Title: Theory / Unimportant / CS / Fundamental / Automata / Finite / Pin{ / C The profinite world / 11 Varieties
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Name: A variety of ordered semigroups/monoids
Definition: closed under division and finite product
Page: 191, 197
Chapter: XI
Section: 1, 3.3
Definition2: satisfying an arbitrary set of profinite identities
Conjdef: A set of ordered semigroups/monoid, equivalently

Name: A variety of semigroups/monoids
Definition: A set of semigroups/monoinds closed under division and finite product
Page: 191, 197
Chapter: XI
Section: 1, 3.3
Definition2: The set of semigroups satisfying an arbitrary set of profinite identities

Name: The variety of all semigroups
Page: 192
Chapter: XI
Section: 1
Notation: \(\mathbf S\)
Index: (1)

Name: The variety of empty and singleton semigroup
Page: 192
Chapter: XI
Section: 1
Context: Variety
Notation: \(\mathbf 1\)
Index: (2)

Definition: minimal containing \(C\).
Page: 192
Chapter: XI
Section: 1
Variables: \(C\) a class of semigroups
Notation: \(\langle C\rangle\)
Kind: Prop
Definition2: it contains exactly the dividors of finite product of elements of \(\mathcal C\)
Conjdef: The variety such that, equivalently

Name: The supremum of \(\mathbf V_1\) and \(\mathbf V_2\)
Definition: \(\langle\mathbf V_1\cup\mathbf V_2\rangle\),
Page: 192
Chapter: XI
Section: 1
Variables: \(\mathbf V_1\) and \(\mathbf V_2\) two varieties
Notation: \(\mathbf V_1\lor\mathbf V_2\)
Definition2: the set of dividors of \(M_1\times M_2\) with \(M_i\in\mathbf V_i\).

Definition: The cardinality of the minimal element of \(\mathbf V\) that separates \(u\) and \(v\)
Page: 192
Chapter: XI
Section: 2
Variables: \(\mathbf V\) a variety
\(u,v\in A^*\)
Notation: \(r_{\mathbf V}(u,v)\)

Page: 192
Section: 2
Notation: \(d_{\mathbf V}(u,v)\)
Variables: \(\mathbf V\) a variety and \(u,v\in A^*\)
Chapter: XI
Definition: \(2^{-r_{\mathbf V}(u,v)}\)
Metric: An ultrametric on \(A^*/\sim_{\mathbf V}\)

Smaller0: \(d_{\mathbf V}(uw,vw)\)
Greater0: \(d_{\mathbf V}(u,v)\)
Page: 192
Chapter: XI
Section: 2
Variables: \(\mathbf V\) a variety
\(u,v,w\in A^*\)
Index: 3 (2)
Kind: Prop
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(d_{\mathbf V}(wu,wv)\)
Greater0: \(d_{\mathbf V}(u,v)\)
Page: 192
Chapter: XI
Section: 2
Variables: \(\mathbf V\) a variety and \(u,v,w\in A^*\)
Index: 3 (2)
Kind: Prop
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Question: Product on \(A^*/\sim_{\mathbf V}\) is
Answer: uniformly continuous.
Page: 193
Chapter: XI
Section: 2
Variables: \(\mathbf V\) a variety
Index: 4 (2)
Kind: Prop

Page: 194
Name: Free pro-\(\mathbf V\)-monoid on \(A\)
Chapter: XI
Section: 2
Variables: \(\mathbf V\) a variety
Kind: , Prop
Index: , 5(1)
Notation: \(\widehat F_{\mathbf V}(A)\)
Definition: least topology on which each morphism \(F_{\mathbf V}(A)\to M\) with \(M\) finite can be extended in a continuous morphism
Compact: yes
Extra: , Compact
Completion of: the metric space \(\left(A^*/\sim_{\mathbf V},d_{\mathbf V}\right)\).

Definition: The natural morphism from \(A^*\) onto \(A^*/\sim_{\mathbf V}\)
Page: 194
Chapter: XI
Section: 2
Variables: \(\mathbf V\) a variety
Notation: \(\pi_{\mathbf V}\)

Question: The number of extension of \(\pi_{\mathbf V}\) on \(\widehat F_{\mathbf V}(A)\) is
Answer: exactly one.
Page: 193
Chapter: XI
Section: 2
Variables: \(\mathbf V\) a variety
Index: 5 (2)
Kind: Prop

Question: The number of uniformly continuous extension of \(\phi\) on \(\widehat F_{\mathbf V}(A)\) is
Answer: exactly one.
Page: 194
Chapter: XI
Section: 2
Variables: \(\mathbf V\) a variety
\(M\in \mathbf V\)
\(\phi:A\to M\).
Index: 6
Kind: Prop

Smaller0: a quotient by a continuous morphism of \(\widehat F_{\mathbf V}(A)\)
Greater0: belongs to \(\mathbf V\)
Page: 194
Chapter: XI
Section: 2
Variables: \(\mathbf V\) a variety,
\(A\) a set
\(M\) a \(A\)-generated monoid
Index: 7
Kind: Prop
Smaller to greater: iff
Prefix Greater: \(M\)
Assuming: \(M\) is finite
Prefix Smaller: \(M\) is

Definition: elements: functions from \(M^A\) to \(M\)
Page: 194
Chapter: XI
Section: 2
Variables: \(A\) an alphabet
\(M\) a monoid
Notation: \(M^{M^A}\)
Definition2: product: pointwise multiplication
Conjdef: The monoid

Definition: [$]\defFun{M^A}{M}{\phi}{\prod_{i=1}^n\phi(a_i)}[/$]
Page: 194
Chapter: XI
Section: 2
Variables: \(A\) an alphabet,
\(w=a_1\dots a_n\in A^*\)
\(M\) a monoid
Context: \(M^{M^A}\)
Notation: \(\tilde w\)

Smaller0: \(\mathbf V\) is generated by \(M\)
Greater0: equal to \(F\)
Page: 195
Chapter: XI
Section: 2
Variables: \(A\) an alphabet
\(M\) a monoid
\(\mathbf V\) a variety
Index: 8
Kind: Prop
Prefix Greater: \(\hat{\mathcal F}_{\mathbf V}(A)\) is
Greater1: a submonoid of \(M^{M^A}\), hence finite
Connect Greater0: and is

Name: Locally finite variety
Definition: A variety \(\mathbf V\) such that \(\widehat F_{\mathbf V}(A)\) is finite for each alphabet \(A\).
Page: 195
Chapter: XI
Section: 2

Name: Locally finite variety which is not finitely generated
Page: 195
Chapter: XI
Section: 2
Example: The variety of idempotents monoids

Name: \(M\) satisfies the profinite identity \(u=v\)
Definition: For each monoid morphism \(\phi:A^*\to M\), \(\widehat \phi(u)=\widehat \phi(v)\).
Page: 195
Chapter: XI
Section: 3.1
Variables: \(A\) an alphabet,
\(u,v\in\widehat {A^*}\)
\(M\) a monoid

Name: Explicite equality
Definition: \(u=v\) such that \(u,v\in A^*\).
Page: 195
Chapter: XI
Section: 3.1
Variables: \(A\) an alphabet
Context: Varieties

Name: Explicite inequality
Definition: \(u\le v\) such that \(u,v\in A^*\)
Page: 195
Chapter: XI
Section: 3.1
Variables: \(A\) an alphabet, and \(M\) a ordered monoid
Context: Varieties

Smaller0: \(M\) is commutative
Greater0: \(M\) satisfies the identity \(xy=yx\)
Page: 196
Chapter: XI
Section: 3.1
Variables: \(M\) a semigroup
Smaller to greater: iff

Smaller0: \(E(M)\) is commutative
Greater0: \(M\) satisfies the identity \(x^{\omega}y^{\omega}=y^{\omega}x^{\omega}\)
Page: 196
Chapter: XI
Section: 3.1
Variables: \(A\) an alphabet
\(M\) a monoid
Smaller to greater: iff

Name: An idempotent
Page: 196
Chapter: XI
Section: 3.1
Variables: \(A\) an alphabet,
\(M\) a monoid
Context: Identity over finite monoid
Notation: \(x^{\omega}\)

Name: The set of identities satisfied by \(M\)
Page: 192
Section: 3.2
Chapter: XI
Index: 9
Variables: \(A\) an alphabet,
\(M\) an (ordered) monoid,
\(u,v\in\widehat{A^*}\)
Kind: Prop
Under: product by a word, on right or left
Under2: morphism
Under3: product of identity

Definition: The set of (ordered) monoids/semigroups satisfiying the identities of \(E\)
Page: 196
Chapter: XI
Section: 3.3
Variables: \(E\) a set of identities
Notation: [$]\llbracket{}E\rrbracket[/$]

Name: \(\mathbf V\) satisfies \(i\)
Definition: Each element of \(\mathbf V\) satisfies \(i\).
Page: 193
Chapter: XI
Section: 3.3
Variables: \(\mathbf V\) a variety
\(i\) an identity
Name2: \(i\) is an identity of \(\mathbf V\)

Smaller0: \(u=v\) is an identity of \(\mathbf V\)
Greater0: \(\pi_{\mathbf V}(u)=\pi_{\mathbf V}(v)\)
Page: 193
Chapter: XI
Section: 3.3
Variables: \(\mathbf V\) a variety
\(u,v\in\widehat{A^*}\)
Index: 11
Kind: Prop
Smaller to greater: iff

Smaller0: both varieties satisfy the same identities on the alphabet \(A\)
Greater0: \(\widehat F_{\mathbf V}(A)\) and \(\widehat F_{\mathbf W}(A)\) are isomorphic
Page: 197
Chapter: XI
Section: 3.3
Variables: \(\mathbf {V,W}\) varieties
\(A\) an alphabet
Index: 12
Kind: Cor
Smaller to greater: iff

Smaller0: \(\mathbf S\) is a variety
Greater0: \(\mathbf S\) can be defined by a set of profinite identities
Page: 197
Chapter: XI
Section: 3.3
Variables: \(\mathbf S\) a class of [ordered] monoids/semigroups
Index: 13
Kind: Theo
Smaller to greater: iff
Name: Reiterman Theorem

Name: A nilpotent semigroup
Definition: \(S^{n}=0\) for some \(n\in\mathbb N\)
Page: 198
Chapter: XI
Section: 4.1
Index: 14
Kind: Prop
Definition2: \(S\) satisfies the identity \(x_1\dots x_{|S|}=y_1\dots y_{|S|}\)
Conjdef: A semigroup such that
Definition3: for every \(e\in E(S)\) and \(s\in S\), \(es=e=se\)
Definition4: \(E(S)=\{0\}\)
Definition5: \(x^\omega y=x^\omega=yx^\omega\) (assuming finiteness)

Name: The variety of nilpotent semigroup
Page: 198
Chapter: XI
Section: 4.1
Context: semigroup variety
Notation: \(\mathbf N\)

Name: The variety of lefty trivial semigroup
Page: 199
Chapter: XI
Section: 4.1
Notation: \(\ell\mathbf 1\),
Notation2: \(\mathbf K\)

Name: The variety of righty trivial semigroup
Page: 199
Chapter: XI
Section: 4.1
Notation: \(r\mathbf 1\)
Notation2: \(\mathbf D\)

Name: A locally trivial semigroup
Definition: \(ese=e\) for each \(e\in E(S)\) and \(s\in S\)
Page: 200
Chapter: XI
Section: 4.1
Variables: \(S\) a semigroup
Index: 47
Kind: Prop
Definition2: the minimal ideal of \(S\) equal \(E(S)\)
Conjdef: A semigroup \(S\) such that, equivalently
Definition3: for \(e,f\in E(S)\) and \(s\in S\), \(esf=ef\)
Definition4: it satisfies the identity \(x_1\dots x_{|S|}xx_1\dots x_{|S|}=x_1\dots x_{|S|}\)
Definition5: it satisfies \(x^\omega yx^\omega=x^\omega\)

Name: The variety of locally trivial groups
Page: 158
Chapter: VII
Section: 4.1
Notation: \(\mathbb L\mathbf 1\)

Name: The variety of locally trivial semigroup
Page: 156
Chapter: VII
Section: 4.1
Notation: \(\mathbb L\mathbf 1\)
Index: 25
Kind: Prop

Name: A semigroup which is locally a group
Definition: for every \(e\in E(S)\), \(eSe\) is a group
Page: 201
Chapter: XI
Section: 4.1
Index: 18
Kind: Prop
Definition2: every idempotent of \(S\) belongs to the minimal ideal of \(S\),
Conjdef: A semigroup \(S\) such that
Definition3: \(S\) satisfies \((x^\omega yx^\omega)=x^\omega\) (assuming finiteness)

Name: The variety of semigroups which are locally a group
Page: 201
Chapter: XI
Section: 4.1
Notation: \(\mathbb L\mathbf G\)

Name: The variety of simple semigroups.
Page: 201
Chapter: XI
Section: 4.4
Context: variety
Notation: \(\mathbf {CS}\)

Name: The variety of groups.
Page: 202
Chapter: XI
Section: 4.2
Context: variety
Notation: \(\mathbf {G}\)

Name: The variety of commutative monoids.
Page: 202
Chapter: XI
Section: 4.2
Context: variety
Notation: \(\mathbf {Com}\)

Name: The variety of aperiodic monoids.
Page: 202
Chapter: VII, X
Section: 4.2, 2
Context: variety
Notation: \(\mathbf{A}\)
Index: , 1

Name: The variety of \(\mathcal J\)-trivial monoids
Page: 202
Chapter: XI
Section: 4.2
Notation: \(\mathbf{J}\)

Name: The variety of \(\mathcal R\)-trivial monoids
Page: 202
Chapter: XI
Section: 4.2
Notation: \(\mathbf{R}\)

Name: The variety of \(\mathcal L\)-trivial monoids
Page: 202
Chapter: XI
Section: 4.2
Notation: \(\mathbf{L}\)

Name: A \(\mathcal J\)-trivial monoid
Definition: satisfying \(y(xy)^\omega=(xy)^\omega=(xy)^\omega x\) (assuming finiteness)
Page: 202
Chapter: XI
Section: 4.2
Index: 23
Kind: Prop
Definition2: such that for each element \(x,y\), \(x\mathcal Jy\) implies \(x=y\)
Conjdef: A monoid, equivalently
Definition3: satisfying \((xy)^\omega=(yx)^\omega\) and \(x^{\omega+1}=x^\omega\)(assuming finiteness)

Name: A \(\mathcal R\)-trivial monoid
Definition: satisfying \((xy)^\omega x=(xy)^\omega\) (assuming finiteness)
Page: 202
Chapter: XI
Section: 4.2
Index: 23
Kind: Prop
Definition2: such that for each element \(x,y\), \(x\mathcal Ry\) implies \(x=y\)
Conjdef: equivalently
DefType: A monoid

Name: A \(\mathcal L\)-trivial monoid
Definition: satisfying \(y(xy)^\omega=(xy)^\omega\) (assuming finiteness)
Page: 202
Chapter: XI
Section: 4.2
Index: 23
Kind: Prop
Definition2: such that for each element \(x,y\), \(x\mathcal Ly\) implies \(x=y\)
Conjdef: A monoid, equivalently

Name: A \(\mathcal J\)-trivial monoid
Definition: \(y(xy)^\omega x=(xy)^\omega=(xy)^\omega x\) (assuming finiteness)
Page: 153
Chapter: VII
Section: 4.2
Index: 23
Kind: Prop
Definition2: \(x^{\omega+1}=x^\omega\) and \((xy)^\omega=(yx)^\omega\) (assuming finiteness)
Conjdef: A monoid satisfying
Definition3: \(x \mathcal Jy\) implies \(x=y\)

Name: \(M\) satisfies the profinite inequality \(u<v\)
Definition: For each monoid morphism \(\phi:A^*\to M\), \(\widehat\phi(u)<\widehat\phi(v)\).
Page: 195
Chapter: XI
Section: 3.1
Variables: \(A\) an alphabet
\(u,v\in\widehat {A^*}\)
\(M\) a monoid

Name: Righty trivial semigroup
Definition: for every \(e\in E(S)\), \(Se\) is trivial
Page: 199
Chapter: XI
Section: 4.1
Index: 16
Kind: Prop
Definition2: \(e\) is a right zero
Conjdef: A semigroup \(S\) such that, equivalently
Definition3: its minimal ideal is a right zero semigroup equal to \(E(S)\)
Definition4: its minimal ideal is a right zero semigroup equal to \(S^{|S|}\)
Definition5: \(xx_1\dots x_{|S|}=x_1\dots x_{|S|}\)
Definition6: satisfying \(yx^\omega=x ^\omega\)

Name: Lefty trivial semigroup
Definition: for every \(e\in E(S)\), \(eS\) is trivial
Page: 199
Chapter: XI
Section: 4.1
Index: 15
Kind: Prop
Definition2: \(e\) is a left zero
Conjdef: A semigroup \(S\) such that, equivalently
Definition3: its minimal ideal is a left zero semigroup equal to \(E(S)\)
Definition4: its minimal ideal is a left zero semigroup equal to \(S^{|S|}\)
Definition5: \(x_1\dots x_nx=x_1\dots x_n\)
Definition6: satisfying \(x^\omega y=x ^\omega\)

Name: A semilattice
Definition: an idempotent commutative monoid
Page: 203
Chapter: XI
Section: 4
Index: ,25
Kind: ,Prop
Definition2: \(\mathcal J\)-trivial idempotent
Conjdef: A monoid, equivalently

Definition: generated by \(U_1\)
Page: 203
Chapter: XI
Section: 4
Notation: \(\mathbf{J_1}\)
Index: 24
Kind: Prop
Definition2: of semilattices
Conjdef: the variety, equivalently

Smaller0: \(M\) is commutative
Greater0: the product of its monogenic submonoids
Page: 202
Chapter: XI
Section: 4
Variables: \(M\) a monoid
Index: 21
Kind: Prop
Prefix Greater: \(M\) divides
Extra: It seems that the proof does not require commutativity

Definition: of idempotent and \(\mathcal R\)-trivial monoids
Page: 203
Chapter: XI
Section: 4
Context: Variety
Notation: \(\mathrm{R_1}\)
Index: 26
Kind: Prop
Definition2: defined by \(xyx=xy\)
Conjdef: Variety, equivalently

Definition: of idempotent and \(\mathcal L\)-trivial monoids
Page: 203
Chapter: XI
Section: 4
Context: Variety
Notation: \(\mathrm{L_1}\)
Index: 26
Kind: Prop
Definition2: defined by \(xyx=yx\)
Conjdef: Variety, equivalently

Definition: satisfying \(((xy)^\omega(yx)^\omega(xy)^\omega)^\omega=(xy)^\omega\)
Page: 203
Chapter: XI
Section: 4
Context: Variety
Notation: \(\mathbb{D}\mathrm{S}\)
Index: 27
Kind: Prop
Definition2: whose \(\mathcal D\)-classes is a semigroup
Conjdef: The variety of monoid \(M\), equivalently
Definition3: every regular \(\mathcal H\)-class is a group
Definition4: for each \(s\) regular and \(s\le_{\mathcal J}t\), then \(ts\mathcal L s\mathcal Rst\)
Definition5: for each idemptoent \(e\) \(M_e=\{s\in M\mid e\le_{\mathcal J}s\}\) is a subsemigroup
Definition6: for each \(e,f\in E(M)\) such that \(e\mathcal Jf\), \(ef\mathcal Je\)
Definition7: \(B^1_2\) does not divide \(M\times M\)

Name: Basketball
Learn: true
Subpart: A= x2 intro
Ax2 couplet
Subpart5: BB'( )
Subpart3: B=
Subpart4: B'= ( )
Subpart9: C= ( )x3
Subpart10: C( )

Name: In Dreams
Learn: true
Rythm: Slowly and freely, with pedal

Name: La complainte de la serveuse automate
Learn: true
Subpart: A=( ) A( )
Subpart2: B= A( )
Subpart6: Ax2
Subpart3: B
Subpart7: x2

Name: Bach BWV927
Learn: true
Rythm: Allegro non legato