⇒∀x∈E^{ns}
, (s(x)∨s'(x)) ⇒ (s(f০x)∨s'(f০x))
 For 0 and 1 it is trivial
 ∀x,y∈E, x=y ⇒ f(x)=f(y)
 ∀x∈E^{ns},(∃y∈E,
s(x,y)) ⇒ (∃z=f(y)∈F,
s(f০x, z))
Thus, for any f ∈Mor_{L}(E,F),
if a ground formula with language L using the only
logical symbols (=,∧,∨,0,1,∃), is true in E, then it is also
true in F. However morphisms may no more preserve structures defined with other
symbols (¬,⇒,∀).
The above cases of 0, 1, ∨ and ∧ are mere particular cases (the nullary and binary cases)
of the following:
 Any union of a family of preserved structures in a concrete category is a preserved structure.
 Any intersection of a family of preserved structures is also a preserved structure.
Rebuilding structures in a concrete category.
The preserved relations of any concrete category can be generated from the following
kinds of "smallest building blocks".
Proposition. In any concrete category, for any choice of tuple t of elements
of some object K, the relation s defined in each object E as
s_{E} = {f০t  f∈Mor(K,E)} is
preserved.
Proof : ∀g∈Mor(E,F), ∀x∈s_{E},
∃f∈Mor(K,E), (x = f০t ∧
g০f∈Mor(K,F)) ∴
g০x = g০f০t
∈ s_{F}.∎
In a small concrete category, the preserved families of relations are precisely all choices of
unions of those : each preserved s equals the union of those with t
ranging over s (with K ranging over all objects).
However the class of relational systems obtained by even giving in this way
"all possible structures" to the objects of an otherwise given concrete category such as
topology, may still admit more morphisms than
those we started with (like a closure).
Categories of typed systems
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 and 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}০f=h_{E}
where h_{E}:E→τ, h_{F}
:F→τ are the functions giving the type of each element.
3.2. Notion of algebra
Algebras. Given an algebraic language L, an Lalgebra is the data
(E,φ) of a set E with an Lalgebraic structure φ : L⋆E →
E, sum of a family of interpretations of each symbol s∈L as an operation
⃗φ(s)∈Op_{E}^{(ns)}.
Again, we shall usually assume a class of Lalgebras with only one choice of algebraic
structure on each set:
s_{E} : E^{ns}
→ E
φ_{E} = ∐_{s∈L}
s_{E} : L⋆E → E
This would be the case if they were defined by restricting
a role played by each s∈L as an n_{s}ary operator
independent of E, but this option may have to be rejected to let the
constant symbols be interpreted in disjoint algebras, taking equal values in the one and distinct
ones in another.
These form 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 the
copy s'∈L' of each s∈L has increased arity
n_{s'} = n_{s}+1, so that
L'⋆E ≡ ∐_{s∈L}
E^{ns}×E ≡
(L⋆E)×E ≡ {(s,x,y)  s∈L ∧
x∈E^{ns} ∧ 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 resulting condition for an f ∈ F^{E} to be a
morphism is equivalent : (∀(x,y)∈E,
(f_{L}(x),f(y))∈F) ⇔
(∀x∈L⋆E, φ_{F}(f_{L}(x))=
f(φ_{E}(x))).
Subalgebras. An Lsubalgebra of an Lalgebra E is an element of
Sub_{L} E = {A⊂E 
φ_{E}[L⋆A]⊂A}. It is also an Lalgebra, with structure
φ_{A} restriction of φ_{E} to L⋆A.
If a formula of the form (∀(variables), formula without
binder) is true in E, then it is true in each A∈ Sub_{L}E.
Images of algebras. For any two Lalgebras E,F,
∀f ∈Mor_{L}(E,F), Im f ∈ Sub_{L}F.
Proof using 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 ∎
(An extension of this result to noninjective morphisms between algebras with infinitary operations,
would require AC)
Let us generalize these concepts to categories of relational L'systems
(E,E) whose structure E ⊂ (L⋆E)×E
no more needs to be functional:
Stable subsets. The concept of stability of a subset A of an
L'system E generalizes that of subalgebra :
A ∈ Sub_{L} E ⇔ (E_{*}(L⋆A)
⊂A) ⇔ (∀(s,x,y)∈E,
Im x⊂A ⇒ y∈A).
We have E ∈ Sub_{L} E.
Stability is no more preserved by direct images by morphisms, but is still preserved by preimages:
Preimages of stable subsets. ∀f∈Mor_{L}(E,F),
∀B∈Sub_{L}F, f *(B) ∈ Sub_{L}
E.
Proof. Let A=f *B.
For Lalgebras,
∀(s,x)∈L⋆A, f০x
∈ B^{ns}∴ f(s_{E}(x))
= s_{F}(f০x) ∈B ∴
s_{E}(x)∈A.
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_{F}(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, we denote
〈A〉_{L,E} or simply 〈A〉_{L}, the
smallest Lstable subset of E including A
(called Lsubalgebra of E generated by A if E is an algebra):
〈A〉_{L}=
∩{B∈Sub_{L}E  A⊂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 or is a generating subset of E if 〈A〉_{L}=E.
Minimal subalgebra. For any L'system E,
its minimal stable subset (or minimal subalgebra for an Lalgebra)
is defined as
Min_{L}E = 〈∅〉_{L,E}
= ∩Sub_{L}E ∈ Sub_{L}E.
An Lalgebra E is minimal when E = Min_{L}
E, or equivalently Sub_{L}E
= {E}.
Proposition. For any L'system E, ∀A∈Sub_{L}E,
 ∀B⊂A, B∈Sub_{L}E ⇔
B∈Sub_{L}A
 Min_{L}A = Min_{L}E
 A = Min_{L}E ⇔ A is minimal.
Proof: Min_{L}E ⊂ Min_{L}A because
Sub_{L} A ⊂ Sub_{L} E;
Min_{L} A ⊂ Min_{L} E
because ∀B∈Sub_{L}E,
A⋂B ∈ Sub_{L}A.
∎
Among subsets of E, other minimal L'systems are included
in Min_{L} E but are not stable.
The stable subset generated by A is the minimal one for the extended language
with A seen as a set of constants:
〈A〉_{L,E}= Min_{L∪A} E.
Injective, surjective algebras. An Lalgebra (E,φ_{E})
will be called injective if φ_{E} is injective, and surjective if
Im φ_{E} = E.
Proposition. For any Lalgebras E, F,
 ∀A⊂E, Im φ_{E}⊂A ⇒
A∈Sub_{L}E.
 Any minimal Lalgebra is surjective.
 Min_{L}E =
φ_{E}[L⋆Min_{L}E] ⊂ Im φ_{E}
 ∀A⊂E, 〈A〉_{L} =
A∪φ_{E}[L⋆〈A〉_{L}] ⊂
A∪Im φ_{E}.

∀f ∈Mor_{L}(E,F),
f [Min_{L}E] = Min_{L}F ∧
∀A⊂E, f [〈A〉_{L}] =
〈f [A]〉_{L}
Proofs:  φ_{E}[L⋆A] ⊂
Im φ_{E}⊂A ⇒ A∈Sub_{L}E
 Im φ_{E} ∈ Sub_{L}E
 Min_{L}E is surjective
 A∪φ_{E}[L⋆〈A〉_{L}]
∈ Sub_{L} 〈A〉_{L}
 ∀B ∈ Sub_{L}F, f
*(B)∈Sub_{L} E ∴ Min_{L}E
⊂ f*(B) ∴ f [Min_{L}E]⊂B.∎
Injectivity lemma. 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.
 For each uniqueness quantifier Q (either ∃! or !),
B = {y∈F  Qx∈E, y =
f(x)} ∈ Sub_{L}F
They are essentially the same but let us write separate proofs:
 ∀(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.
 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). ∎
Schröder–Bernstein theorem.
If there exist injections f: E → F and
g: F→ E then there exists a bijection between E and F.
Proof : replacing F by the bijectively related set Im g, simplifies things to
the case F⊂E.
Then a bijection from E to F can be defined as x ↦
(x∈〈E\F〉_{{f}} ? f(x) : x).∎
3.3. Special morphisms
Let us introduce diverse possible qualifications for morphisms of
relational systems.
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).
Embeddings and isomorphisms
Strong preservation. A relation symbol r interpreted as
r_{E} in E
and r_{F} in F is strongly preserved by a function
f ∈ F^{E}, if both r and ¬r are preserved :
∀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 :
E = f_{L}*(F).
Injectivity is usually added to
the definition of the concept of embedding, 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.
Isomorphism. Between objects E
and F of a concrete category, an isomorphism is a bijective morphism
(f ∈Mor(E,F) ∧ f : E ↔ F)
whose inverse is a morphism (f^{ 1}∈Mor(F,E)). In
the case of relational systems, isomophisms are the bijective embeddings;
injective embeddings are isomorphisms to their images.
Two objects E, F of a category are said to be isomorphic (to each other) if
there exists an isomorphism between them. This is an equivalence predicate, i.e. it works as an equivalence relation on the class of
objects (in this category).
The isomorphism class of an object in a category, is the class of all objects
which are isomorphic to it. Then an isomorphism class of objects in a category, is a class
of objects which is the isomorphism class of some object in it (independently of the choice).
Embeddings of algebras
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 between algebras f ∈ Mor_{L}(E,F),
is injective whenever 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
f_{L}^{1} = (f^{1})_{L}
∴ 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}.
Elementary embeddings
Lembeddings still strongly preserve structures defined with symbols in L
and logical symbols ∧,∨,0,1,¬, and also = in
the case of injective embeddings.
Thus, they also preserve invariant structures whose
formula may use symbols of L and ∧,∨,¬,0,1,∃ but all occurrences
of ∃ precede those of ¬.
Now the full use of firstorder logic comes by removing this restriction on the order of use of
logical symbols: 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.
Isomorphism ⇒ Elementary embedding ⇒ Embedding ⇒ Injective morphism
Elementary equivalence. Different systems are said to be elementarily equivalent,
if they have all the same true ground firstorder formulas. The existence
of an elementary embedding between systems implies that they are elementarily equivalent.
The most usual practice of mathematics ignores the diversity of elementarily
equivalent but nonisomorphic systems, as well as nonsurjective elementary
embeddings. However, they exist and play a special role in the foundations of
mathematics, as we shall see with Skolem's paradox and
nonstandard
models of arithmetic.
Endomorphisms, automorphisms.
An endomorphism of an object E
in a category, is an element of Mor(E,E) = End(E).
For any set E, ∀f∈E^{E}, ∀A⊂E,
A ∈ Sub_{{f}}E ⇔ f ∈ End_{{A}}E.
An automorphism of an object E is an isomorphism
of E to itself:
Automorphism ⇔ (Endomorphism ∧ Isomorphism)
An endomorphism f∈ End(E) may be an embedding but not an automorphism : just an
isomorphism to a strict subset of E.
But if it is an invariant elementary embedding then it is an automorphism:
Im f is also invariant (defined by
∃y∈ E, f(y)=x)
∀x∈ E,
x∈Im f ⇔ f(x)∈ Im f
Im f = E. ∎
3.4. Monoids
Transformations monoids
A transformation of a set E, is a function from E
to itself. The full transformation monoid of E is the set E^{E}
of all transformations of E, seen as an algebra
with two operations: the constant Id, and the binary
operation ০ of composition.
A transformation monoid of E is a set M of transformations of E forming an {Id,০}algebra,
M∈Sub_{{Id,০}}E^{E} :
 Id_{E} ∈ M
 ∀f,g∈M, g০f ∈ M.
The set of endomorphisms of a fixed object in a concrete category is a transformation monoid. Any transformation monoid can
be seen as a concrete category with only one object.
Trajectories
For any set of transformations L ⊂ E^{E}, seen
as a set of function symbols interpreted in E, ∀x∈E,
〈{x}〉_{L} = {f(x)f∈〈L〉_{{Id,০}}}.
As we shall formalize later, this is because both mean the same : the set of all composites
of any number of functions in L, applied to x.
But here is a simple proof, denoting A=〈{x}〉_{L}, M=〈L〉_{{Id,০}} and
K={f(x)f∈M}:
Proof of A ⊂ K
Id_{E}∈M ∴ x∈K
(∀g∈L, ∀y∈K,
∃f∈M, y=f(x)∧g০f∈M ∴
g(y)∈K) ∴ K∈Sub_{L}E
Proof of K ⊂ A
L ⊂ {f∈E^{E} A ∈
Sub_{{f}}E} = End_{{A}}E
∈ Sub_{{Id,০}}E^{E}
M⊂End_{{A}}E
∀f∈M,
x ∈ A ∈ Sub_{{f}}E
∴ f(x)∈A. ∎
The trajectory of an element x∈E by a transformation monoid
M of E, is the set it generates:
〈{x}〉_{M} = {f(x)f∈M} ⊂ E
Monoids
A monoid is an algebra behaving like a transformation
monoid, but without specifying a set which its elements may
transform. As both symbols Id and ০ lose their
natural interpretation, they are respectively renamed as e and •.
Namely, the concept of monoid is the theory made of
 One type
 Two
operation symbols
 a constant symbol e of "identity";
 a binary operation • of "composition"
 Axioms
 Associativity : ∀x,y,z, x • (y • z)
= (x • y) • z so that either term can be more simply
written x • y • z.
 Identity : ∀x, x • e = x
= e • x
Both equalities in the last axiom may be considered separately,
forming two different concepts
 a left identity of a binary
operation • is an element e such that ∀x,
e • x = x
 a right identity of • is an element e' such that ∀x,
x • e' = x
If both a left identity and a right identity exist then they are
equal : e = e • e' = e' which makes
it the identity of • (the unique element satisfying each
identity condition). The existence of a right identity implies the
uniqueness of the left identity, but without right identity, several
left identities may coexist (and similarly switching left and right).
From any associative operation on a set E we can form a
monoid by adding the identity e as an extra element, E'=E⊔{e}, to which the interpretation of • is extended as
determined by the identity axioms (preserving associativity), but where any identity element
which could exist in E loses its status of identity element in E'.
As the identity axiom ensures the surjectivity of •, every embedding
between monoids is injective.
Cancellativity
An element x is called left cancellative for an operation • if the left
composition by x is injective: ∀y,z,
x•y=x•z ⇒ y=z.
Similarly it is
right cancellative if x•z=y•z ⇒ x=y.
If a right identity e is left cancellative then it is the unique right identity :
e • e' = e = e • e ⇒ e' = e.
An operation is called cancellative if all elements are cancellative on both sides.
Any submonoid of a cancellative monoid is cancellative. Not all monoids are cancellative:
the monoid of addition in {0,1, several} is not cancellative as 1+several = several+several.
Submonoids and morphisms of monoids
Any {e,•}subalgebra of a monoid is a monoid, thus called a
submonoid.
Modifying of the formalization of monoid by replacing the status of e as a constant
by ∃e in the identity axiom, would weaken the concepts of submonoids
and morphism (allowing more of them) as follows.
For any monoid (M,e,•), any set X with a binary operation
▪, and any function f ∈ Mor_{{•}}(M,X),
 its image is a monoid (A,a,▪) where A = Im f
and a = f(e)
 f is a morphism of monoid from M
to this monoid A.
If the target forms a monoid (X,e',▪)
then (by uniqueness of the identity in A)
a = e' ⇔
e' ∈ A ⇔ A ∈ Sub_{{e, ▪}} X
but these equivalent formulas may still be false, unless a is cancellative on one side
(a▪a = a = a▪e' ⇒ a = e').
Antimorphisms. The opposite of a monoid is the monoid
with the same base set but where composition is replaced by its transpose. An
antimorphism from (M,e,•) to (X,e',▪) is a morphism
f from one
monoid to the opposite of the other (or equivalently viceversa):
f(e)=e'
∀a,b∈M, f(a • b) =
f(b) ▪ f(a)
3.5. Actions of monoids
Left actions
After monoids, introduced as sets of transformations, were deprived of this role,
it can be given back to them as follows.
A left action of a monoid (M,e, •) on a set X, is an operation ⋅ :
M×X → X such that
 ∀x∈X, e ⋅ x = x;
 ∀a,b∈M, ∀x∈X,
(a • b) ⋅ x = a ⋅ (b ⋅ x).
Seing M as a set of function symbols, an Malgebra (X, ⋅)
satisfying these axioms is called an Mset.
In curried view, a left action of M on X is a
{e,•}morphism from M to the full transformation monoid X^{X}.
Effectiveness and free elements
A left action is said effective if this morphism is
injective:∀a, b ∈ M, (∀x∈ X,
a·x = b·x) ⇒ a=b
letting e
and • be defined by (= unique for) the axioms on the given action, like the axioms for functions ensured the sense
of the definitions of Id and ০ from the
function
evaluator.
An element x∈X of an Mset, is free
if the function it defines from M to X
is injective. The existence of a free element implies that the
action is effective:
(∃x∈X,
∀a≠b∈M, a·x≠b·x)
⇒ (∀a≠b∈M, ∃x∈X,
a·x≠b·x)
General example.
Any transformation monoid M of a set E acts by restriction on any Mstable
subset A of E, i.e. any preserved A⊂E in a concrete category where
M = End E. Thus, the monoid of endomorphisms of any typed system E=
∐_{i∈I} E_{i},
acts on every type E_{i} it contains.
Acts as algebraic structures
Let M, X be given structures of Malgebras by any
operations • : M×M → M and ⋅ : M×X → X.
Then denoting
∀x∈X, h_{x} = (M∋a ↦ a ⋅ x), we have
directly from definitions
h_{x}(e) = x ⇔ e ⋅ x = x
h_{x} ∈ Mor_{M}(M,X) ⇔
∀a,b∈M, (a • b) ⋅ x = a ⋅ (b ⋅ x)
h_{e} = Id_{M} ⇔
(∀a∈M, a • e = a) ⇒
∀g∈Mor_{M}(M,X),
g=h_{g(e)}.
So in the formula ∀g∈X^{M}, ∀x∈X,
g=h_{x} ⇔
(g∈Mor_{M}(M,X) ∧ g(e)=x)
the ⇒ expresses the axioms of action; the converse is implied by the last axiom of monoid
beyond the copies of both axioms of action, which comes as a particular case once X
is replaced by M : (Id_{M} ∈
Mor_{M}(M,M) ∧ Id_{M}(e)=e)
⇒ h_{e} = Id_{M}
Right actions
A right action of a monoid M on a
set X, is an operation ⋅ : X × M → X
such that
 ∀x∈X, x ⋅ e = x;
 ∀a,b∈M, ∀x∈X, (x
⋅ a) ⋅ b = x ⋅ (a • b)
It defines an antimorphism from M to X^{X}.
Commutants
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).
Such A,B are said to commute with each other as each element of A
commutes with each element of B.
A binary operation # in a set E, is called
commutative when C(E) = E, i.e.
∀x,y∈E, x#y
= y#x.
Proposition. For any associative
operation # on a set E, ∀A⊂E,
 C(A) ∈ Sub_{#}F
 If A⊂C(A) and 〈A〉_{#}=E then # is commutative
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
⇒ E=C(A)
⇒ A⊂C(E) ∈ Sub_{#}F ⇒
C(E) = E.
Centralizers
In monoids, commutants are called centralizers; they are submonoids, as e
commutes with all elements, which can be seen in any transformation monoid M⊂E^{E}as
∀A⊂M, C_{M}(A) = M ∩
End_{A} E.
This concept will be later generalized to clones
of operations with all arities.
If A⊂C(A) and 〈A〉_{{e,•}}=E
then # is commutative.
When 2 monoids acting on the same set X commute with each other
in X^{X}, this commutation appears as an
associativity law when seeing them as acting on a
different side: a∈M left acting on X commutes with b∈N
right acting on X when ∀x∈X,
(ax)b = a(xb)
Let us verify that both axioms of monoid suffice to gather all
properties of transformation monoids, and even all properties of
monoids of endomorphisms.
Representation theorem. For any monoid M there exists a language
L of function symbols and an Lalgebra X
such that the monoid End_{L} X is
isomorphic to M.
The proof is linked to the above formulas on acts, as will be used in 3.7 for
a wider result; let us write it separately.
Let L and X be two copies of M.
Give L the right action on X copied from the
composition in M (whose axioms of monoid give those of
action).
Let f ∈ Mor(M, X^{X})
represent the left action of M on X also copied
from the composition in M.
Im f ⊂ End_{L} X by associativity
of the operation from which both actions on opposite sides are
copied.
f is injective because the copy k of e in X
is a free element.
End_{L} X ⊂ Im f because
∀g∈End_{L}X, ∃u∈M,
g(k)=uk ∴ (∀x∈X, ∃s∈L,
ks=x ∴ g(x) = g(ks) =
g(k)s=uks=ux)
∴ g=f(u) ∎
The bijections identifying L and X as copies of M,
finally do not play any special role: while they are definable from
k, this k itself may be not unique in the role its
plays here.
Trajectories by commutative monoids
Let a monoid M act on a set X, and let k∈X.
The trajectory Y of k by M is
stable by M, thus defines a morphism of monoid from M to
Y^{Y} with image a transformation monoid N of Y.
Forgetting M and X, we have a monoid N with an effective action
on Y generated by k.
Now if N is commutative (which is the case if M is commutative) then k
is free for the action of N (thus Y can be seen as a copy of N).
The proof is easy and left as an exercise.
3.6. Invertibility and groups
Permutation groups
A permutation of a set E is a bijective
transformation of E.
The set ⤹E ={f ∈ E^{E} Inj f ∧ Im
f=E} = {f ∈ E^{E}f:E↔E}
of all permutations of E, is a transformation monoid of E called the
symmetric group of E.
A permutation group G of a set E, is a transformation monoid of
E such that G ⊂
⤹E ∧ ∀f∈G, f^{ 1}∈ G.
It will be seen as an algebra with one more operation : the inversion function f↦f^{ 1}. So
it is a {Id,०, ^{1}}subalgebra of ⤹E.
Unlike full transformation monoids and symmetric groups, the concepts of
transformation monoid and permutation group make sense independently of the
powerset, as sets of transformations with stability requirements expressed as firstorder axioms,
ignoring the containing sets E^{E} or ⤹E.
Trajectories are usually called orbits in the case of a permutation group.
As noted in the example in 2.7,
the binary relation on E defined by trajectories by a transformation monoid
M of E, (y∈〈{x}〉_{M})
is a preorder; it is an equivalence relation if M is a permutation group.
Inverses
In a monoid (M,e,•), the formula x•y=e
is read "x is a left inverse of y", and "y
is a right inverse of x".
Seeing M as a transformation monoid by left action on itself, this x•y=e
is interpreted as relating transformations :
∀z,t∈M, y•z = t ⇒ x•t = z
We say x is right invertible
when a right inverse y exists; similarly, y is left invertible.
As right invertible functions are surjective and left invertible functions are injective, the left invertibility of y is equivalent to saying the right composition by y is surjective ({z•yz∈M} = M)
and implies that y is left cancellative:x•y =
e ⇒ ∀z∈M, z•x•y = z
∀z,t∈M, (y•z = y•t ∧ x•y=e) ⇒
(z = x•y•z = x•y•t
= t)
An element x both left invertible and right invertible is called invertible.
Then any left inverse and any right inverse of x are equal,
and thus unique, called the inverse of x and written x^{1}:
y•x = e = x•z ⇒ y
= y•x•z = z
If a left invertible element y is also right cancellative then it is invertible: x•y=e ⇒ y•x•y
= e•y ⇒ y•x=e.
This characterization of invertible elements also makes sense for an element x of an
Mset X: saying that x is both generating and free, means that the morphism h_{x}∈ Mor_{M}(M,X) is both surjective
and injective, thus an isomorphism between the Msets M
and X. Then we might still say it has an inverse in the form of an Mmorphism
from X to M.
If x commutes with an invertible element y then it also commutes with its inverse z:
x•y=y•x ⇔ x=y•x•z ⇔ z•x=x•z
An element x of a monoid is called involutive
if it is its own inverse (equivalently on one or both sides): x•x=e.
This qualifies an element of a monoid (such as a
transformation), regardless the replacement of this monoid by a submonoid
containing it (a transformation monoid).
Groups
A group is a monoid where all elements are invertible.
For a transformation monoid, being a permutation group is equivalent to being a group.
The set of invertible elements in any monoid, is a group :
 it is a submonoid : if x and y are invertible then x•y is invertible,
with inverse y^{1}•x^{1} (like in 2.6).
 Inverses are invertible (and by uniqueness of the inverse, inversion is an involutive transformation of any group).
As any group is cancellative, any submonoid of a group is also cancellative. This has a partial converse (not very easy to prove): any commutative cancellative monoid can be embedded into a commutative group.
We shall soon see the example of the monoid ℕ embedded in the group ℤ.
Subgroups. The concept of subgroup of a group, is equivalently defined as
 a submonoid which is a group, or
 a subalgebra for the language of groups {e, •,^{1}} extending the one {e, •} of monoids,
with the function symbol ^{1} of inversion.
The interpretation of inversion is determined from those of • and e by the axiom
(the last axiom of group beyond the axioms of monoid)
∀x, x•x^{1} =
x^{1}•x = e
The admission of inversion as a symbol, has no effect on morphisms: any morphism of monoid
f from a group G to a monoid M preserves the inversion relation,
thus its image G' is a group (subgroup of the group of invertible elements of M), and
f is a group morphism from G to G'.
Thus, an action of a group G on a set
X, can be equivalently conceived as an action of monoid, or as a group morphism from G
to the symmetric group of X.
By the representation theorem,
any group is isomorphic to some permutation groups,
among which the group of automorphisms of an algebra.
As inversion is an antimorphism,
it switches any left action ▪ of G on X into a right
action • by ∀x∈ X, ∀g∈G,
x•g = g^{1}▪x.
In a group, the subgroup generated by a subset A, coincides with the
submonoid G generated by A∪A where A={x^{1}x∈A} .
Proof: to check that G is stable by inversion, notice that the definition of G is
stable by inversion (which is involutive), thus G = G.
3.7. Categories
Categories (also called abstract categories for insistence) differ from concrete
categories, by forgetting that objects
are sets (ordered by inclusion) and that morphisms are functions. The concept of monoid
was the particular case of an abstract category with only one object. The general
case of categories is similarly formalized as the data of:
 A class of "objects" of that category, regarded as pure elements (ignoring any inclusion order);
the category is called small if this class is a set;
 to any objects A,B is given a set Mor(A,B) of
«morphisms from A to B»; these are regarded
as pairwise disjoint sets of pure elements;
 to any object A is given a morphism 1_{A}∈Mor(A,A);
 to any 3 objects A,B,C is given a
composition operation we shall abusively denote by the same
symbol • : Mor(B,C)×Mor(A,B)→Mor(A,C) ;
satisfying the axioms
 For any objects A,B, ∀x∈Mor(A,B),
x•1_{A} = x = 1_{B}•x
 For any objects A,B,C,D, ∀x∈Mor(A,B), ∀y∈Mor(B,C),
∀z∈Mor(C,D), (z•y)•x = z•(y•x)
In an abstract category, for any objects E and F, an isomorphism f
from E to F is an f∈Mor(E,F) such that
∃g∈Mor(F,E), g•f=1_{E} ∧
f•g=1_{F}. This g is unique, called the inverse of f.
(This generalizes both isomorphisms in
concrete categories, and invertible elements in monoids)
Representation theorem. Any small category can be interpreted
as that of all morphisms in some given list of typed algebras.
Let us fix the set of types as a copy of the set of
objects : from each object X we make a type X'
(not giving to this bijective correspondence any special status).
Each object M is interpreted as a system where each type X' is
interpreted as the set Mor(X',M).
As a language, let us take all morphisms between types: the set of
function symbols from type X' to type Y' is defined as
Mor(Y',X') (with reverse order, as symbols act on the right).
The proof goes on just like with
monoids.∎
Functions defined by composition
In any category, any f ∈ Mor(E,F) defines functions
by currying composition with other morphisms to or from another object X:
let us denote (almost following wikipedia but adapted to our concept of concrete category)
 Hom(X, f) = (Mor(X, Dom
f)∋g↦ f•g),
with target Mor(X,F) for any target F of f.
 Hom_{F}(f, X) = (Mor(F,
X)∋g↦ g•f), with target
Mor(E, X). Simplified as Hom(f,X) in abstract
categories where f determines F.
The former respects composition, while the latter 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, g•f)
Hom_{F}(f, X) ০
Hom_{G}(g, X) =
Hom_{G}(g•f, X)
The concepts of cancellativity and invertibility are generalized to categories as follows.
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),
g•f=h•f ⇒ 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.
Sections, retractions. When g•f=1_{E} we say that f
is a section of g, and that g is a retraction of f.
 A morphism f∈Mor(E,F) is a section
(or section in F if the category is concrete), if
1_{E}∈Im(Hom_{F}(f,E)),
i.e. ∃g∈Mor(F,E), g•f=1_{E}.
Then f is monic and for all objects X we have
Im(Hom_{F}(f,X)) = Mor(E,X). 
A morphism g∈Mor(F,E)
is a retraction (or retraction on E if the
category is concrete), if 1_{E}∈Im(Hom(E,g)),
i.e. ∃f∈Mor(E,F), g•f=1_{E}.
Then g is epic and for all objects X we have
Im(Hom(X, g)) = Mor(X, F).
Proof: if g•f=1_{E} then for all objects X,
Hom_{F}(f,X) ০ Hom_{E}(g,X)
= Hom_{E}(1_{E},X)
= Id_{Mor(E,X)}, thus
 Hom_{E}(g,X) is injective (g is epic)
 ∀h∈Mor(E,X), h =
h•g•f
= Hom_{F}(f,X)(h•g).
Similarly, Hom(X,g) ০ Hom(X,f) =
Hom(X,1_{E}) = Id_{Mor(X,E)} thus
f is monic and Im(Hom(X,g)) = Mor(X,F).∎
If f is an isomorphism then Hom(X,f) and Hom(X,g)
are bijections, inverse of each other, between Mor(X,E) and Mor(X,F).
Any epic section is an isomorphism. Any monic retraction is an isomorphism.
These dependencies between qualities of morphisms, can be mapped as follows:
Isomorphism ⇔ (Retraction ∧ Monomorphism) ⇔ (Section ∧ Epimorphism)
Retraction ⇒ Surjective morphism ⇒ Epimorphism
Section ⇒ Embedding ⇒ Injective morphism ⇒ Monomorphism
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.
When they exist, all such objects are
isomorphic, by a unique isomorphism between any two of them:
For any initial objects X, Y, ∃f∈Mor(X,Y),
∃g∈Mor(Y,X), g•f ∈ Mor(X,X) ∧
1_{X} ∈ Mor(X,X) ∴ g•f = 1_{X}.
Similarly, f•g = 1_{Y}.
Thus f is an isomorphism, unique because Mor(X,Y)
is a singleton.∎
By this unique isomorphism, X and Y may be treated as identical to
each other. We 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:
 Singletons are final objects (where all relations are
constantly true); this also goes in categories of algebras.
In the case of multitype systems, final objects are made of one singleton per type.
 The only initial object is the empty set (where any nullary
relation, i.e. boolean constant, is false).
Exercise. Given two fixed sets K and B, consider the category where
 Objects are all (X,φ) where X is a set and φ: X×K→B ;
 Mor((X,φ),(Y,φ') = {f∈Y^{X} 
∀a∈X,∀k∈K, φ(a,k) = φ'(f(a),k)}.
Does it have an initial object ? a final object ?
Categories of acts
From a concrete category C, let C' be the category where
 Objects are all (X,x)
where X is an object of C and x∈X
 Mor((X,x),(Y,y)) =
{f∈Mor(X,Y)  f(x)=y}.
Then we have
 If C is the category of Msets for a monoid (M,e, •) then,
seeing M as an Mset interpreting • as left action, (M, e) is
an initial object of C' ; initial objects are the (X,x)
where x is a free and generating element of X.
 Conversely, for any initial object (M,e) of C', if that exists, there is a unique
monoid structure (M,e,•) with an action on every other object
X of C (beyond • on M itself), such that for all objects X,
Y of C we have Mor(X,Y) ⊂ Mor_{M}(X,Y)
and Mor(M,X) = Mor_{M}(M,X).
Proof. 1. see properties of acts
as algebraic structures and inverses.
2. Defining ∀x∈X, h_{x}∈Mor(M,X) ∧
h_{x}(e)=x, provides an Mstructure on
each X which interprets each a∈M in X as defined by the tuple (e,a).
So they are preserved: Mor(X,Y) ⊂ Mor_{M}(X,Y),
which implies the axioms of Macts.
The composition in M coming as this
Mstructure for M=X, satisfies the same axioms.
The last axiom of monoid, h_{e} = Id_{M} results from
Id_{M}∈Mor(M,M) and ensures the reverse inclusion:
∀g∈Mor_{M}(M,X),
g=h_{g(e)} ∴ g∈Mor(M,X). ∎
This monoid (M,e,•) is essentially the opposite of the monoid
End(M). Indeed for all a, b∈M, h_{a},
h_{b}∈End(M) and
h_{a}(h_{b}(e))=h_{a}(b)=b•a.
3.8. Algebraic terms and term algebras
Algebraic drafts
This section aims to formalize as systems
in a set theoretical framework, the concept of algebraic term, particular case of terms
(first intuitively
introduced in 1.5), with a purely algebraic language L
(no predicate, logical symbols nor binders) and a set V of variable symbols,
still assuming only one type (the generalization to many types is easy).
For convenience we need to start with a wider class of systems: let us call Ldraft
any 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, with domain O_{D} = Im D = D\V
called the set of occurrences in D (which differs from our first intuitive
notion of occurrences), and its complement V_{D} = D∩V
is the set of used variables of D;
 〈V_{D}〉_{L}= D (wellfoundedness condition).
Let us denote ∀x∈O_{D},
Ψ_{D}(x) = (σ(x), l_{x})
∈ L⋆D where σ∈L^{OD} and
l_{x}∈D^{nσ}^{(x)}.
Equivalent formulations of wellfoundedness are
∀A⊂O_{D}, (∀x∈O_{D},
Im l_{x} ⊂ A∪V ⇒ 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} = ∅
A ground draft is a draft with no variable, i.e. V_{D}=∅. Thus,
Ψ_{D}: D→ L⋆D and Sub_{L}D
= {D}.
Variables in a draft may be reinterpreted as constants: extending Ψ_{D}
by Id_{VD} : V_{D} → V,
forms a ground (L∪V)draft.
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 a subdraft. Moreover,
any union of subdrafts is also a subdraft (unlike for subalgebras with an operation
with arity >1, as from arguments in different subalgebras the result may escape their union).
The subdraft cogenerated by a subset of a draft, is the intersection of all subdrafts
that include it. A term is a draft cogenerated by a single element called its root.
Each x in a draft D defines a term T_{x} with
root x, subdraft of D cogenerated by {x}.
Each draft D is ordered by x ≤ y ⇔ x∈T_{y}.
It is obviously a preorder.
Proof of antisymmetry (uniqueness of the root):
∀x∈O_{D}, V_{D} ⊂
A={y∈Dx∉T_{y}} which is a subdraft
by transitivity of ≤.
x∉A ∴ ∃z∈O_{D}\A,
Im l_{z}⊂A.
A∪{z} is a subdraft, thus T_{z}⊂A∪{z}
by definition of T_{z}.
z∈O_{D}\A ∴
x∈T_{z} ∴ x=z.
Thus x is determined by A. ∎
More properties of this order will be seen for natural numbers in 3.6, and in the
general case with 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
σ_{E}০f_{OD}
= σ_{D}
∀x∈O_{D},
l_{f(x)}=f০l_{x}
Another concrete category is that of drafts with variablespreserving morphisms, where V
is fixed and morphisms f from a draft D are subject to
f_{VD} = Id_{VD}.
This is equivalently expressed reinterpreting variables as constants, as the category of ground
(L∪V)drafts.
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},
also expressible as
∀x∈O_{D}, f(x) =
σ(x)_{E}(f০l_{x})
Any interpretation v∈E^{V} of variables in an algebra E
determines an interpretation of any draft D in E. To simplify formulations,
restricting v to V_{D} reduces the problem to the case
V_{D}=V.
Theorem. For any Ldraft D with V_{D}=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. Let us now prove 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 ∎
Operations defined by terms
Any element t of an Ldraft D defines a Vary
operation symbol, interpreted in each Lalgebra E by ∀v∈E^{V},
t_{E}(v) = h(t)
for the unique h∈Mor_{L}(D,E)
such that h_{VD}=v_{VD}.
This was introduced as the operation defined
by a term, namely the Lterm with root t, which can replace D here
without modifying the interpretations of t.
These operations are preserved by all Lmorphisms, thus can be added to L
without changing the category of Lalgebras. This is deducible in two ways
from seen preservation
properties in concrete categories:
A partial proof, namely a schema of one proof for each concretely written term, reexpresses
the term defining an operation as a formula defining a relation (its graph),
using symbols ∃ and ∧.
This repeatedly uses the preservation property for each occurrence of symbol it contains.
 The full proof (for abstractly conceived terms regardless their size) sees them as relations
defined by tuples (Id_{V},t).
Condensed drafts
A draft D is condensed if, equivalently, 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 exists an algebra where their interpretations differ.
1.⇒2. if D≠∅ (otherwise replace D by a singleton), ∃φ∈D^{L⋆D},
φ০Ψ_{D} = Id_{OD}, i.e. Id_{D}
interprets D in (D,φ).
2.⇒3.
3.⇒1. ∀x,y∈O_{D}, if Ψ_{D}(x) =
Ψ_{D}(y) then x,y have the same interpretation in every algebra.
If L only has symbols with arity 0 or 1 then every Lterm is condensed.
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}. With this is usually assumed that
V=V_{D}=E\Im φ_{E}, i.e.
variables of term algebras are usually used as an isolated set rather than as a
subset of any larger set of available variables. This term algebra is ground if
V=∅, i.e. E=Im φ_{E}. So, a ground term
algebra is an algebra both minimal and injective, and thus also bijective.
If L has no constant then ∅ is a ground term Lalgebra.
If L only has constants, then ground term Lalgebras are the copies of L.
From any injective Lalgebra (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.
Whenever present as object,
any ground term Lalgebra is a final object in any category of ground
Ldrafts, and an initial object in any category of Lalgebras.
In any variablespreserving category of Ldrafts for a given V,
any term Lalgebra (E,φ_{E}) with
E\Im φ_{E}=V, when present, is a final object.
Proposition. For any ground term Lalgebra K
and any injective Lalgebra M, the unique
f∈Mor_{L}(K,M) is injective.
Proof 1. By the injectivity lemma,
{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 Lalgebra, so the morphism f between initial Lalgebras
K and Im f is an isomorphism.
Role of term algebras as sets of all terms
As any draft can be seen as a family of terms, any term algebra (final draft)
F precisely plays the more role of the "set of all terms"
(with the given list V of variable symbols), as it contains exactly one element image
of each term (operation symbol defined by a term) in the sense that two terms everywhere
interpreted as the same operation have the same image.
Namely, any
Lterm T with root t and V_{T}⊂V, is
represented in F by the image Im f with root f(t) of
the f∈Mor(T,F) such that
f_{VT} = Id_{VT}.
Im f has the same interpretation as T in any Lalgebra E extending any
v∈E^{V} because the unique
g∈Mor_{L}(F,E) and
h∈Mor_{L}(T,E) extending v, are related by
h=g০f, thus
h(t)=g(f(t)).
Im f is condensed, and f is injective if and only if T is condensed.
For any subset A of an Lalgebra E, the Lsubalgebra
〈A〉_{L} of E is the set of
values of all Lterms with variables interpreted in A; it is the image of the interpretation
in E, of a term algebra whose set of variables is a copy of A.
The unary term Lalgebra M, essentially determined by L, represents
all function symbols definable by unary Lterms. As a particular case of
category of acts, its interpretations
define actions of monoid preserved by morphisms across
all Lalgebras. Conversely if L is made of function symbols then any
Mmorphism is an Lmorphism, because each s∈L plays the
same role as s_{M}(e)∈M.
A free monoid on a set X is a unary term Xalgebra
seeing X as made of function symbols; or equivalently,
a monoid M such that X ⊂ M and
 〈X〉_{{e,•}} = M
 Any function from X to any monoid
is extensible as an {e,•}morphism (unique due to the previous condition).
This equivalence is deducible from the property of trajectories
and the representation theorem.
The image of M by any morphism of monoid is the submonoid generated by the image of L.
3.9. Integers and recursion
The set ℕ
Any theory aiming to describe the system ℕ of natural numbers is called an Arithmetic.
There are several of them, depending on the logical framework and the choice of
language. Let us start with the set theoretical arithmetic, which is the most precise
as it determines ℕ up to isomorphism, i.e. it is a definition of an isomorphism class of systems in
a given universe.
The use of algebra in this formalization may make it look circular, as our study of
algebras used natural numbers as
arities of operation symbols. But arithmetic only uses operation symbols with arity
0, 1 or 2, for which previous definitions might be as well written without reference to numbers.
Definition. ℕ is a ground term algebra with
two symbols: a constant symbol 0 ("zero"), and a unary symbol S.
This S is called the successor. Its meaning is to add one
unit (Sn=n+1) as will be seen below.
This definition can be explicited as the following list of 3 axioms on this
{0,S}algebra :
(H0) 
∀n∈ℕ, Sn ≠ 0

: 0 ∉ Im S 
(Inj) 
∀n,p∈ℕ, Sn =
Sp ⇒ n = p 
: S is injective 
(Ind) 
∀A⊂ℕ, (0∈A ∧ ∀n∈A,Sn∈A)
⇒ A=ℕ 
(induction) : ℕ is minimal. 
We can define 1=S0, 2=SS0...
The existence of a ground term {0,S}algebra is an equivalent form of the
axiom of infinity, which
completes the set theory we progressively introduced from the beginning to the powerset (with optionally the
axiom of choice), to form what is
essentially the standard foundation of mathematics as practiced by most mathematicians.
It is most conveniently expressed by an insertion of arithmetic into set theory,
in the form of the constant symbols of the set ℕ, its element 0 and its
transformation S, and the above axioms (from which more symbols such as
+ and ⋅ can then be defined). It will
be seen to imply the existence of term algebras of any language.
Fixing ℕ in its class does not cause any uncertainty thanks to the essential uniqueness
of initial {0,S}algebras.
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 number n represents a
term with symbols 0 and S respectively interpreted as a and f in E.
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)) 
Reinterpreting the constant 0 as a variable, makes ℕ a unary term
{S}algebra, where each number n is a unary term S^{n} with n
occurrences of S, 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 and f^{ 2} = f০f.
Generally for any f∈E^{E},
g∈E^{X}, the sequence (h_{n}) in
E^{X} recursively defined by (h_{0}=g) and
(∀n∈ℕ, h_{Sn} = f০h_{n})
is h_{n}=f^{ n}০g.
Addition
Like any unary term algebra, ℕ forms a monoid (ℕ, 0, +),
whose actions are the sequences (f^{ n}) for any transformation f.
It is commutative as it is generated by a singleton, {1} (which commutes with itself). Thus the side won't
matter, but conventions basically present it as acting on the right, defining addition as
n+p = S^{p}(n), or recursively as
n + 0 = n
∀p∈ℕ, n+S(p) = S(n+p).
Thus, n+1 = S(n+0) = Sn.
Like in the general case, the action formula ∀n,p∈ℕ, f^{ n+p} = f^{
p}০f^{ n} is deduced from
(f^{n+0}=f^{n} ∧ ∀p∈ℕ,
f^{n+Sp} = f^{S(n+p)
} = f০f^{n+p})
∴
∀a∈E, ∀f∈ E^{E}, (p↦f^{
n+p}(a))∈Mor(ℕ,(E,f^{n}(a),f)),
from which associativity comes as (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), or recursively as
∀x∈ℕ, x⋅0 = 0
∀x,y∈ℕ, x⋅(Sy) = (x⋅y)+x
Then generally, ∀f∈E^{E}, f^{ x⋅y}
= (f^{ x})^{y}.
Inversed recursion and relative integers
By induction using commutativity (n+1 = 1+n), ∀f,g∈
E^{E}, g০f = Id_{E}
⇒ ∀n∈ℕ, g^{n}০f^{ n} = Id_{E}.
Thus if f is a permutation then f^{ n} is also a permutation, with inverse
(f^{ n})^{1} = (f^{ 1})^{n}.
Commutativity was just here to show that composing n times is insensitive to sides reversal,
as (f^{ n})^{1} has the more direct recursive definition
(f^{ Sn})^{1} =
(f^{n})^{1}০f.
The system of (relative) integers ℤ is defined as the {0,S}algebra where
 the set ℤ is defined as ℕ ∪ ℕ, where elements of ℕ (natural numbers) are
called positive, and ℕ= {nn∈ℕ} is a copy of ℕ
called the set of negative integers (where n is the opposite of n),
only intersecting ℕ at 0 = 0.
 S_{ℤ} is the permutation extending S_{ℕ} by
∀n∈ℕ, S(Sn)= n, thus letting Gr S_{ℤ} be
the union of Gr S_{ℕ} with its transposed copy in ℕ.
Proposition. ℤ is a commutative group, and for any permutation f of a set E,
there exists a unique (f^{ n})_{n∈ℤ} which is,
equivalently, a {0,S}morphism from ℤ to (E^{E}, Id_{E},
f), or an action of ℤ on E interpreting 1 as f.
Proof: the {0,S}morphism condition requires on ℕ the same n ↦
f^{ n} as above; on ℕ, it recursively defines f^{ n} =
(f^{ 1})^{n}, namely
 f^{ 0} = Id_{E} = f^{ 0}
 ∀n∈ℕ, f^{ n} = f০f^{ Sn},
equivalently f^{ 1}০f^{ n} = f^{ Sn}
This makes (ℤ,0,S) an initial object in the category of {0,S}algebras
(E,a,f) where f is a permutation. This category is derived as described with categories of acts
from that of
{S}algebras (E,f) where f is a permutation.
Therefore, ℤ is a monoid acting by (f^{ n})_{n∈ℤ} on every set E
with a permutation f.
This monoid is a commutative group because it is generated by {1, 1}, which commute and are the
inverse of each other : (1)+1=0=1+(1).
The formula of its inverse, (n)+n = 0 = n+(n) can be deduced either
from symmetry ((n)+n∈ℕ ⇔ n+(n)∈ℕ) and commutativity, or from the
above result.
3.10. Arithmetic with addition
Firstorder theories of arithmetic
Our first formalization of ℕ
was based on the framework of set theory, using the powerset to determine the isomorphism class of ℕ.
This allowed recursion, from which addition and multiplication could be defined.
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 as the set theoretical framework was involved to justify recursive definitions,
the successor function no more suffices to define addition and multiplication in firstorder logic.
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, can finally
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 we shall use set
theoretical notations in such ways that they can be read as abbreviations of works in
firstorder logic: as long as we only consider subsets of ℕ* defined by firstorder formulas
in this arithmetical language, (Ind)
and (Min) can be read as abbreviations of schemas of statements, 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}).
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) in set theory (ignoring our previous definition of ℕ)
Let M = Min_{{0,S}}ℕ.
∀x∈M, M ∈ Sub(ℕ,x,S) ∧ f_{x} =
(M∋y↦x+y) ∈ Mor_{{S}}(M,ℕ).
Im f_{x} = f_{x} [〈0〉_{{S}}] =
〈x〉_{{S}} ⊂ M.
As M is stable by + and contains 1, it equals ℕ.∎
(As ∧ Min) ⇒ (Ind) in firstorder logic
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.∎
So, all possible axiom pairs are equivalent:
(A1 ∧ Min) ⇔ (As ∧ Min) ⇔ (As ∧ Ind) ⇔ (A1 ∧ Ind), and imply commutativity.
Now (F) ⇔ (∀x∈ℕ*, ∀y∈ℕ, x+y ≠ y)
because x+0 = x ≠ 0.
(Inj ∧ Ind ∧ A1) ⇒ (F) because ∀x∈ℕ*,
{y∈ℕ  x+y ≠ y} ∈ Sub_{{0,S}}ℕ .
For the converse, we need to use the order relation.
The order relation
In any model of Presburger arithmetic,
let us define binary relations ≤ and < as x<y ⇔ ∃z∈ℕ*, y =
x+z
x≤y ⇔ ∃z∈ℕ, y = x+z
(A1 ∧ Ind) implies that
 < is transitive
 x≤y ⇔ (x<y ∨ x=y)
 x<y ⇔ x+1≤y
 ∀A⊂ℕ, A≠∅ ⇒ ∃x∈A, ∀y∈A,
x≤y (meaning a schema of formulas in Presburger arithmetic)
 ∀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}. Or using 3, A={x∈ℕ
∀y∈ℕ, x<y ∨ x=y ∨ y<x}
⇒ (0∈A ∧ ∀x∈A, x+1∈A)
 y = x+n ⇒ y+z = x+z+n
∎
Now the full system (A1 ∧ Ind ∧ F) implies that  < is a strict total order, associated
with the total order ≤
 ∀x,y,z∈ℕ, x<y ⇔ x+z
< y+z
 + is cancellative (which gives (Inj) as a particular case).
Proof. (F) means that < is irreflexive, which with transitivity (1.) implies
that it is a strict order, which is total by 5.
There must be one true formula among (x<y),
(x
= y), (y<x), which by 6. respectively imply
(x+z<y+z),
(x+z = y+z), (y+z<x+z).
But only one of the latter can be true, thus each implication must be an equivalence.
Cancellativity on one side extends to the other side by commutativity.
∎
Finally, by 4., every nonempty subset A of ℕ has a smallest element
(unique by antisymmetry), written min A.
This order coincides with the order
between terms in the common particular case of the set theoretical ℕ, as will
be clear from the properties of generated preorders.
Arithmetic with order
It is possible to express a firstorder arithmetic with language {0,S, ≤}, more
expressive than {0,S} but less than Presburger arithmetic, in the sense that
addition cannot be defined from ≤.
There, ≤ is related to S by the following property (which determines it in set theory,
but no more in bare arithmetic due to the poverty of its
interpretation of induction by an axiom schema):
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 in set theory (not firstorder arithmetic) by induction
on n; its uniqueness for a fixed n is deduced by induction on x.
Trajectories of recursive sequences
For any recursive sequence u∈Mor(ℕ,(E,a,f)), the trajectory
K = Im u = {f^{ n}(a)  n∈ℕ} of an
action of ℕ on E is generated by a, which is a free element for the image of
ℕ as a transformation monoid of K thanks to the commutativity of +.
Therefore K can be seen as a commutative monoid, whose description
coincides with the above arithmetic without axiom (F), where the roles of the neutral element 0
and the generator 1 are respectively played by a and f(a). However for convenience, let us focus on the set theoretical viewpoint on the remaining case, when
u is not injective so that K is not a copy of ℕ. Then K must be a
noninjective {0,S}algebra: there must be a pair in {0,S}⋆K
mapped to a singleton, but we shall see that such a pair is unique.
Let y the minimal number such that ∃x<y, u_{x} =
u_{y}. This x is unique because y is minimal.
{u_{n}  n<y} ∈ Sub_{{0,S}}K,
thus =K.
As Inj f_{K} ⇔ x=0 ⇔ a∈ Im
f_{K} ⇔ f^{y}(a)=a
⇔ (f_{K})^{y} = Id_{K},
a trajectory K where these are true is called a cycle of f with period
y; the restriction of f to K is then a permutation of K. Then replacing
a by another element of K would leave both K and
y unaffected.
Now if f is a permutation then every cycle of f is also a cycle of f^{ 1}
with the same period.
Set theory and foundations
of mathematics
1. First foundations of
mathematics
2. Set theory (continued)
3. Algebra 1
4. Model Theory