About the geometry groupoid of a balanced equational variety

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About the geometry groupoid of a balanced equational variety

Laurent Poinsot

LIPN - UMR CNRS 7030

Université Paris XIII, Sorbonne Paris Cité - Institut Galilée

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Balanced equations

Let Σbe a signature (graded set over the integers). Anequationu =v on the free algebra Σ[X](on a setX) isbalanced when both termsu,v have the same set of variables.

For instance, associativity (x₀∗x₁)∗x₂=x₀∗(x₁∗x₂), left 1∗x₀ =x₀ or right x₀∗1=x₀ unit, commutativity x₀∗x₁=x₁∗x₀, quasi-inverse (in an inverse semigroup) x₀^?·x0·x₀^? =x0, all are balanced equations.

Counter-example : left (or right) inverse x₀⁻¹∗x₀ =1.

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Balanced equations

For instance, associativity (x₀∗x₁)∗x₂=x₀∗(x₁∗x₂), left 1∗x₀ =x₀ or right x₀∗1=x₀ unit, commutativity x₀∗x₁=x₁∗x₀, quasi-inverse (in an inverse semigroup) x₀^?·x₀·x₀^? =x₀, all are balanced equations.

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Balanced equations

For instance, associativity (x₀∗x₁)∗x₂=x₀∗(x₁∗x₂), left 1∗x₀ =x₀ or right x₀∗1=x₀ unit, commutativity x₀∗x₁=x₁∗x₀, quasi-inverse (in an inverse semigroup) x₀^?·x₀·x₀^? =x₀, all are balanced equations.

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Balanced variety

An equational variety Vof Σ-algebras is said to bebalancedwhen it is determined by balanced equations on Σ[N].

For instance, the varieties of semigroups, of monoids or also of inverse semigroups (in commutative or non-commutative version) are balanced, while that of groups is not.

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Balanced variety

An equational variety Vof Σ-algebras is said to bebalancedwhen it is determined by balanced equations on Σ[N].

For instance, the varieties of semigroups, of monoids or also of inverse semigroups (in commutative or non-commutative version) are balanced, while that of groups is not.

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Substitution operations

Given a signatureΣ, a set X and a termt∈Σ[N], P. Dehornoy defined Subst_X(t) :={σ(t)ˆ ∈Σ[X] : σ:N→Σ[X]}.

For instance, Subst_X(n) = Σ[X]for every n∈N.

Let (s,t) be a balanced pair of terms (i.e.,s =t is a balanced equation in Σ[N]). One defines a map ρ^(s,t):Subst_X(s)→Subst_X(t) by

ρ^(s,t)(ˆσ(s)) := ˆσ(t) .

(This is a well-defined function since all the variables in t occur ins, hence the value of σ(t)ˆ is entirely determined by that ofσ(sˆ ).)

More generally, given a position p, one defines thetranslatedρ^(s,t)p ofρ^(s,t) that acts in a similar way on sub-terms at position p. In particular

ρ^(s,t) =ρ^(s,t).

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Substitution operations

ρ^(s,t)(ˆσ(s)) := ˆσ(t) .

ρ^(s,t) =ρ^(s,t).

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Substitution operations

Let (s,t) be a balanced pair of terms (i.e.,s =t is a balanced equation in Σ[N]).

One defines a mapρ^(s,t):Subst_X(s)→Subst_X(t) by ρ^(s,t)(ˆσ(s)) := ˆσ(t) .

ρ^(s,t) =ρ^(s,t).

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Substitution operations

ρ^(s,t)(ˆσ(s)) := ˆσ(t) .

ρ^(s,t) =ρ^(s,t).

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Substitution operations

ρ^(s,t)(ˆσ(s)) := ˆσ(t) .

ρ^(s,t) =ρ^(s,t).

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Substitution operations

ρ^(s,t)(ˆσ(s)) := ˆσ(t) .

More generally, given a position p, one defines thetranslatedρ^(s,t) ofρ^(s,t)

In particular ρ^(s,t) =ρ^(s,t).

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Substitution operations

ρ^(s,t)(ˆσ(s)) := ˆσ(t) .

ρ^(s,t) =ρ^(s,t).

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P. Dehornoy’s geometry monoid

Now let us fix a set R ⊆Σ[N]² of balanced equations: some relations.

According toHSP Theorem of G. Birkhoff, thefully invariant, i.e., invariant under endomorphisms, congruence ∼=, onΣ[N]generated by R (it is balanced because R is so) determines a unique balanced equational variety V of Σ-algebras. For instance,

R ={((0∗1)∗2,0∗(1∗2)),(0∗e,0),(e∗0,0)}determines the variety of monoids.

Let us consider the sub-monoidG_R(V) (also denoted byG(V)) of partial bijections of Σ[X]generated by ρ^(s,t)p : Σ[X]→Σ[X],(s,t)∈R or (t,s)∈R, and the positionsp.

This object G(V), introduced by P. Dehornoy, was called themonoid of geometry of the varietyV.

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P. Dehornoy’s geometry monoid

According toHSP Theorem of G. Birkhoff, the fully invariant, i.e., invariant under endomorphisms, congruence ∼=, on Σ[N]generated by R (it is balanced because R is so) determines a unique balanced equational variety V ofΣ-algebras.

For instance,

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P. Dehornoy’s geometry monoid

According toHSP Theorem of G. Birkhoff, the fully invariant, i.e., invariant under endomorphisms, congruence ∼=, on Σ[N]generated by R (it is balanced because R is so) determines a unique balanced equational variety V ofΣ-algebras. For instance,

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P. Dehornoy’s geometry monoid

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P. Dehornoy’s geometry monoid

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Remark

P. Dehornoy also defined anoriented version of G(V) by only considering the generators of the form ρ^(s,t)p for (s,t)∈R, and positions p

: It is relevant to studyrewrite term system.

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Remark

P. Dehornoy also defined anoriented version of G(V) by only considering the generators of the form ρ^(s,t)p for (s,t)∈R, and positions p :

It is relevant to studyrewrite term system.

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Properties

G(V) is aninversemonoid.

Let recall that an inverse monoid is a monoid with a unary operation (−)^?:M →M that satisfies the relations(xy)^? =y^?x^?, 1^? =1,xx^?x =x, (x^?)^? =x,x^?xy^?y =y^?yx^?x, et(xy)^? =y^?x^?. Inverse monoids form a variety (with monoid homomorphisms that commute with the involutions (−)^?).

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Properties

Let recall that an inverse monoid is a monoid with a unary operation (−)^?:M →M that satisfies the relations(xy)^?=y^?x^?, 1^? =1,xx^?x =x, (x^?)^? =x,x^?xy^?y =y^?yx^?x, et(xy)^? =y^?x^?.

Inverse monoids form a variety (with monoid homomorphisms that commute with the involutions (−)^?).

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Properties

Let recall that an inverse monoid is a monoid with a unary operation (−)^?:M →M that satisfies the relations(xy)^?=y^?x^?, 1^? =1,xx^?x =x, (x^?)^? =x,x^?xy^?y =y^?yx^?x, et(xy)^? =y^?x^?. Inverse monoids form a variety (with monoid homomorphisms that commute with the involutions (−)^?).

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Geometry ?

Geometric property (Dehornoy)

The monoid G(V) acts onΣ[X]and the homogeneous space Σ[X]/G(V) (set of orbits) associated with this action is the free algebra V[X]in Von X.

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Relation between G(V) and ∼ =

P. Dehornoy proved the following result :

Theorem (Dehornoy)

For every (non void) partial bijection θin G(V), there is a pair of balanced terms (s_θ,t_θ)∈Σ[N]²,unique up to renaming of variables, such that s_θ∼=t_θ and

θ=ρ^(s^θ^,t^θ⁾ .

Moreover for each set of relations R andR⁰ that generate ∼=, G_R(V)∼=G_R⁰(V).

Hence the geometry monoid G(V)is essentially the fully invariant congruence ∼=on Σ[N]generated by the relationsR, and thus is largely independent of the choice of R and the set X, and, by HSP Theorem of G. Birkhoff, is intrinsically related to the variety Vitself.

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Relation between G(V) and ∼ =

P. Dehornoy proved the following result : Theorem (Dehornoy)

θ=ρ^(s^θ^,t^θ⁾ .

Hence the geometry monoid G(V)is essentially the fully invariant congruence ∼=on Σ[N]generated by the relationsR, and thus is largely independent of the choice of R and the set X, and, by HSP Theorem of G. Birkhoff, is intrinsically related to the variety Vitself.

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Relation between G(V) and ∼ =

θ=ρ^(s^θ^,t^θ⁾ .

Hence the geometry monoid G(V)is essentially the fully invariant congruence ∼=on Σ[N]generated by the relationsR,

and thus is largely independent of the choice of R and the set X, and, by HSP Theorem of G. Birkhoff, is intrinsically related to the variety Vitself.

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Relation between G(V) and ∼ =

θ=ρ^(s^θ^,t^θ⁾ .

Hence the geometry monoid G(V)is essentially the fully invariant congruence ∼=on Σ[N]generated by the relationsR, and thus is largely

and, by HSP Theorem of G. Birkhoff, is intrinsically related to the variety Vitself.

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Relation between G(V) and ∼ =

θ=ρ^(s^θ^,t^θ⁾ .

Hence the geometry monoid G(V)is essentially the fully invariant congruence ∼=on Σ[N]generated by the relationsR, and thus is largely independent of the choice of R and the set X, and, by HSP Theorem of G.

Birkhoff, is intrinsically related to the variety Vitself.

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Groupoid structure on G(V)

For θ1, θ2∈G(V), one defines (classical construction) therestricted product θ₂·θ₁ :=θ₂◦θ₁ if, and only if,dom(θ₂) =im(θ₁).

Remark

There is an isomorphism of categories between inverse semigroups and inductive ordered groupoids (C. Ehresmann, M. Lawson).

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Groupoid structure on G(V)

For θ1, θ2∈G(V), one defines (classical construction) therestricted product θ₂·θ₁ :=θ₂◦θ₁ if, and only if,dom(θ₂) =im(θ₁).

Remark

There is an isomorphism of categories between inverse semigroups and inductive ordered groupoids (C. Ehresmann, M. Lawson).

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Action of a groupoid

Let Gbe a groupoid, and let us denote by Bijthe groupoid of sets with bijections.

Anaction ofG is just a functorF fromGto Bij.

If the class Ob(G) of objects ofG is a small set, then one can define ΣF :=F

A∈Ob(G)F(A).

Let us assume to simplify thatF(A)∩F(B) =∅,A6=B. Theorbit O(x) of x ∈F(A) is defined by {F(f)(x) :d₀(f) =A}⊆Σ_F.

Given x,y ∈ΣF, one defines x ∼_F y if, and only if, there exists an arrow f such that x ∈F(d₀(f))andF(f)(x) =y.

One shows that ∼_F is anequivalence relation on Σ_F andx ∼_F y if, and only if,O(x) =O(y).

One defines the orbit spaceΣF/G asΣF/∼_F.

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Action of a groupoid

Let Gbe a groupoid, and let us denote by Bijthe groupoid of sets with bijections. An action ofG is just afunctor F fromGto Bij.

If the class Ob(G) of objects ofG is a small set, then one can define ΣF :=F

A∈Ob(G)F(A).

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Action of a groupoid

If the class Ob(G)of objects of G is a small set, then one can define ΣF :=F

A∈Ob(G)F(A).

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Action of a groupoid

A∈Ob(G)F(A).

Let us assume to simplify thatF(A)∩F(B) =∅,A6=B.

Theorbit O(x) of x ∈F(A) is defined by {F(f)(x) :d₀(f) =A}⊆Σ_F.

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Action of a groupoid

A∈Ob(G)F(A).

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Action of a groupoid

A∈Ob(G)F(A).

Givenx,y ∈ΣF, one defines x ∼_F y if, and only if, there exists an arrow f such that x ∈F(d₀(f)) andF(f)(x) =y.

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Action of a groupoid

A∈Ob(G)F(A).

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Action of a groupoid

A∈Ob(G)F(A).

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Other definition of an action

Mark V. Lawson introduced the notion of a groupoid action as follows.

Let us assume that E is a set with a map π:E →Ob(G).

A (left) action G on (E, π) is a map from the fibered product

Arr(G) d0×_π E toE, denoted by (f,x)7→f ·x, that satisfies a certain number of axioms.

Theorem (PL, 2014)

Both versions of the definition of a groupoid action are equivalent.

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Other definition of an action

Mark V. Lawson introduced the notion of a groupoid action as follows. Let us assume that E is a set with a map π:E →Ob(G).

Theorem (PL, 2014)

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Other definition of an action

Theorem (PL, 2014)

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Other definition of an action

Theorem (PL, 2014)

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Renaming of variables

The symmetric group S(X) acts onΣ[X]by renaming of variables:

π·t := ˆπ(t).

The value of π·t only depends onπ_|_var(t). Hence one can consider an action by the infinite symmetric group S_∞(X) (permutations ofX that fix all but finitely many members of X).

But actually it is not important that π is a permutation on the wholeX. So one can restrict to endo-functions of X which are bijective only on a finite set, and consider two such functions as equal as soon as they coincide on the finite set: germs of local bijections.

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Renaming of variables

π·t := ˆπ(t).

The value of π·t only depends onπ_|_var(t).

Hence one can consider an action by the infinite symmetric group S_∞(X) (permutations ofX that fix all but finitely many members of X).

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Renaming of variables

π·t := ˆπ(t).

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Renaming of variables

π·t := ˆπ(t).

But actually it is not important that π is a permutation on the whole X.

So one can restrict to endo-functions of X which are bijective only on a finite set, and consider two such functions as equal as soon as they coincide on the finite set: germs of local bijections.

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Renaming of variables

π·t := ˆπ(t).

But actually it is not important that π is a permutation on the whole X. So one can restrict to endo-functions of X which are bijective only on a finite set,

and consider two such functions as equal as soon as they coincide on the finite set: germs of local bijections.

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Renaming of variables

π·t := ˆπ(t).

But actually it is not important that π is a permutation on the whole X. So one can restrict to endo-functions of X which are bijective only on a finite set, and consider two such functions as equal as soon as they coincide on the finite set: germs of local bijections.

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Groupoid of germs of local bijections

Let X be a set, and let us denote by P_fin(X)the set of finitesubsets ofX.

One defines a categoryLocBij(X) oflocal bijections (but not partial) on X: the objects are the members of P_fin(X), an arrow from AtoB is a pair (σ,A) whereσ:X →X such that σ_|_A:A→B is a bijection(hence in particular σ(A) =B and|A|=|B|).

For finite subsets A,B ⊆X of the same cardinal number, let

(σ,A)≡_A,B (τ,A) if, and only if,σ_|_A =τ_|_A. The family(≡_A,B)_A,B is a congruence.

The quotient category Germ_∞(X)is actually a groupoid, with

[σ,A]⁻¹= [(σ_|_A)⁻¹, σ(A)], called the groupoid of germs of bijections ofX.

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Groupoid of germs of local bijections

Let X be a set, and let us denote by P_fin(X)the set of finitesubsets ofX. One defines a categoryLocBij(X) oflocal bijections (but not partial) on X

: the objects are the members of P_fin(X), an arrow from AtoB is a pair (σ,A) whereσ:X →X such that σ_|_A:A→B is a bijection(hence in particular σ(A) =B and|A|=|B|).

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Groupoid of germs of local bijections

Let X be a set, and let us denote by P_fin(X)the set of finitesubsets ofX. One defines a categoryLocBij(X) oflocal bijections (but not partial) on X: the objects are the members of P_fin(X),

an arrow from AtoB is a pair (σ,A) whereσ:X →X such that σ_|_A:A→B is a bijection(hence in particular σ(A) =B and|A|=|B|).

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Groupoid of germs of local bijections

Let X be a set, and let us denote by P_fin(X)the set of finitesubsets ofX. One defines a categoryLocBij(X) oflocal bijections (but not partial) on X: the objects are the members of P_fin(X), an arrow from AtoB is a pair (σ,A) whereσ:X →X such that σ_|_A:A→B is a bijection(hence in particular σ(A) =B and|A|=|B|).

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Groupoid of germs of local bijections

For finite subsets A,B ⊆X of the same cardinal number, let (σ,A)≡_A,B (τ,A)if, and only if, σ_|_A =τ_|_A.

The family (≡_A,B)_A,B is a congruence.

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Groupoid of germs of local bijections

(σ,A)≡_A,B (τ,A)if, and only if, σ_|_A =τ_|_A. The family(≡_A,B)_A,B is a congruence.

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Groupoid of germs of local bijections

(σ,A)≡_A,B (τ,A)if, and only if, σ_|_A =τ_|_A. The family(≡_A,B)_A,B is a congruence.

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Action of Germ

_∞

(X ) on Σ[X ]

Let[σ,A]·t := ˆσ_|_A(t) for each termt such thatvar(t) =A.

This defines a (left) action Germ∞(X) on (Σ[X],var).

The orbit O(t) is just the set of terms obtained fromt byrenaming of variables, and so s ∼t if, and only if, s andt are equal up to renaming of variables.

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Action of Germ

_∞

(X ) on Σ[X ]

Let[σ,A]·t := ˆσ_|_A(t) for each termt such thatvar(t) =A. This defines a (left) action Germ∞(X) on (Σ[X],var).

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Action of Germ

_∞

(X ) on Σ[X ]

The orbit O(t) is just the set of terms obtained fromt byrenaming of variables,

and so s ∼t if, and only if, s andt are equal up to renaming of variables.

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Action of Germ

_∞

(X ) on Σ[X ]

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Action on a balanced congruence

If ∼=is a balanced (i.e., u∼=v ⇒var(s) =var(t)) and fully invariant (i.e., invariant under all endomorphisms) congruence of Σ[X],

then Germ_∞(X) also acts on ∼=by adiagonalaction [σ,A]·(u,v) := (ˆσ|_A(u),σˆ|_A(v)) for every u ∼=v such that var(u) =A=var(v).

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Action on a balanced congruence

then Germ_∞(X) also acts on ∼=by adiagonalaction

[σ,A]·(u,v) := (ˆσ|_A(u),σˆ|_A(v)) for every u ∼=v such that var(u) =A=var(v).

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Action on a balanced congruence

then Germ_∞(X) also acts on ∼=by adiagonalaction [σ,A]·(u,v) := (ˆσ|_A(u),σˆ|_A(v)) for every u ∼=v such that var(u) =A=var(v).

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Notations

Let X be a set and∼= be a fully invariant balanced congruence onΣ[X].

Notations

O(t) andO denote respectively the orbit oft ∈Σ[X] and any orbit under the action of the groupoid of germs of bijections, O⁽²⁾(s,t) andO⁽²⁾ denote respectively the orbit of(s,t)with s ∼=t, and any orbit under the (diagonal) action of Germ∞(X) on ∼=.

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Notations

O(t) andO denote respectively the orbit oft∈Σ[X] and any orbit under the action of the groupoid of germs of bijections,

O⁽²⁾(s,t) andO⁽²⁾ denote respectively the orbit of(s,t)with s ∼=t, and any orbit under the (diagonal) action of Germ∞(X) on ∼=.

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Notations

O(t) andO denote respectively the orbit oft∈Σ[X] and any orbit under the action of the groupoid of germs of bijections, O⁽²⁾(s,t) andO⁽²⁾ denote respectively the orbit of(s,t)with s ∼=t, and any orbit under the (diagonal) action of Germ∞(X) on ∼=.

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Reflexive directed graph

One defines a structure of (small) directed graph:

- The set of verticesisΣ[X]/Germ∞(X), - The set of edgesis ∼=/Germ∞(X),

- Thesourceandtarget maps: let us define∂0(s,t) =s and∂1(s,t) =t for all s ∼=t, then there is a unique (well-defined) map

d_i: ∼=/Germ∞(X)→Σ[X]/Germ∞(X)such that di(O⁽²⁾(s,t)) =O_∂_i_(s,t)

i =0,1.

- Theloops: the map i: Σ[X]→∼=/Germ∞(X), defined by

i(t) :=O⁽²⁾(t,t), passes to the quotient to provide a map that satisfies ι(O(t)) =O⁽²⁾(t,t) for every t ∈Σ[X].

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Reflexive directed graph

- The set of verticesisΣ[X]/Germ∞(X),

- The set of edgesis ∼=/Germ∞(X),

i =0,1.

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Reflexive directed graph

i =0,1.

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Reflexive directed graph

- Thesourceandtarget maps: let us define∂0(s,t) =s and∂1(s,t) =t for all s ∼=t,

then there is a unique (well-defined) map d_i: ∼=/Germ∞(X)→Σ[X]/Germ∞(X)such that

di(O⁽²⁾(s,t)) =O_∂_i_(s,t)

i =0,1.

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Reflexive directed graph

d_i: ∼=/Germ∞(X)→Σ[X]/Germ∞(X)such that d_i(O⁽²⁾(s,t)) =O_∂_i_(s,t) i =0,1.

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Reflexive directed graph

d_i: ∼=/Germ∞(X)→Σ[X]/Germ∞(X)such that d_i(O⁽²⁾(s,t)) =O_∂_i_(s,t)

i =0,1.

- Theloops:

the map i: Σ[X]→∼=/Germ∞(X), defined by

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Reflexive directed graph

d_i: ∼=/Germ∞(X)→Σ[X]/Germ∞(X)such that d_i(O⁽²⁾(s,t)) =O_∂_i_(s,t)

i =0,1.

- Theloops: the mapi: Σ[X]→∼=/Germ∞(X), defined by

i(t) :=O⁽²⁾(t,t), passes to the quotient to provide a map that satisfies ι(O(t)) =O⁽²⁾(t,t) for every t∈Σ[X].

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Structure of multiplicative graph

Deformation of the groupoid structure of an equivalence relation

- Let us define m:{((s,t),(r,s⁰))∈∼=²:s⁰ ∈ O(s)} →Σ[X]×Σ[X]by m((s,t),(r,s⁰)) := (r,[σ,var(s)]·t)

where [σ,var(s)]·s =s⁰. - One observes thatim(m)⊆∼=.

- It can be shown that there is a unique well-defined map

γ: (∼=/Germ_∞(X))_d₀ ×_d₁ (∼=/Germ_∞(X))→(∼=/Germ_∞(X))such that

γ(O⁽²⁾(s,t),O⁽²⁾(r,s⁰)) =O⁽²⁾m((s,t),(r,s⁰)) .

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Groupoid structure

Therefore ∼=/Germ_∞(X) is a (small) category.

Moreover, for every orbit O⁽²⁾(s,t)∈∼=/Germ_∞(X) has an inverse namelyO⁽²⁾(t,s): this provides a structure of groupoid.

This defines the geometry groupoidGeom(V) of the balanced variety V determined by∼=.

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Groupoid structure

Therefore ∼=/Germ_∞(X) is a (small) category. Moreover, for every orbit O⁽²⁾(s,t)∈∼=/Germ∞(X)has an inverse namely O⁽²⁾(t,s)

: this provides a structure of groupoid.

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Groupoid structure

Therefore ∼=/Germ_∞(X) is a (small) category. Moreover, for every orbit O⁽²⁾(s,t)∈∼=/Germ∞(X)has an inverse namely O⁽²⁾(t,s): this provides a structure of groupoid.

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Groupoid structure

Therefore ∼=/Germ_∞(X) is a (small) category. Moreover, for every orbit O⁽²⁾(s,t)∈∼=/Germ∞(X)has an inverse namely O⁽²⁾(t,s): this provides a structure of groupoid.

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Refinement of G. Birkhoff’s HSP Theorem

Theorem (PL, 2014)

The map V7→Geom(V)is a Galois connection between the lattice of balanced sub-varieties of Σ-algebras and a sub-poset of small groupoids.

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Groupoid action on Σ[X ]

Let X be a set.

Lemma (Dehornoy)

For all terms s,t∈Σ[N],Subst_X(s) =Subst_X(t)if, and only if, O(s) =O(t).

It follows that one can define Subst_X(O(t)) :=Subst_X(t) independently of the choice of the representative t.

For all X, one defines an action ofGeom(V) by the functorF_X given by F_X(O) :=Subst_X(O)and

F_X(O⁽²⁾(s,t)) : Subst_X(O(s))→Subst_X(O(t)),F_X(O⁽²⁾(s,t)) :=ρ^(s,t) (does not depend on the choice of (s,t)).

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Groupoid action on Σ[X ]

Let X be a set.

Lemma (Dehornoy)

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Groupoid action on Σ[X ]

Let X be a set.

Lemma (Dehornoy)

For all X, one defines an action ofGeom(V) by the functorF_X given by F_X(O) :=Subst_X(O)

and

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Groupoid action on Σ[X ]

Let X be a set.

Lemma (Dehornoy)

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Recover G(V) from Geom(V)

When X is not void F_X is faithfulandinjective on objects.

It follows that theimage ofF_X (i.e., the classes of sets

F_X(Σ[N]/Germ∞(N)) and of bijectionsF_X(∼=/Germ∞(N))) is a sub-groupoid of Bij: more precisely it is the groupoid associated to the geometry monoid of P. Dehornoy.

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Recover G(V) from Geom(V)

It follows that theimage ofF_X (i.e., the classes of sets F_X(Σ[N]/Germ∞(N))and of bijections F_X(∼=/Germ∞(N)))

is a sub-groupoid of Bij: more precisely it is the groupoid associated to the geometry monoid of P. Dehornoy.

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(89)

Recover G(V) from Geom(V)

F_X(Σ[N]/Germ∞(N))and of bijections F_X(∼=/Germ∞(N))) is a sub-groupoid of Bij

: more precisely it is the groupoid associated to the geometry monoid of P. Dehornoy.

(90)

Recover G(V) from Geom(V)

F_X(Σ[N]/Germ∞(N))and of bijections F_X(∼=/Germ∞(N))) is a sub-groupoid of Bij: more precisely it is the groupoid associated to the geometry monoid of P. Dehornoy.

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Remark

The equivalence relation ∼induced by the action ofGerm_∞(X) on Σ[X] (i.e., u∼v if, and only if, O(s) =O(t))is not in general a congruence,

so that the orbit space Σ[X]/Germ∞(X) is not canonically equipped with a structure of a Σ-algebra.

For instance, letx,y ∈X,x 6=y. Thenx ∼x andx ∼y butx∗x 6∼x∗y.

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Remark

The equivalence relation ∼induced by the action ofGerm_∞(X) on Σ[X] (i.e., u∼v if, and only if, O(s) =O(t))is not in general a congruence, so that the orbit spaceΣ[X]/Germ∞(X) is not canonically equipped with a structure of a Σ-algebra.

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Remark

The equivalence relation ∼induced by the action ofGerm_∞(X) on Σ[X] (i.e., u∼v if, and only if, O(s) =O(t))is not in general a congruence, so that the orbit spaceΣ[X]/Germ∞(X) is not canonically equipped with a structure of a Σ-algebra.

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A Σ-algebra structure on the orbit space

Nevertheless, in case of X =N, it is possible to define aΣ-algebra structure (Lawson)

: letf ∈Σ(n), andu₁,· · · ,u_n∈Σ[N], we let f⁰(u₁,· · ·,u_n) :=O(f(v₁,· · ·,v_n))

for v_i ∈ O(u_i),i =1,· · ·,n such that var(v_i)∩var(v_j) =∅,i 6=j (it is possible since the set of variables is infinite). Of course this definition does not depend on the choice of such that var(v_i)∩var(v_j) =∅.

One shows that there exists one, and only one, well-defined map such that f¯(O(u₁),· · · ,O(u_n)) =O(f(v₁,· · · ,v_n))with v_i ∈ O(u_i),i =1,· · · ,n such that var(v_i)∩var(v_j) =∅,i 6=j.

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About the geometry groupoid of a balanced equational variety