Part 3 : Algebra 1

3.1. Morphisms of relational systems and concrete categories

For simplicity, let us focus the study on systems with only one type.

For any number n and any set E, let En abbreviate the product EVn = E×...×E (n times), that is the set of n-tuples of elements of E.
The sets of all n-ary relations and of all n-ary operations in E are defined as Languages. A language is a set L of "symbols", with the data of the intended arity ns∈ℕ of each symbol sL. It may be For any language L and any set E, let LE = ∐sL Ens
A relational language L, aims to be interpreted in a set E as a family of relations, which belongs to

sL ℘(Ens) ≅ ℘(LE)

Let us now conceive an L-system as a pair (E,E) made of a set E with an L-structure ELE.
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 take a class of L-systems where each E is the intersection of LE with a fixed class of (s,x), denoted as s(x) because the ns-ary relation sE interpreting each symbol sL in each system E is somehow independent of E:

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

Morphism. Between any 2 L-systems E,F, we define the set of L-morphisms from E to F as

MorL(E,F) = {fFE|∀sL,∀xEns, s(x)⇒ s(fx)}
= {fFE|∀(s,x)∈E, (r,fx)∈F}.

For any function f, let fL = (L⋆Domf ∋(s,x) ↦ (s,fx)). This gives shorter definitions for sets of morphisms
MorL(E,F) = {fFE| fL[E]⊂F} = {fFE| EfL*F}.

Concrete categories

The concept of concrete category is what remains of a kind of systems with their morphisms, when we forget which are the structures that the morphisms are preserving (as we saw that this structures list can be extended without affecting the sets of morphisms). Let us introduce a slightly different (more concrete) version of this concept than the one usually found elsewhere: here, a concrete category will be the data of
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 interpreted in a given concrete category is said to be preserved if all morphisms of the category are also morphisms for this symbol. According to 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.

Rebuilding structures in a concrete category.

Starting now with any concrete category, its possible preserved families of relations (one relation in each object) can be produced from some sorts of "smallest building blocks" as follows.

Proposition. In any concrete category, for any choice of tuple t of elements of some object K, the relation 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.∎

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 running over s (with K ranging over all objects).
This can be easily deduced from the fact that any union of preserved structures in a concrete category is a preserved structure (not only finite unions but unions of families indexed by any set). Any intersection of a family of preserved structures is also a preserved structure.

However, the case of topology will show that even giving "all these structures" to the objects of a given concrete category, the resulting category of relational systems may admit more morphisms than those we started with (like a closure).

Preservation of some defined structures

In any given category of L-systems, any further invariant structure whose defining formula only uses symbols in L and logical symbols ∧,∨,0,1,=,∃ is preserved (where 0, 1, ∨ and ∧ are particular cases of unions and intersections we just mentioned).
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 (¬,⇒,∀).

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

Algebras. Given an algebraic language L, an L-algebra is a set E with an interpretation of each sL as an ns-ary operation in E.
Again, let us assume a fixed class of L-algebras E where each s is interpreted as the restriction sE of an ns-ary operator s independent of E,
sE = (Ensxs(x)).
These can be packed as one function

φE = ∐sL sE = ((s,x) ↦ s(x)) : LEE.

Such a class of algebras forms 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| φFfL = 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 to each sL corresponds s'L' with increased arity ns' = ns+1, so that
L'E ≡ ∐sL Ens×E ≡ (LEE
also expressible as the set of triples (s,x,y) such that sL, xEns and yE.
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, (fL(x),f(y))∈F) ⇔ (∀xLE, φF(fL(x))= fE(x))).

Subalgebras. A subset AE of an L-algebra E will be called an L-subalgebra of E, if φE[LA]⊂A.
Then the restriction φA of φE to LA gives it a structure of L-algebra.
The set of all L-subalgebras of E will be denoted SubL E. It is nonempty as E ∈ SubL E.

For any formula of the form (∀(variables), some formula without any binder), its truth in E implies its truth in each A∈SubL E.

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

The proof uses the finite choice theorem with (AC 1)⇒(6):
∀(s,y)∈ L⋆Im f, ∃xEns, fx = ysF(y) = f(sE(x)) ∈ Im f
Thus trying to exend this result to algebras with infinitary operations, would require the axiom of choice, otherwise it anyway still holds for injective morphisms.

Let us generalize the concept of algebra, to any L'-systems (E,E), where E ⊂ (LEE needs not be functional. They form the same kind of categories previously defined, with a different notation (through the canonical bijection depending on the choice of distinguished argument) by which more concepts can be introduced.

Stable subsets. The concept of subalgebra is generalized as that of stability of a subset A of E by L :

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

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.

Let A=f *B. Proof for L-algebras:
∀(s,x)∈LA, fxBnsf(sE(x)) = sF(fx) ∈BsE(x)∈A.
Proof for L'-systems:
∀(x,y)∈E, (fL(x),f(y))∈F∴ (xLAfL(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) = s(fx) = g(y). ∎

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

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

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

Subalgebra generated by a subset.AE, the L-subalgebra of E generated by A, written 〈AL,E or more simply 〈AL, is the smallest L-subalgebra of E including A:

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

For fixed E and L, this function of A is a closure with image SubLE.
We say that A generates E if 〈AL=E.

Minimal subalgebra. For any L-algebra (or other L'-system) E, its minimal subalgebra (or minimal stable subset) is MinLE =〈⌀〉L,E = ∩SubLE ∈SubLE.

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

Proposition. For any L-algebra 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).
We can redefine generated subalgebras in terms of minimal subalgebra with a different language: 〈AL,E= MinLA E where A is seen as a set of constants.

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[L⋆MinLE] ⊂ Im φE
  4. AE, 〈AL = A∪φE[L⋆〈AL] ⊂ A∪Im φE.
  5. f ∈MorL(E,F), f [MinLE] = MinLF ; more generally ∀AE, f [〈AL] = 〈f [A]〉L
Proofs:
  1. φE[LA] ⊂ Im φEAA∈SubLE
  2. Im φE ∈ SubLE
  3. MinLE is surjective
  4. A∪φE[L⋆〈AL] ∈ 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) ⊂ LBQzLE, φF(fL(z)) = y.
    As φFfL = 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.

Replacing F by the bijectively related set Im g, simplifies things to the case FEg=IdF.
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.

Embeddings and isomorphisms

Strong preservation. A function fFE is said to strongly preserve a relation symbol or formula r interpreted in each of E and F, if it preserves both r and ¬r :

xEnrxrEfxrF.
Embeddings. An f ∈MorL(E,F) is called an L-embedding if it strongly preserves all structures : ∀rL,∀xEnrxrEfxrF.

Isomorphism. An isomophism between algebras is a bijective embedding. Generally, an isomorphism between objects E and F of a concrete category, is a morphism (f ∈Mor(E,F)) that is bijective (f : EF) and whose inverse is a morphism (f -1∈Mor(F,E)).
Two objects E, F are said to be isomorphic if there exists an isomorphism between them.

Injective embeddings are isomorphisms to their images.

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

fL-1 = (f-1)L ∴ φEfL-1 = f -1f০φEfL-1 = f -1০φFfLfL-1 = f -1০φF.

More generally for relational systems, 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.

Elementary embeddings

Embeddings still strongly preserve structures defined using the symbols in L and the logical symbols ∧,∨,0,1,¬, and also = in the case of injective embeddings.
Thus, they also preserve invariant structures defined using symbols of L and ∧,∨,¬,0,1,∃ where any occurrence of ¬ comes after (inside) any occurrence of ∃.

Now the full use of first-order logic comes by removing this restriction on the order of use of logical symbols: an f ∈ MorL(E,F) is called an elementary embedding (or elementary L-embedding) if it (strongly) preserves all invariant structures (defined by first-order 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 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

An endomorphism of an object E in a category, is a morphism from E to itself. Their set is written End(E)=Mor(E,E).

For any set E, ∀fEE, ∀AE, A ∈ Sub{f} Ef ∈ End{A}E.

Automorphisms. An automorphism of an object E is an isomorphism of E to itself.

So we have: Automorphism ⇔ (Endomorphism ∧ Isomorphism)
However an endomorphism of E which is an isomorphism to a strict subset of E, is not an automorphism.

If f∈ End(E) is an invariant elementary embedding then it is an automorphism:
Im f is also invariant (defined by ∃yE, f(y)=x)
xE, x∈Im ff(x)∈Im f
Im f = E. ∎

Quotient systems

For any relational language L, any L-system (E,E) and any equivalence relation R on E, the quotient set E/R has a natural L-structure defined as  RL[E].
It is the smallest L-structure on E/R such that R∈ Mor(E, E/R).

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 {Id, ০}-algebra, with two operation symbols: the constant Id, and the binary operation ০ of composition.
More generally, a transformation monoid G of E is an {Id, ০}-algebra G∈Sub{Id, ০}EE, of transformations of E: These axioms were those directly subjecting the set of endomorphisms of a fixed object in a concrete category. In particular, they are the exact axioms subjecting (defining the concept of) a concrete category with only one object E.

Permutation groups

A permutation of a set E is a bijective transformation of E.
The set ⤹E ={f EE| Inj f ∧ Im f=E} = {f EE|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 transformation monoid of E such that

G ⊂ ⤹E ∧ ∀fG, f -1G.

It will be seen as an algebra with one more operation : the inversion function ff -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 satisfying the above first-order stability axioms, ignoring the containing sets EE or ⤹E.

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}}.

Intuitively, this is because they mean the same : the set of all composites of any number of functions in L, applied to x. This will be formalized later. But here is a simple proof, denoting K={f(x)|f∈〈L{Id, ০}}:
Proof of 〈{x}〉LK
IdE∈〈L{Id, ০}xK
(f∈〈L{Id, ০},∀gL, gf∈〈L{Id, ০}g(f(x))∈K)K∈SubLE
Proof of K ⊂ 〈{x}〉L
L ⊂ {fEE| 〈{x}〉L ∈ Sub{f} E} = End{〈{x}〉L}E ∈ Sub{Id, ০}EE ∴ ∀f∈ 〈L{Id, ০}, 〈{x}〉L ∈ Sub{f} Ef(x)∈〈{x}〉L. ∎
The trajectory of an element xE by a transformation monoid G of E, is the set it generates (usually called its orbit when G is a permutation group):

〈{x}〉G = {f(x)|fG} ⊂ E

As noted in the example in 2.7, the binary relation on E defined by (y∈〈{x}〉G) is a preorder; it is an equivalence relation if G is a permutation group.

Monoids

A monoid is an algebra behaving like a transformation monoid, but without specifying a set which its elements may transform, by which both symbols Id and ০ got their interpretation. This abstractness can be formalized by renaming these symbols, respectively as e and •.
Namely, the concept of monoid is the theory made of

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 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} (its identity property defines how composition extends to it, preserving associativity). Any identity element in E loses its status of identity in E'.

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 xz=yzx=y.
If a right identity e is left cancellative then it is the unique right identity : ee' = e = eee' = e.
The operation • is called cancellative if all its 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 sub-monoid.
From the concept of monoid, replacing the use of e as a constant by ∃e in the axiom, would weaken or generalize the concepts of submonoids and morphism as follows.
For any monoid (M, e, •) and any set with a binary operation (X, ▪), if a function preserves composition f ∈ Mor(M,X) then If the target forms a monoid (X,▪, e') then (by uniqueness of the identity in A)

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').
Every embedding between monoids is injective (as the identity element ensures the surjectivity of composition).

3.5. Actions of monoids

Left actions

Now let us give back to monoids their role as sets of transformations.
A left action of a monoid (M,e, •) on a set X, is an operation ⋅ from M×X to X satisfying the axioms: This turns X into an M-algebra (where M is seen as a set of function symbols), called an M-set.

Effectiveness and free elements

A left action of M on X can be seen by currying as a {e, •}-morphism from M to XX. A left action is said effective if this morphism is injective:

a, bM, (∀xX, a·x = b·x) ⇒ a=b

letting the axioms of action be usable as definitions of e and • from the action, in the same way the definitions of Id and ০ are re-expressible (from their initial expressions via the axioms for function) as determined by 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. The monoid of endomorphisms of any typed system E= ∐iI Ei, acts on every type Ei it contains, by the morphism of monoid from End(E) to EiEi defined by restricting to Ei each endomorphism of E.

Right actions

The opposite of a monoid is the monoid with the same base set but where composition is replaced by its transposed. This symmetry of the concept of monoid, leads to the similar concept of right action of a monoid M on a set X: it is an operation ⋅ : X × MX such that
It defines a function f : MXX which is not a morphism but an anti-morphism, i.e. a morphism from one monoid to the opposite of the other (or equivalently vice-versa):

f(e)=IdX
a,bM, f(ab) = f(b) ০ f(a)

Commutants

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

C(A) = {xE|∀yA, x#y = y#x}.

This is a Galois connection: ∀A,BE, BC(A) ⇔ AC(B).
A binary operation # in a set E, is called commutative when C(E) = E, i.e. ∀x,yE, x#y = y#x.

Proposition. For any associative operation # on a set E, ∀AE,

  1. C(A) ∈ Sub#F
  2. If AC(A) and 〈A#=E then # is commutative
Proof:
  1. x,yC(A), (∀zA, (x#y)#z = x#z#y = z#(x#y)) ∴ x#yC(A)
  2. AC(A)∈ Sub#FE=C(A) ⇒ AC(E) ∈ Sub#FC(E) = E.

Centralizers

In monoids, commutants of subsets are sub-monoids (as e commutes with all elements). There, the word "centralizer" is used instead of "commutant". This concept will be later generalized further, from this unary case (acting as sets of transformations) to clones of operations with all arities.
The centralizer of any GEE, is its monoid of endomorphisms EndG E.
The above commutativity result works with the weakened assumption 〈A{e,•}=E.
When 2 actions of monoids on the same set X commute with each other, it can be formally convenient to see them acting on a different side: the commutation between aM acting on the left and bN acting on the right on X is written

xX, (ax)b = a(xb)

which formally looks like an associativity law.

Remark. 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).

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 L-algebra X such that the monoid EndL X is isomorphic to M.

Proof:

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, XX) represent the left action of M on X also copied from the composition in M.
Im f ⊂ EndL 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.
EndL X ⊂ Im f because
g∈EndLX, ∃uM, g(k)=uk ∴ (∀xX, ∃sL, ks=xg(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 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

Inverses

In a monoid (M, e, •), the formula xy=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 xy=e becomes an invertibility of transformations :

z,tM, yz = txt = 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 the surjectivity of the right composition by y ({xy|xM} = 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 both left invertible and right invertible is called invertible. Then any left inverse y and any right inverse z must be equal, and thus unique, simply called the inverse of x and written x-1.
Proof: yx = e = xzy = yxz = z

If a left invertible element y is also right cancellative then it is invertible: xy=eyxy = eyyx=e.
This 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 can still make sense of its invertibility by saying it has an inverse M-morphism from X to M.
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 (equivalently on one or both sides): xx=e. This qualifies an element such as a transformation, regardless the choice of sub-monoid containing it (a transformation monoid).

Groups

A group is a monoid where all elements are invertible.
The set of invertible elements in any monoid, is a 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 The interpretation of inversion is determined from those of • and e by the axiom (whose truth in a group implies its truth in every subgroup)

x, xx-1 = x-1x = 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 preserves the inversion relation, thus its image G' is a group, and f is a morphism of groups 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 anti-morphism, it switches any left action ▪ of G on X into a right action • by ∀xX, ∀gG, x•g = g-1▪x.

The subgroup of a group generated by a subset A, coincides with the submonoid G generated by A∪-A where -A={x-1|xA} .

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: satisfying the axioms Isomorphisms in abstract categories are the generalization of invertible elements : an isomorphism f from E to F is an f∈Mor(E,F) such that ∃g∈Mor(F,E), gf= IdEfg= IdF (for concrete categories, this condition is equivalent to the one we gave).

Like in monoids, the inverse of any isomorphism (= invertible morphism) is unique.

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)
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.

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: we say that 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) = 1Mor(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

Initial and final objects

In any category, an object X is called an initial object (resp. a final object) if for all objects Y the set Mor(X,Y) (resp. Mor(Y,X)) is a singleton.
Such objects have this remarkable property: when they exist, all such objects are isomorphic, by a unique isomorphism between any two of them.

Proof: For any initial objects X and Y, ∃f∈Mor(X,Y), ∃g∈Mor(Y,X), gf ∈Mor(X,X) ∧ fg ∈ Mor(Y,Y).
But 1X ∈ Mor(X,X) which is a singleton, thus 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. 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: Exercise. Given two fixed sets X and Y, consider the category whose objects are all (E,φ) where E is a set and φ: E×XY, and the morphisms from (E,φ) to (E',φ') are all f : EE' such that ∀aE,∀xX, φ(a,x) = φ'(f(a),x).
Does it have an initial object ? a final object ?

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 (M, e) is an initial object of C' (when seen as acting on itself by •); initial objects are the (X,x) where x is a free and generating element of X.
  2. Conversely, if C' has an initial object (M, e) then we can use e to define a binary operation on M making it a monoid with neutral element e acting on all objects of C, and all morphisms of C are M-morphisms between these M-sets (but there may exist other M-morphisms from objects other than M)
These are direct consequences of the remark on monoid acts.
For 2., the composition in M is defined as the particular case of the M-algebra structure on M satisfying axioms of M-acts.
The monoid M can be seen as a transposed copy of the monoid End(M). Indeed for all a, bM, ha, hb∈End(M) and ha(hb(e))=ha(b)=ba.
The preservation of these interpreted function symbols, letting morphisms of C remain M-morphisms, is a particular case of preservation of relations defined by tuples.

3.8. Algebraic terms and term algebras

Algebraic drafts

As the concept of term was only intuitively introduced in 1.5, let us now formalize the case of terms using a purely algebraic language (without logical symbols), called algebraic terms, as mathematical systems in a set theoretical framework.
For convenience, let us work with only one type (the generalization to many types is easy) and introduce a class of systems more general than terms, that we shall call drafts. Variables will have a special treatment, without adding them as constants in the language.

Given an algebraic language L, an L-draft will be an L'-system (D,D) where D⊂ (LDD, such that:

Let us denote ∀xOD, ΨD(x)=(σ(x), lx) ∈ LD where s(x)∈L and lxDnσ(x). We can also denote σD(x)=σ(x) to let σD be a function with domain OD.
Well-foundedness can be equivalently written in any of these forms

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

The set of used variables of (D,D), those which effectively occur, is VD⋂⋃xOD Im lx. Unused variables can be added or removed in D while keeping D fixed (by changing DOD∪(⋃xOD Im lx)), so that their presence may be irrelevant.
A ground draft is a draft with no variable, i.e. VD=∅. Thus, ΨD: DLD and SubLD = {D}.
More generally a draft is ground-like if it has no used variable (Dom DLOD).

Sub-drafts and terms

Drafts do not have interesting stable subsets (by well-foundedness), but let us introduce another stability concept for them.
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 also a sub-draft; the sub-draft co-generated by a subset is the intersection of all sub-drafts that include it.
A term is a draft co-generated by a single element which is its root.
Moreover, any union of sub-drafts is also a sub-draft (which was not the case for sub-algebras because an operation with arity >1 whose arguments take values in different sub-algebras may give a result outside their union).

There is a natural order relation on each draft D defined by xy ⇔ x∈ (the term with root y). It is obviously a preorder. Its antisymmetry is less obvious; a proof for integers will be given in 3.6, while the general case will come from properties of 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= fL০ΨD)

where the equality condition can be split as

σEf|OD = σD
xOD, lf(x)=flx

Another kind of category of drafts can be considered, with objects also L-drafts but with a common set of variables (VD=VE=V) and taking smaller sets of morphisms: the variables-preserving morphisms, i.e. moreover satisfying f|V = IdV.
But for any element t in any draft, the term T co-generated by {t} has as set of variables TV (which is the set of used variables of T unless T={t}⊂V) generally smaller than V, so the admission of terms defined as subsets co-generated by singletons in such a category requires this loosening of the condition. This naturally simplifies when reformulating such categories as those of ground (LV)-drafts: in each draft, variable symbols are replaced (reinterpreted) by constant symbols added to the language, so ΨE is extended by IdV, to form a ground (LV)-draft.

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= φEfL০ΨD, which can also be written

xOD, f(x) = σ(x)E(flx)

Theorem. For any L-draft D with set of variables 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).

The uniqueness comes from a previous proposition.

Proof of existence.
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(hLD(x))))) ∈ K (see conditional operator)
D*(LC) ⊂ C
C=D

Operations defined by terms

A new operation symbol can be defined by any element t of an L-draft D : it is the V-ary operation defined by the L-term with root t, that is co-generated by t in D. Its interpretation in each E is defined by ∀vEV, tE(v) = h(t) for the unique h∈MorL(T,E) such that h|TV=v.

As a particular case of a relation defined by a tuple, here (IdV,t), these operations are preserved by all morphisms between L-algebras, and can thus be added to L without changing the sets of morphisms.

Another way to see it is as a particular case of conservation of relations defined by formulas in categories of relational systems, since any term defining an operation can be re-expressed as a formula defining the graph of this operation, using logical symbols ∃ and ∧. The advantage now that it is established for the general case of abstractly conceived terms no matter their size, instead of concretely written terms on which the conservation property must be repeatedly used for each occurrence of symbol it contains.

Term algebras

An L-algebra (EE) is called a term algebra if it is injective and 〈E\Im φEL = E. Thus it is also an L-draft with ΨE = φE-1. As such, it is ground if moreover E=Im φE. So, a ground term algebra is an algebra both minimal and injective, and thus also bijective.

If L does not contain any constant then ⌀ is a ground term L-algebra.
If L only contains constants, then ground term L-algebras are the copies of L.
The existence of term algebras in other cases will be discussed in the next section; let us admit it for now.

Whenever present as object, any ground term L-algebra is a final object in any category of ground L-drafts, and an initial object in any category of L-algebras. In any variables-preserving category of L-drafts with a fixed set V of variables (category of ground (LV)-drafts), any term L-algebra F, when present, is a final object.

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 a previous result, {xK | ∀yK, f(x) = f(y) ⇒ x=y} ∈ SubLK, thus = K.
Proof 2. The subalgebra Im f of M 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

Any term algebra F plays the role of the "set of all terms" with its list V of variable symbols, for the following reason:
Each element of F bijectively defines a term in F as the sub-draft of F it co-generates, thus where it is the root.
For any L-term T with root t and variables ⊂V, the unique f∈Mor(T,F) such that f|TV = IdTV represents it in F as the term Imf with root f(t).
Then its interpretation in any L-algebra E extending any vEV, is determined by the unique g∈MorL(F,E) extending v, as gf∈Mor(T,E), with result g(f(t)).
The same for terms whose set of variables V' is interpreted in E by the composite of a function from V' to V, with one from V to E (instead of having V'V).
For any subset A of an L-algebra E, any term algebra FA whose set of variables is a copy of A, represents the set of all L-terms with variables interpreted in A. Then, the L-subalgebra 〈AL of E is the image of the interpretation of FA in E, i.e. the set of all values of these terms.

The monoid of unary terms

Let M be a unary term L-algebra (unary = with one variable; this algebra is essentially unique for each L). Then its elements play the roles of the function symbols defined by all possible unary L-terms, and preserved by morphisms across all L-algebras, including M itself. Therefore, thanks to a previous remark, M is natually a monoid acting on all L-algebras.
If L is only made of function symbols then for any L-algebra E, the image of M in EE is the transformation monoid 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 ∧ ∀nA,SnA) ⇒ 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 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) 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)),
from which associativity comes as (a+b)+n = Sn(Sb(a)) = Sb+n(a) = a+(b+n).

Multiplication

Multiplication in ℕ can be defined as xy = (Sx)y(0), or recursively as

x∈ℕ, x⋅0 = 0
x,y∈ℕ, x⋅(Sy) = (xy)+x

Then generally, ∀fEE, f xy = (f x)y.

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 commutative because it is generated by {-1, 1}, which commute: (-1)+1=0=1+(-1).
It is a group: (-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

First-order theories of arithmetic

Our first formalization of ℕ was based on the framework of set theory, where it used the powerset to characterize ℕ in an "essentially unique" way (specify its isomorphism class). It allowed recursion, from which we could define addition and multiplication.

Let us now consider formalizations in the framework of first-order logic, thus called 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 without the set theoretical framework we used to express and justify the definiteness of recursive definitions, the successor function no more suffices to define addition and multiplication. This leaves us with several 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

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,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 let us regard our present use of set theoretical notations as mere abusive 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. In particular, (Min) will be meant as abbreviating a schema of axioms with A ranging over all classes in this theory.
(A1) is a particular case of
x,y,z∈ℕ*, x + (y+z) = (x+y)+z (As) : + is associative

Conversely, (A1 ∧ Min) ⇒ (As) :
Let A={a∈ℕ* |∀x,y ∈ℕ*, x+(y+a) = (x+y)+a}. ∀a,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).

(Ind) ⇒ (Min) :
A⊂ℕ*, the set A0= A∪{0} satisfies 0∈A0 and
(1∈A ∧ (∀x,yA, x+yA)) ⇒ (S0∈A0 ∧ (∀xA, Sx=x+1 ∈AA0)) ⇒ A0=ℕ.∎
(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((M,0,S),(ℕ,x,S)).
Images of minimal algebras by morphisms are included in any subalgebras: Im fxM.
As M is stable by + and contains 1, it equals ℕ.∎
(As ∧ Min) ⇒ (Ind) in 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.∎
(F) ⇔ (∀x∈ℕ*, ∀y∈ℕ, x+y y) because x+0 = x ≠ 0.
(Inj ∧ Ind ∧ A1) ⇒ (F) : ∀x∈ℕ*,
x+0 ≠ 0
y∈ℕ, x+y yx+y+1 ≠ y+1.∎
For the converse, we need to use the order relation.

The order relation

From the operation of addition in Presburger arithmetic, let us define binary relations ≤ and < on ℕ and show that they form an order and its strict order (it coincides with the order between terms in the common particular case of the set theoretical ℕ, thanks to the properties of generated preorders) :

x<y ⇔ ∃z∈ℕ*, y = x+z
xy ⇔ ∃z∈ℕ, y = x+z

For this, here are successive consequences of (Ind ∧ A1) :
  1. < is transitive
  2. xy ⇔ (x<yx=y)
  3. x<yx+1≤y
  4. A⊂ℕ, A≠∅ ⇒ ∃xA, ∀yA, xy (to be interpreted as a schema of formulas if we study 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}
  6. y = x+n y+z = x+z+n
(for 5. it is also possible to more directly prove for A={x∈ℕ |∀y∈ℕ, x<yx=y y<x} that 0∈A and ∀xA, x+1∈A)

Now, (F) means that < is irreflexive, thus a strict total order thanks to 1. and 5.

Moreover it implies ∀x,y,z∈ℕ, (x<yx+z < y+z) and (x = yx+z = y+z). The last formula gives (Inj) as a particular case, and means cancellativity, as sides can be switched thanks to commutativity (deduced from the same assumptions as shown above).

Proof: the direct implications were shown above; the converses are deduced from there as < is a strict total order : one of the 3 formulas (x<y), (x = y), ( y<x) must be true while only one of (x+z<y+z), (x+z=y+z), (y+z<x+z) can.∎
Finally, ≤ is a total order with strict order < and every nonempty subset A of ℕ has a smallest element (unique by antisymmetry), written min A.

Arithmetic with order

It is possible to express a first-order arithmetic with language {0,S, ≤}, stronger than {0,S} but weaker than Presburger arithmetic, in the sense that addition cannot be defined from ≤.
Namely, it can be based on the characteristion of the order by the property:
For all n ∈ℕ, the set {x∈ℕ | nx} is the unique A⊂ℕ such that
x∈ℕ, xA ⇔ (x = n ∨ ∃yA, Sy=x).
Its existence in ℘(ℕ) can be deduced by induction on n; its uniqueness for a fixed n is deduced by induction on x.

Trajectories of recursive sequences

A trajectory K = Im u where u= (f n(a))n∈ℕ is a copy of ℕ as soon as it is an injective {0,S}-algebra.
Let us look at the rest of cases (there will turn out to be only one pair in {0,S}⋆K mapped to a singleton).
Let y the minimal number such that ∃x<y, ux= uy. This x is unique because y is minimal.
The restriction of u to Vy is injective; its image being stable, equals K.
As Inj f|Kx=0 ⇔ a∈ Im f|Kfn(a)=a ⇔ (f|K)n = 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.
Picking another element a in such a 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