Part 3 : Algebra 1

3.1. Morphisms of relational systems and concrete categories

Let us formalize the concept of system, focusing for simplicity on those with only one type. For any number n∈ℕ and any set E, let us denote We then denote the set of all operations OpE = ⋃n∈ℕ OpE(n), and that of all relations RelE = ∐n∈ℕ ℘(En).

Languages

A language L is a set of symbols, with the data of the intended arity ns∈ℕ of each sL. For any set E, let

LE = ∐sL Ens

A relational language is a language L of relation symbols, where each sL aims to be interpreted in any set E as an ns-ary relation. These form a family called an L-structure on E, element of

sL ℘(Ens) ≅ ℘(LE)

Relational systems

A relational system with language L, or L-system, is the data (E,E) of a set E with an L-structure ELE.

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 L-structure on each set, so that E can be treated as implicit, determined by E. Precisely, let us imagine given a class of L-systems where each E is the intersection of LE with a fixed class of (s,x), denoted as a predicate s(x) for how it is naturally curried: each symbol sL is interpreted in each system E as the ns-ary relation sE somehow independent of E,

sE = {xEns | s(x)} = E(s)
E = {(s,x)∈LE | s(x)} = ∐sL sE.

For any function f : EF, let Lf : LELF defined by (s,x) ↦ (s,fx).
Im Lf = LIm f by finite choice with (AC 1)⇒(6), as arities of symbols are finite (otherwise it still goes for injective f, or using AC).
BF, (Lf)*(LB) = L(f*(B))

Morphism. Between any L-systems E,F, we define the set MorL(E,F) ⊂ FE of L-morphisms from E to F by ∀fFE,

f ∈ MorL(E,F) ⇔ ∀(s,x)∈E, (r,fx)∈F
⇔ (∀sL,∀xEns, s(x) ⇒ s(fx))
Lf[E]⊂FE⊂(Lf)*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)
satisfying the following axioms:
The last condition is easily verified for L-morphisms : ∀(s,x)∈E, (s,fx)∈F ∴ (s,gfx)∈G.
A relational symbol s with the data of an interpretation sEEns 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), ∀xsE, fxsF. From definitions, each symbol in a language L is preserved in any category of L-systems.

A category is small if its class of objects is a set.

Preservation of some defined structures

In any given category of L-systems, 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 L-morphism f∈MorL(E,F), Thus, for any f ∈MorL(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:

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 n-tuple t of elements of some object K, the relation s defined in each object E as sE = {ft | f∈Mor(K,E)} is preserved.

Proof : ∀g∈Mor(E,F), ∀xsE, ∃f∈Mor(K,E), (x = ftgf∈Mor(K,F)) ∴ gx = gftsF.∎
From these definitions it might happen between objects EF that sEsFEn but we shall not face this in our use.

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:

3.2. Notion of algebra

Given an algebraic language L, an L-algebra is the data (E,φ) of a set E with an L-algebraic structure φ : LEE, sum of a family of interpretations of each symbol sL as an operation ⃗φ(s)∈OpE(ns).
Again, we shall usually consider a class of L-algebras with only one choice of algebraic structure on each set:

sE : EnsE
φE = ∐sL sE : LEE

This would be the case if the sE were the restrictions of a common ns-ary operator to each E, but this would not allow to interpret constant symbols r and s in algebras E and F with rE=sE but rFsF.
These form a concrete category with the following concept of morphism.

Morphisms of algebras. For any L-algebras E, F,

MorL(E,F) = {fFE | ∀(s,x)∈LE, sF(fx) = f(sE(x))} = {fFE| φFLf = f০φE}.

When cL is a constant (i.e. nc =0), this condition on f says f(cE)=cF.

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 sL has increased arity ns' = ns+1, so that
L'E ≡ ∐sL Ens×E ≡ (LEE ≡ {(s,x,y) | sLxEnsyE}.
Each ns-ary operation sE defines an ns'-ary relation s'EGr sE. These are packed as an L'-structure
E = Gr φE ≡ ∐sL s'E.
The resulting condition for an fFE to be a morphism is equivalent :
(∀(x,y)∈E, (Lf(x),f(y))∈F) ⇔ (∀xLE, φF(Lf(x))= fE(x))).

Subalgebras. A subset AE of an L-algebra E is called stable by L or an L-subalgebra of E, if φE[LA]⊂A. It is also an L-algebra, with structure φA restriction of φE to LA. The set of L-subalgebras of E is written SubL E = {AE | φE[LA]⊂A}.

If a formula of the form (∀(variables), formula without binder) is true in E, then it is true in each A∈SubLE.

Images of algebras. For any two L-algebras E,F, ∀f ∈MorL(E,F), Im f ∈ SubLF.

Proof : φF[LIm f] = φF[Im Lf] = Im (f০φE) ⊂ Im f

Stable subsets of systems

The concept of subagebra is generalized to relational L'-systems (E,E) with possibly non-functional structure E ⊂ (LEE, as the following condition of stability for subsets A (which no more makes them algebras) :

A ∈ SubL E ⇔ (E*(LA) ⊂A) ⇔ (∀(s,x,y)∈E, Im xAyA).

We have E ∈ SubL E. Stability is no more preserved by direct images by morphisms, but is still preserved by preimages:

Preimages of stable subsets.f∈MorL(E,F), ∀B∈SubLF, f *(B) ∈ SubL E.

Proof. Let A=f *B.
For L-algebras, ∀(s,x)∈LA, fxBnsf(sE(x)) = sF(fx) ∈BsE(x)∈A.
For L'-systems, ∀(x,y)∈E, (Lf(x),f(y))∈F∴ (xLALf(x)∈LBf(y)∈ByA).∎

Proposition. For any L'-system E and any L-algebra F,

f,g∈MorL(E,F), {xE|f(x)=g(x)}∈ SubLE.

Proof : ∀(s,x,y)∈E, fx=gxf(y) = sF(fx) = g(y). ∎

Intersections of stable subsets.X ⊂ SubLE,X ∈ SubL E where ∩X {xE|∀BX, xB}.

Proof: ∀(x,y)∈E, xLX ⇒ (∀BX, xLByB) ⇒ y∈∩X. ∎

Other way: E*(LX) = E*(∩BX LB) ⊂∩BX E*(LB) ⊂∩X.

Subalgebra generated by a subset.AE, we denote 〈AL,E or simply 〈AL, the smallest L-stable subset of E including A (called L-subalgebra of E generated by A if E is an algebra):

AL = {xE | ∀B∈SubLE, ABxB} = ∩{B∈SubLE | AB} ∈ SubLE

For fixed E and L, the function A↦〈AL is the closure, with image SubLE, from the relation ∈ between E and SubLE:
XE, ∀Y⊂SubLE, X ⊂ ∩Y ⇔ (∀BY, XB) ⇔ Y ⊂ {B∈SubLE | XB}.
We say that A generates E or is a generating subset of E if 〈AL=E.

Minimal subalgebra. For any L'-system E, its minimal stable subset (or minimal subalgebra for an L-algebra) is defined as
MinLE = 〈∅〉L,E = ∩SubLE ∈ SubLE.

An L-algebra E is minimal when E = MinL E, or equivalently SubLE = {E}.

Proposition. For any L'-system E, ∀A∈SubLE, Proof: MinLE ⊂ MinLA because SubL A ⊂ SubL E;
MinL A ⊂ MinL E because ∀B∈SubLE, AB ∈ SubLA. ∎

Among subsets of E, other minimal L'-systems are included in MinL 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: 〈AL,E= MinLA E.

Injective, surjective algebras.

An L-algebra (EE) will be called injective if φE is injective, and surjective if Im φE = E.

Proposition. For any L-algebras E, F,

  1. AE, Im φEAA∈SubLE.
  2. Any minimal L-algebra is surjective.
  3. MinLE = φE[LMinLE] ⊂ Im φE
  4. AE, ⋃xA〈{x}〉L ⊂ 〈AL = A∪φE[LAL] ⊂ A∪Im φE
  5. f ∈MorL(E,F), f [MinLE] = MinLF ∧ ∀AE, f [〈AL] = 〈f [A]〉L
Proofs:
  1. φE[LA] ⊂ Im φEAA∈SubLE
  2. Im φE ∈ SubLE
  3. MinLE is surjective
  4. A∪φE[LAL] ∈ SubLAL
  5. B ∈ SubLF, f *(B)∈SubL E ∴ MinLEf*(B) ∴ f [MinLE]⊂B.∎

Injectivity lemma. If E is a surjective algebra and F is an injective one then ∀f ∈MorL(E,F),

  1. A= {xE | ∀yE, f(x) = f(y) ⇒ x=y} ∈ SubLE.
  2. For each uniqueness quantifier Q (either ∃! or !), B = {yF | QxE, y = f(x)} ∈ SubLF
They are essentially the same but let us write separate proofs:
  1. ∀(s,x)∈LA, ∀yE,
    f(sE(x)) = f(y) ⇒ (∃(t,z)∈φE(y), sF(fx) = f(sE(x)) = f(y) = f(tE(z)) = tF(fz) ∴ (s=tfx=fz) ∴ x=z)sE(x) = y.
  2. As φF is injective, ∀y∈φF[LB], ∃!: φF(y) ⊂ LBQ zLE, φF(Lf(z)) = y.
    As φFLf = f০φE and φE is surjective, we conclude QxE, y = f(x). ∎

Schröder–Bernstein theorem. If there exist injections f: EF and g: FE then there exists a bijection between E and F.

Proof : replacing F by the bijectively related set Im g, simplifies things to the case FE.
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 between relational systems.

Quotient systems

For any relational language L, any L-systems (E,E) and (F,F) and any f : EF we have f∈Mor(E,F) ⇔ Lf[E]⊂F.
If f : EF and Lf[E] = F then f will be called a projection from E to F.
The role it gives to F is that of quotient of E by ∼f. Namely, for any equivalence relation R on E, the quotient set E/R has a natural L-structure E/R defined as LR[E]. It is the smallest L-structure on E/R such that R∈Mor(E, E/R).
If R ⊂ ∼f then (f ∈ Mor(E,F) ⇔ f/R ∈ Mor(E/R,F).

Given an algebraic language L, an equivalence relation R on E is said to be compatible with an L'-structure E if the quotient structure is a functional graph. If E is an algebra structure then Dom(E/R) = L(E/R) so that the compatibility condition means that the quotient is also an algebra.
For any L'-systems (E,E) and (F,F), any f : EF, BF and A= f*(B) we have
(f∈Mor(E,F) ∧ B∈SubLF) ⇒ A∈SubLE
(FfL[E] ∧ A∈SubLE) ⇒ B∈SubLF
Lf[E]=F ⇒ (B∈SubLFA∈SubLE)

Embeddings and isomorphisms

Strong preservation. A relation symbol r interpreted as rE in E and rF in F is strongly preserved by a function fFE, if both r and ¬r are preserved :

xEnr, xrEfxrF.

Embeddings. An f ∈ MorL(E,F) is called an L-embedding if it strongly preserves all structures : E = Lf*(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 non-injective embeddings.

Isomorphism. Between objects E and F of a concrete category, an isomorphism is a bijective morphism (f ∈Mor(E,F) ∧ f : EF) 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).

For any relational system E and any subset AE, defining the structure of A by restricting that of E to A,

Embeddings of algebras

Every injective morphism f between algebras is an embedding :

∀(s,x,y)∈L'E, f(y) = sF(fx) = f(sE(x)) ⇒ y=sE(x).

Any embedding between algebras f ∈ MorL(E,F), is injective whenever Im φE = E or some sE 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

(Lf)-1 = L(f -1) ∴ φELf -1 = f -1f০φELf -1 = f -1০φFLfLf -1 = f -1০φF.

Elementary embeddings

L-embeddings 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 first-order logic comes by removing this restriction on the order of use of logical symbols: an elementary embedding (or elementary L-embedding) is a morphism that (strongly) preserves all invariant structures (defined by first-order formulas with language L).
An elementary subsystem of a system E is an FE interpreting L by restriction, such that IdF is an elementary embedding from F to E, i.e. any formula with parameters in F takes the same Boolean value whether all its bound variables range in F or all range in E.

Isomorphism ⇒ Elementary embedding ⇒ Embedding ⇒ Injective morphism

Elementary equivalence. Different systems are said to be elementarily equivalent, if they have all the same true ground first-order 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 non-isomorphic systems, as well as non-surjective elementary embeddings. However, they exist and play a special role in the foundations of mathematics, as we shall see with Skolem's paradox and non-standard models of arithmetic.

Endomorphisms, automorphisms.

An endomorphism of an object E in a category, is an element of Mor(E,E) = End(E). It is nontrivial if it differs from IdE.
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 any endomorphism which is an invariant elementary embedding is an automorphism:
Im f is also invariant (defined by ∃yE, f(y)=x)
xE, x∈Im ff(x)∈ Im f
Im f = E. ∎

The Galois connection (End, Inv)

For any set E and any n∈ℕ, the preservation relation (∀xR, fxR) between fEE and R∈RelE(n) gives a Galois connection (End, Inv(n)) between their powersets, where Inv(n)(M) is the set of n-ary relations invariant by MEE. In particular, Inv(1) M = SubM :

MEE, ∀F⊂℘(E), M ⊂ EndFE ⇔ (∀fM, ∀AF, f[A] ⊂ A) ⇔ F ⊂ SubME.

Gathering all arities forms the Galois connection (End, Inv) with

Inv = ∏n∈ℕ Inv(n) : ℘(EE) → ∏n∈ℕ ℘(RelE(n)) ≅ ℘(RelE).

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 EE 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}EE : 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.

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 •. So, the concept of monoid is the theory with only one type and

Both equalities in the last axiom may be considered separately, forming two different concepts

If both a left identity and a right identity exist then they are equal : e = ee' = e' which makes it the identity of • (the unique identity element on either side). The existence of a right identity implies the uniqueness of any left identity, but without right identity, several left identities may coexist (and similarly with sides reversed).
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.
Any {e,•}-subalgebra of a monoid is a monoid, thus called a submonoid.

Cancellativity

An element x is called left cancellative for an operation • if the left composition by x is injective: ∀y,z, xy = xzy = z.
Similarly it is right cancellative if yx = zxy = z.
If a right identity e is left cancellative then it is the unique right identity : ee' = e = eee' = e.
An operation is called cancellative if all elements are cancellative on both sides.
For example the monoid of addition in {0,1, several} is not cancellative as 1+several = several+several.
Any submonoid of a cancellative monoid is cancellative.

Commutants and centralizers

The commutant of any subset AE for a binary operation • in E, is defined as

C(A) = {xE|∀yA, xy = yx}.

This is another Galois connection : ∀A,BE, BC(A) ⇔ AC(B). Such A,B are said to commute as each element of A commutes with each element of B.

If • is associative then ∀AE, C(A) ∈ SubF. (Proof: ∀x,yC(A), (∀zA, xyz = xzy = zxy) ∴ xyC(A))

Commutants in monoids are called centralizers. They are more precisely sub-monoids as obviously eC(A).
This can be understood for transformation monoids MXX by the fact it is an intersection of submonoids: AM, CM(A) = M ∩ EndA X.
This concept will be later generalized to clones of operations with all arities.

A binary operation • in a set E, is called commutative when C(E) = E, i.e. x,yE, xy = yx.

If AC(A) and 〈A=E then • is commutative
Proof: AC(A)∈ SubFE=C(A) ⇒ AC(E) ∈ SubFC(E) = E.∎
In the case of monoids the conditions AC(A) and 〈A{e,•}=E suffice.

Other concepts of submonoids and morphisms

Modifying 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 morphism of composition f ∈ Mor{•}(M,X), If the target forms a monoid (X,e',▪) then (by uniqueness of the identity in A)

f ∈ Mor{e}(M,X) ⇔ a = e'e'AA ∈ Sub{e, ▪} X

but these equivalent formulas may still be false, unless a is cancellative on one side (aa = a = ae'a = e').

Anti-morphisms. The opposite of a monoid is the monoid with the same base set but where composition is replaced by its transpose. An anti-morphism from (M,e,•) to (X,e',▪) is a morphism f from one monoid to the opposite of the other (or equivalently vice-versa):

f(e) = e'
a,bM, f(ab) = f(b)▪f(a)

3.5. Actions of monoids

Left actions

After monoids were deprived of their role as sets of transformations, 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×XX such that Seing M as a set of function symbols, an M-algebra (X, ⋅) satisfying these axioms is called an M-set.
In curried view, a left action of M on X is a {e,•}-morphism from M to the full transformation monoid XX.

Right actions

A right action of a monoid M on a set X, is like a left action with sides switched: it is an operation ⋅ : X × MX such that
It amounts to a left action of the opposite monoid, and defines an anti-morphism from M to XX.

The commutation of 2 submonoids of XX looks like an associativity formula when written as acting by opposite sides on X: aM left acting on X commutes with bN right acting on X when

xX, (ax)⋅b = a⋅(xb)

Effectiveness and free elements

A left action of M on X is said effective if this morphism from M to XX is injective (thus an embedding):

a,bM, (∀xX, a·x = b·x) ⇒ a=b

letting e and • be definable from (i.e. unique for) the given action, like the axioms for functions ensured the sense of the definitions of Id and ০ from the function evaluator.
An element xX of an M-set, 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:

(∃xX, ∀abM, a·xb·x) ⇒ (∀abM, ∃xX, a·xb·x)

General example. Any transformation monoid M of a set E acts by restriction on any M-stable subset A of E, i.e. any preserved AE in the concrete category M with object E. Thus, the monoid of endomorphisms of any typed system E= ∐iI Ei, acts on every type Ei it contains.

Acts as algebraic structures

Let M, X be given structures of M-algebras by any operations • : M×MM and ⋅ : M×XX. Then denoting ∀xX, hx = (Maax), we have directly from definitions

hx(e) = xex = x
hx ∈ MorM(M,X) ⇔ ∀a,bM, (ab)⋅x = a⋅(bx)
he = IdM ⇔ (∀aM, ae = a) ⇒ ∀g∈MorM(M,X), g=hg(e).

So in the formula

gXM, ∀xX, g=hx ⇔ (g∈MorM(M,X) ∧ g(e)=x)

the ⇒ expresses the axioms of left action of M on X; the ⇐ is implied by (∀aM, ae = a). This last axiom of monoid (beyond those of left action of M on itself), comes conversely as a particular case of this ⇐ when X=M :

(IdM ∈ MorM(M,M) ∧ IdM(e)=e) ⇒ he = IdM

Trajectories

For any set of transformations LEE, seen as a set of function symbols interpreted in E,

xE, 〈{x}〉L = {f(x)|f∈〈L{Id,০}}
XE, 〈XL = ⋃f∈〈L{Id,০} f[X] = ⋃xX 〈{x}〉L

Denoting A = 〈XL, M = 〈L{Id,০} and K = {f(x) | (f,x)∈M×X}, the claim is that A = K. As we shall formalize later, both A and K mean the set of all composites of any number of functions in L, applied to any xX.
Proof of AK
IdEMXK
gL, ∀yK, ∃(f,x)∈M×X, y=f(x) ∧ gfMg(y) = gf(x)∈K
K ∈ SubLE
Proof of KA
L ⊂ {fEE| A ∈ Sub{f}E} = End{A}E ∈ Sub{Id,০}EE
M ⊂ End{A}E
fM, XA ∈ Sub{f}Ef[X] ⊂ A. ∎
The trajectory of an element xE by a transformation monoid M of E, is the set it generates:

〈{x}〉M = {f(x)|fM} ⊂ E

For a monoid M, Inv M is made of unions of trajectories of tuples. Other cases come down to this as ∀LEE, Inv L = Inv 〈L{Id,০}, so that closures can be written

EndInv(n)L E = {gEE| ∀xEn, ∃f∈〈L{Id,০}, fx=gx}.
EndInv L E = {gEE| ∀n∈ℕ,∀xEn, ∃f∈〈L{Id,০}, fx=gx}.

Trajectories by commutative monoids

Let a monoid M act on a set X, and let kX. The trajectory Y of k by M is stable by M, thus defines a morphism of monoid from M to YY 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 = {fEE| Inj f ∧ Im f = E} = {fEE| f : EE} 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 {Id,০, -1}-subalgebra of ⤹E, i.e. a transformation monoid of E made of permutations and stable by inversion.

While the concepts of full transformation monoid and symmetric group depend on the powerset, those of transformation monoid and permutation group can be defined independently of it, as first-order theories with 2 types.

Trajectories are usually called orbits in the case of a permutation group.
For any transformation monoid or action of a monoid M on a set E, the relation defined by its trajectories (∐xE 〈{x}〉M) is a preorder on E (as seen in example in 2.7). If M is a group then this preorder is an equivalence relation, whose partition of E is the set of orbits.

Inverses

In a monoid (M,e,•), the formula xy = e is read "x is a left inverse of y", or "y is a right inverse of x"; this x is right invertible (it has a right inverse), and y is left invertible.
Seeing M as a transformation monoid by left action on itself, this xy = e is interpreted as relating transformations :

z,tM, yz = txt = z

As right invertible functions are surjective and left invertible functions are injective, the left invertibility of y means that the right composition by y is surjective ({zy|zM} = M) and implies that y is left cancellative:

xy = e ⇒ ∀zM, zxy = z
z,tM, (yz = ytxy = e) ⇒ (z = xyz = xyt = t)

An element x is called invertible if it is so both sides. Then its left and right inverses are equal, thus unique, called the inverse of x and written x-1:

yx = e = xzy = yxz = z

If a left invertible element y is also right cancellative then it is invertible: xy=eyxy = eyyx=e.
If x commutes with an invertible element y then it also commutes with its inverse z:

xy = yxx = yxzzx = xz

An element x of a monoid is called involutive if it is its own inverse: xx = e. In particular e is involutive.

Groups

The concept of group is the theory obtained by adding to that of monoid, equivalently Indeed this latter axiom determines the interpretation of -1 from those of • and e.
Permutation groups are the transformation monoids which are groups (in the first above sense).
A subgroup of a group is equivalently, a sub-monoid which is a group (not all are: the sub-monoid ℕ of the group ℤ is not a sub-group), or a {e, •,-1}-subalgebra.
The set of invertible elements in any monoid M, is a group: Its subgroups are all the submonoids of M which are groups, which may then be called the subgroups of M.
Between groups, a group morphism is equivalently an {e,•,-1}-morphism, or an {e,•}-morphism, as the latter also preserve inversion. More generally any {e,•}-morphism from a group to a monoid preserves the inversion relation {(x,y) | xy = e = yx}, thus its image is a group.

In a group, the subgroup generated by a subset A, coincides with the submonoid G generated by A∪-A where -A = {x-1|xA}. (To check that G is stable by inversion, notice that the definition of G is stable by inversion, which is involutive, thus -G = G.)

Any submonoid of a group is cancellative. This has no general converse, but some partial ones, such as: any commutative cancellative monoid has an embedding to a commutative group (this is not very easy to prove).

Special actions

The above characterization of invertible elements also makes sense for an element x of an M-set X: saying that x is both generating and free, means that the morphism hx∈ MorM(M,X) is both surjective and injective, thus an isomorphism between the M-sets M and X. Then we might still say it has an inverse in the form of an M-morphism from X to M.
Now this can qualify actions (M-sets) themselves: let us call an action monogenic if it is a trajectory (it is generated by a single element), and free if it is generated by the set of its free elements.
Let us call it regular if it is both monogenic and free. Then there is a free element that generates it, i.e. it is M-isomorphic to M.
Proof: a generator being generated by the set of free elements, must be in the trajectory of one of them, which is thus also generating. (On the other hand, a monogenic action may have free elements without being free).

An action of a group G on a set X, is equivalently an action of monoid, or a group morphism from G to the symmetric group of X.

As inversion is an anti-morphism, it switches any left action ▪ of G on X into a right action • by ∀xX, ∀gG, x•g = g-1x.

If an action of group is monogenic then every element is generating ; if it is free then all elements are free, so that all parts of its partition into orbits are regular.

3.7. Categories

Monoids were introduced as an abstraction of transformation monoids, which are concrete categories with one object. Generalizing this to pluralities of objects, a category (also called abstract category for insistence) differs from a concrete category, by forgetting that objects are sets and that morphisms are functions. It is made of: satisfying both axioms, of identity and associativity: Generalizing isomorphisms in concrete categories, an isomorphism f between objects E and F in an abstract category, is an f∈Mor(E,F) such that ∃g∈Mor(F,E), gf=1Efg=1F. This g is unique, called the inverse of f and written f -1.

Again, an automorphism of an object E, is an isomorphism from E to itself. Their set Aut(E) is the group of invertible elements of the monoid End(E)=Mor(E,E).

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)
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, gf)
HomF(f, X) ০ HomG(g, X) = HomG(gf, 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:

g,h∈Mor(X,E), fg = fhg = h.

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), gf = hfg = h.

In our concept of concrete category, we must specify F, saying f∈Mor(E,F) is F-epic, or an F-epimorphism, if all HomF(f,X) are injective.

In any concrete category, all injective morphisms are monic, and any morphism with image F is F-epic. 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 gf = 1E we say that f is a section of g, and that g is a retraction of f.

Proof: if gf=1E then for all objects X, HomF(f,X) ০ HomE(g,X) = HomE(1E,X) = IdMor(E,X), thus Similarly, Hom(X,g) ০ Hom(X,f) = Hom(X,1E) = IdMor(X,E) thus f is monic and Im(Hom(X,g)) = Mor(X,F).∎

A morphism f is an isomorphism if and only if Hom(X,f) : Mor(X,E) ↔ Mor(X,F); its inverse is then Hom(X, f -1).

In any category, Isomorphism ⇔ (Retraction ∧ Monomorphism) ⇔ (Section ∧ Epimorphism)
In concrete categories, Section ⇒ Injective morphism ⇒ Monomorphism
In categories of relational systems, Retraction ⇒ Quotient ⇒ Surjective morphism ⇒ Epimorphism

Representation theorem

Theorem. Any small category is isomorphic to that of all morphisms in a family of typed algebras.

The proof essentially repeats the formulas on acts as algebraic structures, transposed. From the given small category, a family of typed algebras is formed as follows. Each u ∈ Mor(E,F) acts on each type t by Hom(t,u) : tEtF. These for all u and t define f : Mor(E,F) → FE, so that the system of these for all E, F respects identities and compositions (they form an action in a sense generalized from monoids to small categories). Let us prove that f : Mor(E,F) ↔ MorL(E,F).
Im f ⊂ MorL(E,F) by associativity (f commutes with the action of L).
The existence of an isomorphism ktE ensures that f is injective (as k is epic) and MorL(E,F) ⊂ Im f:
g∈MorL(E,F), u = g(k)•k-1 ∈ Mor(E,F) ⇒ (∀xE, ∃sL, ks = xg(x) = g(ks) = g(k)•s = uks = ux) ⇒ g = f(u) ∎

In particular for any monoid M there is a language L of function symbols and an L-algebra X such that EndL X is isomorphic to M.
Any group is isomorphic to a permutation group, namely the group of automorphisms of an algebra.

3.8. Initial and final objects

In any category, an object X is called an initial object if all sets Mor(X,Y) are singletons. Of course any object isomorphic to an initial object is also an initial object, as all isomorphic objects have the same properties. But conversely all initial objects (if they exist) 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), gf ∈ Mor(X,X) ∧ 1X ∈ Mor(X,X) ∴ gf = 1X.
Similarly, fg = 1Y. 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. Initial objects are said to be essentially unique.
Similarly, an object X is called a final object if all sets Mor(Y,X)) are singletons.
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: Exercise. Given two fixed sets K and B, consider the category where Does it have an initial object ? a final object ?

Embeddings in concrete categories

In categories of relational systems, any section is an embedding, but without converse in general.
For any subset A of an object of such a category,

(There exists a section with image A) ⇔ (for any object N, Mor(A,N) = {g|A | g∈Mor(E,N)}) ⇒ (any embedding with image A is a section).

For images A of embeddings which are not sections, that formula for Mor(A,N) would not generally fit the concept of morphisms for any relational structure on A. Also in some categories of systems, not all subsets A of systems E are images of embeddings, because they do not fit as objects, particularly if the restricted structure on A fails at some chosen axiom. For example categories of algebras do not accept all subsets as sub-algebras.

Let us generalize the concept of embedding to any concrete category C, making it work the same as in categories of relational systems beyond the case of sections (but unlike sections, it will require checking the whole category). For any subset A of an object E of C, let To-A be the category where

The injectivity of f implies the uniqueness of any To-A-morphism to f.

Now a morphism f in C is an embedding if it is a final object of any To-A. Then it is also a final object of To-(Im f). It is an embedding onto A if moreover A = Im f.
All such embeddings, even non-injective ones, are monic : Section ⇒ Embedding ⇒ Monomorphism.

While C may have been given with pairwise disjoint objects ("forgetting" the canonical injections between them), any image A = Im f of an injective embedding f may receive a role of object added to C (we may call it a sub-object), as an additional representative of an existing isomorphism class in C, by copying to A through f (seen as an isomorphism) the sets of morphisms involving Dom f (independently of the choice of f because of its essential uniquess as a final object of To-A). But for an image A of non-injective embedding, the new object must stay another set with a surjection to A.

Products in concrete categories

For any family of objects (Ei)iI of a concrete category, their product, if it exists, is defined as an object P = ∏iI Ei with sets of morphisms to it defined by

For any object F, Mor(F,P) ≅ ∏iI Mor(F,Ei)

Here, the inclusion (Inc), or rather the canonical injection, of Mor(F,P) into ∏iI Mor(F,Ei) for all F, is equivalent to ∀iI, πi∈Mor(P,Ei).
Any data of L-algebra structures φi on each Ei defines the one φP on P, by

(∀iI, πi∈MorL(P,Ei)) ⇔ (∀iI, φiLi) = πi০φP) ⇔ φP = ∏iI φiLi)

thus (Inc) suffices to determine the algebraic structure of P, and imply the reverse inclusion.
In the more general case, assuming (Inc), the reverse inclusion not only defines Mor(F,P), but also determines all Mor(P,F) from the category if it allows such a product as an object, by making essentially unique any coexisting products of a given family of objects in a given category. This essential uniqueness is verified in the following more general case.

Products in categories

A product of any family of objects (Ei)iI in a category, is a final (thus essentially unique) data (P, φ) of an object P with φ∈∏iI Mor(P,Ei), i.e. making bijective all f ↦ (φif)iI :

For all object F, ∏iI Hom (Fi) : Mor(F,P) ↔ ∏iI Mor(F,Ei)

Empty products (I=∅) are the final objects. The concept of embedding is a variant of the unary case (itself trivial).

In concrete categories, for any (P, φ) and any F, (∏φ : P ↪ ∏iI Ei) ⇒ Inj ∏iI Hom (Fi) but products as just defined may exist without admitting as such the set theoretical one (P, π) where P = ∏iI Ei and ∏π = IdP.

Categories of acts

From a concrete category C, let C' be the category where Then we have
  1. If C is the category of M-sets for a monoid (M,e, •) then, seeing M as an M-set 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.
  2. 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) ⊂ MorM(X,Y) and Mor(M,X) = MorM(M,X).
Proof. 1. by properties of acts as algebraic structures and inverses, as x is free and generating in X if and only if (X,x) is isomorphic to (M, e).
2. Defining ∀xX, hx ∈ Mor(M,X) ∧ hx(e) = x, provides an M-structure on each X which interprets each aM in X as defined by the tuple (e,a). So they are preserved: Mor(X,Y) ⊂ MorM(X,Y), which implies the axioms of M-acts.
The composition in M coming as this M-structure for M = X, satisfies the same axioms.
The last axiom of monoid, he = IdM comes from the uniqueness of he obeying its definition, and ensures the reverse inclusion:
g∈MorM(M,X), g = hg(e)g ∈ Mor(M,X). ∎
This monoid (M,e,•) is essentially the opposite of the monoid End(M). Indeed for all a, bM we have ha ∈ End(M), hb ∈ End(M) and hahb(e) = ha(b) = ba.

3.9. Algebraic terms

Algebraic drafts

The concept of algebraic term with a purely algebraic language L and a set V of variable symbols (no predicate, logical symbols nor binders), which was first intuitively introduced in 1.5), is going to be formalized as systems in a set theoretical framework. For convenience let us work with one type (the generalization to many types is easy), and start with a wider class of systems.
Let us call L-draft any L'-system (D,D) where D⊂ (LDD, such that:

Let us denote ∀xOD, ΨD(x) = (σ(x), lx) ∈ LD where σ∈LOD and lxDnσ(x).
Equivalent formulations of well-foundedness are

AOD, (∀xOD, Im lxAVxA) ⇒ A=OD
AD, (VDAD*(LA) ⊂A) ⇒ A = D
AD, VDAD ⇒ ∃xOD\A, Im lxA
AOD, A≠∅ ⇒ ∃xA, A∩ Im lx = ∅

A ground draft is a draft with no variable, i.e. VD=∅. Thus, ΨD: DLD and SubLD = {D}.
Variables in a draft may be reinterpreted as constants: extending ΨD by IdVD : VDV, forms a ground (LV)-draft.

Sub-drafts and terms

Drafts do not have interesting stable subsets (by well-foundedness), but another stability concept: a subset AD is a sub-draft of D (or a co-stable subset of D) if, denoting OA = AOD and ΨA = ΨD|OA, we have (Im ΨALA), i.e. ∀xOA, Im lxA.
Indeed, it remains well-founded, as can be seen on the last formulation of well-foundedness.

Like with stable subsets, any intersection of sub-drafts is a sub-draft. Moreover, any union of sub-drafts is also a sub-draft (unlike for sub-algebras with an operation with arity >1, which from arguments in different sub-algebras may give a result escaping their union).

The sub-draft co-generated by a subset of a draft, is the intersection of all sub-drafts that include it. A term is a draft co-generated by a single element that is its root. Each x in a draft D defines a term Tx with root x, sub-draft of D co-generated by {x}.
Each draft D is ordered by xyxTy. It is obviously a preorder. Proof of antisymmetry (uniqueness of the root):
xOD, VDA={yD|xTy} which is a sub-draft by transitivity of ≤.
xA ∴ ∃zOD\A, Im lzA.
A∪{z} is a sub-draft, thus TzA∪{z} by definition of Tz.
zOD\AxTzx=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 well-founded relations in the study of Galois connections.

Categories of drafts

As particular relational systems, classes of L-drafts form concrete categories. Between two L-drafts D,E,

f ∈MorL(D,E) ⇔ (f[OD]⊂OE ∧ ΨEf|OD= Lf০ΨD)

where the equality condition can be split as

σEf|OD = σD
xOD, lf(x)=flx

Another concrete category is that of drafts with variables-preserving morphisms, where V is fixed and morphisms f from a draft D are subject to f|VD = IdVD. This is equivalently expressed reinterpreting variables as constants, as the category of ground (LV)-drafts.

Intepretations of drafts in algebras

For any L-draft D and any L-algebra E, an interpretation of D in E is a morphism f∈MorL(D,E), i.e. f|OD= φELf০ΨD, also expressible as

xOD, f(x) = σ(x)E(flx)

Any interpretation vEV of variables in an algebra E determines an interpretation of any draft D in E. To simplify formulations, restricting v to VD reduces the problem to the case VD=V.

Theorem. For any L-draft D with VD=V and any L-algebra E, any vEV is uniquely extensible to an interpretation of D:
∃!h∈MorL(D,E), h|V = v, equivalently ∃!hEOD, vh ∈ MorL(D,E).

Uniqueness is deduced from well-foundedness : ∀g,h∈MorL(D,E), g|V = v = h|VV⊂ {xD|g(x) = h(x)} ∈ SubLDg=h.
Let us now prove existence (using conditional operator).
S = {AD | VA ∧ Im ΨALA}
vK = ⋃AS {f∈MorL(A,E) | f|V =v}
f,gK, B = Dom f ∩ Dom g ⇒ (f|BKg|BK) ⇒ f|B=g|B
fK Gr f = Gr h
C= Dom h = ⋃fK Dom fS
hK
(CD*(LC) ∋ x↦ (xC ? h(x) : φE(LhD(x))))) ∈ K
D*(LC) ⊂ C
C=D

Operations defined by terms

Any element t of an L-draft D defines a V-ary operation symbol, interpreted in each L-algebra E by ∀vEV, tE(v) = h(t) for the unique h∈MorL(D,E) such that h|VD = v|VD. This formalizes the operation defined by a term, namely the L-term with root t in D (which can replace D here without modifying the interpretations of t).

This interpreted operation symbol being the structure defined by (IdV,t), is preserved by all L-morphisms, thus can be added to L without changing the category of L-algebras.
Symbols sL come back as the particular cases of the terms they form themselves where Ψ(t) = (s, IdV).
For the case of "small" (concretely written) terms, this preservation also has a schema of one proof for each term: re-expressing the term as a formula defining a relation (graph of the operation) using symbols ∃ and ∧, we can use the preservation of structures defined by such formulas.

3.10. Term algebras

Condensed drafts

A draft D is condensed if, equivalently,
  1. D is functional, i.e. ΨD is injective;
  2. D has an injective interpretation in some algebra;
  3. For any two distinct elements of D there is an algebra interpreting them differently.
1.⇒2. if D≠∅ (otherwise replace D by a singleton), ∃φ∈DLD, φ০ΨD = IdOD, i.e. IdD interprets D in (D,φ).
2.⇒3.
3.⇒1. ∀x,yOD, if ΨD(x) = ΨD(y) then x,y have the same interpretation in every algebra.

Any draft D can be reduced to a condensed draft as follows.

Give ℘(D) the L-algebra structure (s,u) ↦ {xOD | σ(x)=slx∈∏u}.
Then the interpretation of D in ℘(D) which sends any variable x to {x}, is the curried form of the only equivalence relation on D which quotients it into a condensed draft.
Let us call condensation of D this projection of D to a condensed draft. This is the (not typographically convenient) way to confuse the repeated sub-terms of a given term (or draft), which will have the same meaning in all algebras.
This equivalence relation is included in that of any interpretation of D in any algebra, thus quotienting interpretations of D.

We can also compare separately given terms by reducing them to this case as any disjoint union of drafts (only keeping variables in common) is a draft.

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

Term algebras

An L-algebra (EE) is called a term algebra if it is injective and 〈E\Im φEL = E. Thus it is also a condensed L-draft with ΨE = φE-1. Another usual assumption is that V=VD=E\Im φE, i.e. used variables of a term algebra are usually not regarded as a subset of any larger set of available variables. This term algebra is ground if V=∅, i.e. E=Im φE. So, a ground term L-algebra is an algebra both minimal and injective, and thus also bijective.

If L has no constant then ∅ is a ground term L-algebra.
If L only has constants, then ground term L-algebras are the copies of L.
From any injective L-algebra (EE) and VE \ Im φE one can form the term algebra 〈VL. In particular the existence of an injective algebra implies that of a ground term algebra.

Whenever present as object, any ground term L-algebra is an initial object in any category of L-algebras, and a final object in any category of ground L-drafts. It is thus essentially unique for the given L.
Conversely any initial L-algebra (EE) is a ground term algebra: Similarly, term L-algebras (EE) with E\Im φE=V are the final objects in any variables-preserving category of L-drafts for a given V (this is deducible from the above by replacing variables by constants).

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

Proof 1. By the injectivity lemma, {xK | ∀yK, f(x) = f(y) ⇒ x=y} ∈ SubLK, thus = K.
Proof 2. Im f ∈ SubLM is both injective and minimal, thus a ground term L-algebra, so the morphism f between initial L-algebras K and Im f is an isomorphism.

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 representative image of each term (operation symbol defined by a term), i.e. any two "equivalent terms" (defining the same operation) have the same image. Namely, any L-term T with root t and VTV, is represented in F by the image of the f∈Mor(T,F) such that f|VT = IdVT, with root f(t).
This f plays the role of condensation (so is injective if and only if T is condensed), respecting the interpretation in any L-algebra E extending any vEV, as the unique g∈MorL(F,E) and h∈MorL(T,E) extending v, are related by h=gf, thus h(t)=g(f(t)).

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

Free monoids

If it exists, any unary term L-algebra M (essentially determined by L), is a monoid acting on all L-algebras, whose actions are preserved by all L-morphisms (MorL ⊂ MorM as they form a category of acts). It serves as set of all (function symbols defined by) unary L-terms. The set of function symbols from L is canonically injected there by j(s) = sM(e) so that in any L-algebra E, ∀xE, j(s)⋅x = sE(x).
Conversely if L is made of function symbols then essentially LM, thus MorL = MorM.
The monoid structure of M was defined from its L-structure; now let us take the monoid structure as primitive and review alternative descriptions from it, of the situation when L was made of function symbols. These function symbols can be replaced by (reinterpreted as) constant symbols, as these 2 interpretations can be defined from each other by terms using the monoid structure: the function defines the constant as s = s(e), while the constant defines the function as ∀xM, s(x) = sx.
For any set X, let us call X-monoid any (X∪{e,•})-algebra M seeing X as a set of constant symbols by some j:XM, such that (e,•) is a monoid structure on M.
Denoting X1 the copy of X seen a set of function symbols, the following statements on M are equivalent; such an M is called a free monoid on X.
  1. M is a unary term X1-algebra with variable e, interpreting the copy x'∈X1 of each xX as ∀yM, x'M(y) = j(x)•y
  2. For any X1-algebra E there is a unique left action ⋅ of M on E such that ∀xX, ∀yE, j(x)⋅y = xE(y)
  3. M is an initial object in the category of X-monoids.
(The proof of this equivalence remains to be completed, using the property of trajectories and the representation theorem)
1. is equivalent to: for any X1-algebra E and any xE there is a unique hx ∈ MorX1(M,E) such that hx(e)=x. That is the curried form of the action ⋅ : M×EE.
The uniqueness of the morphism to other X-monoids is expressed by 〈X{e,•} = M.

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

3.11. Integers and recursion

The set ℕ

An arithmetic is any theory describing the system ℕ of natural numbers. There are diverse such theories, depending on the choice of a logical framework, then of a language and axioms. First is the set theoretical one, which is the most precise as it determines the isomorphism class of ℕ in the given universe :

Definition. ℕ is a ground term algebra with two symbols: a constant symbol 0 called zero, and a unary symbol S called the successor.

The essential uniqueness of such systems removes any uncertainty attached to fixing a choice of ℕ among them, at least once any irrelevant questions are made inexpressible by seeing ℕ as a set of pure elements.
The existence of a ground term {0,S}-algebra is our first expression 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 will imply the existence of term algebras with any language.
The above use of algebraic concepts in the definition of ℕ may make it look circular, as our study of algebras used natural numbers as arities of operation symbols. But these definitions may also be written without reference to numbers when used symbols have small arities only (for arithmetic, arities are first just 0 and 1, then later 2).
Namely, the most convenient expression of the axiom of infinity consists in inserting arithmetic into set theory, by first inserting the language of {0,S}-algebra in the form of 3 constant symbols: ℕ (as a set) and the images of both algebraic symbols 0 and S, with axioms 0∈ℕ and S:ℕ→ℕ; then making this a ground term algebra by the following 3 axioms :
(H0) n∈ℕ, Sn ≠ 0 : 0 ∉ Im S
(Inj) n,p∈ℕ, Sn = Sp n = p : S is injective
(Ind) A⊂ℕ, (0∈A ∧ ∀nA,SnA) ⇒ A=ℕ (induction) : ℕ is minimal.

More constant symbols can be defined from there: 1=S0, 2=S1=SS0, 3=S2=SSS0, ...

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 uE such that u ∈ Mor(ℕ,(E,a,f)), where (E,a,f) is the {0,S}-algebra E interpreting 0 as aE and S as fEE :

u0=a
n∈ℕ, uSn = f(un).

As un 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))
Re-interpreting the constant 0 as a variable, makes ℕ a unary term {S}-algebra, where each number n is a unary term Sn with n occurrences of S, interpreted in each {S}-algebra (E,f) as the function f nEE, recursively defined by

f 0 = IdE
n∈ℕ, f Sn = ff n

In particular, f 1=f and f 2 = ff.
Generally for any fEE, gEX, the sequence (hn)n∈ℕ in EX recursively defined by (h0=g) and (∀n∈ℕ, hSn = fhn) is hn=f ng.

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 = Sp(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 pf n is deduced from
(fn+0=fn ∧ ∀p∈ℕ, fn+Sp = fS(n+p) = ffn+p) ∴ ∀aE, ∀fEE, (pf n+p(a))∈Mor(ℕ,(E,fn(a),f)),
In this view, associativity appears as (a+b)+n = Sn(Sb(a)) = Sb+n(a) = a+(b+n). From now on, use of associativity will be implicit, omitting parenthesis.

Multiplication

By the concept of cardinals of finite sets (counting their elements) that will be defined in 4.1., multiplication in ℕ may be defined as the cardinal of a product, making obvious its basic properties, including commutativity. Without this, multiplication can be defined by the following recursion, which needs to treat sides differently until commutativity is deduced. Let us choose the side that fits common language, though it is opposite (transpose) to the usual one from the literature on the axioms of arithmetic:
x∈ℕ, 0⋅x = 0
x,y∈ℕ, (Sx)⋅y = (xy)+y
which can be summed up as xy = (Sy)x(0). Then generally, ∀fEE, f xy = (f y)x.
x∈ℕ, x⋅0 = 0 is deduced by induction.
Right distributivity ∀x,y,z∈ℕ, (x+y)⋅z = xz + yz comes by induction on y, or as (Sz)x+y = (Sz)y০(Sz)x.
Left distributivity ∀x,y,z∈ℕ, x⋅(y+z) = xy + xz comes by induction on x using commutativity of +. In particular this gives ∀x,y∈ℕ, xSy = (xy)+x, so that the transpose of multiplication, satisfying the same recursive definition, coincides with it : multiplication is commutative.

Inversed recursion and relative integers

By induction using commutativity (n+1 = 1+n),

f,gEE, gf = IdE ⇒ ∀n∈ℕ, gnf n = IdE.

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 = (fn)-1f.

The system of (relative) integers ℤ is defined as the {0,S}-algebra where 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 (EE, IdE, f), or an action of ℤ on E interpreting 1 as f.

Proof: the {0,S}-morphism condition requires on ℕ the same nf n as above; on -ℕ, it recursively defines f -n = (f -1)n, namely 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.

3.12. Presburger Arithmetic

First-order 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 review first-order theories describing ℕ as their only type, called theories of first-order arithmetic. As first-order 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 first-order 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 first-order logic. This leaves us with several non-equivalent versions of first-order arithmetic depending on the choice of the primitive language, thus non-equivalent versions of the axiom schema of induction whose range of expressible classes depends on this language:

Presburger arithmetic

A minimal formalization describes the set ℕ* of nonzero natural numbers, with symbols 1 (constant) and + (binary operation), and axioms :
x,y∈ℕ*, x + (y+1) = (x+y)+1 (A1) : + is associative on 1
A⊂ℕ*,(1∈A ∧ ∀x,yA, x+yA) ⇒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 first-order logic: as long as we only consider subsets of ℕ* defined by first-order 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,bA,
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∋xx+1.
These definitions directly imply (H0).

Let (Ind1) be the statement ∀A⊂ℕ*, (1∈A ∧ (∀xA, x+1 ∈A)) ⇒ A=ℕ*.
(Ind1) ⇒ (Ind) ; (Ind) ⇒ (Ind1) using A∪{0}.
One may instead use {x∈ℕ|x+1∈A} once noted that (Ind) ⇒ Im S = ℕ*.
(Ind1) ⇒ (Min)
(As ∧ Min) ⇒ (Ind) in set theory (ignoring our previous definition of ℕ):
Let M = Min{0,S}ℕ.
xM, M ∈ Sub(ℕ,x,S) ∧ fx = (Myx+y) ∈ Mor{S}(M,ℕ).
Im fx = fx [〈0〉{S}] = 〈x{S}M.
As M is stable by + and contains 1, it equals ℕ.∎
(As ∧ Min) ⇒ (Ind) as directly convertible to first-order logic:
Let A∈Sub{0,S}ℕ, and B = {y∈ℕ* |∀xA, x+yA}.
y,zB, (∀xA, x+yx+y+zA) ∴ y+zB.
(∀xA, x+1 ∈A) ⇔ 1∈B ⇒ ((Min)⇒ B=ℕ*).
0∈A ⇒ (∀yB, 0+yA) ⇒ BA.∎
So, all possible axiom pairs are equivalent: (A1 ∧ Min) ⇔ (As ∧ Min) ⇔ (As ∧ Ind) ⇔ (A1 ∧ Ind), and imply commutativity.

Parity

A number n∈ℕ is even if ∃x∈ℕ, n=x+x; it is odd if ∃x∈ℕ, n=x+x+1.
Obviously, if n is even then n+1 is odd; if n is odd then n+1 is even thanks to commutativity. Thus by induction, any number is either even or odd. But to show that it cannot be both and that this x is unique, also requires the use of (Inj). These are left as a possible exercise. But instead of (Inj) we proposed axiom (F). This raises the question of their equivalence.
(F) ⇔ (∀x∈ℕ*, ∀y∈ℕ, x+y y) because x+0 = x ≠ 0.
(Inj ∧ Ind ∧ A1) ⇒ (F) because ∀x∈ℕ*, {y∈ℕ | x+yy} ∈ 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
xy ⇔ ∃z∈ℕ, y = x+z

(A1 ∧ Ind) implies that
  1. < is transitive
  2. xy ⇔ (x<yx=y)
  3. x<yx+1≤y
  4. A⊂ℕ, A≠∅ ⇒ ∃xA, ∀yA, xy (meaning a schema of formulas in Presburger arithmetic)
  5. x,y∈ℕ, xyyx
  6. x<yx+z < y+z
Proofs :
  1. using (As), x < y < z ⇒ (∃n,p∈ℕ*, z = y+p = x+n+p) ⇒ x < z.
  2. obvious from definitions;
  3. thanks to (Ind), ℕ is a bijective {0,S}-algebra;
  4. xy ⇒ (x+1≤yx=y)
    0∈{x∈ℕ |∀yA, xy}=B
    xB, x+1∈BxA
    AB=∅ ⇒ (B=ℕ ∴ A=∅)
  5. from 4. with A={x,y}. Or using 3, A={x∈ℕ |∀y∈ℕ, x<yx=y y<x} ⇒ (0∈A ∧ ∀xA, x+1∈A)
  6. y = x+ny+z = x+z+n
Now the full system (A1 ∧ Ind ∧ F) implies that 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 first-order 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 | nx} is the unique A⊂ℕ such that
x∈ℕ, xA ⇔ (x = n ∨ ∃yA, Sy=x).
Its existence in ℘(ℕ) can be deduced in set theory (not first-order 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 non-injective {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, ux = uy. This x is unique because y is minimal.
{un | n<y} ∈ Sub{0,S}K, thus =K.
As Inj f|Kx=0 ⇔ a∈ Im f|Kfy(a)=a ⇔ (f|K)y = IdK, 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