Definition 2.6 .

The cord algebra of K 𝐾 K italic_K is the quotient ring

Cord ⁡ ( K ) = 𝒜 / ℐ , Cord 𝐾 𝒜 ℐ \operatorname{Cord}(K)=\mathcal{A}/\mathcal{I}, roman_Cord ( italic_K ) = caligraphic_A / caligraphic_I ,

where ℐ ℐ \mathcal{I} caligraphic_I is the two-sided ideal of 𝒜 𝒜 \mathcal{A} caligraphic_A generated by the following “skein relations”:

  1. (i)

    = 1 - μ 1 𝜇 \raisebox{-12.9pt}{\includegraphics[height=30.1pt]{figures/skein1}}=1-\mu = 1 - italic_μ

  2. (ii)

    = μ ⋅ ⋅ 𝜇 \raisebox{-12.9pt}{\includegraphics[height=30.1pt]{figures/skein3iwrapped}}=% \mu\cdot\raisebox{-12.9pt}{\includegraphics[height=30.1pt]{figures/skein3i}} = italic_μ ⋅ and = ⋅ μ ⋅ 𝜇 \raisebox{-12.9pt}{\includegraphics[height=30.1pt]{figures/skein3cwrapped}}=% \raisebox{-12.9pt}{\includegraphics[height=30.1pt]{figures/skein3c}}\cdot\mu = ⋅ italic_μ

  3. (iii)

    = λ ⋅ ⋅ 𝜆 \raisebox{-12.9pt}{\includegraphics[height=30.1pt]{figures/skein2a}}=\lambda% \cdot\raisebox{-12.9pt}{\includegraphics[height=30.1pt]{figures/skein2b}} = italic_λ ⋅ and = ⋅ λ ⋅ 𝜆 \raisebox{-12.9pt}{\includegraphics[height=30.1pt]{figures/skein2c}}=\raisebox% {-12.9pt}{\includegraphics[height=30.1pt]{figures/skein2d}}\cdot\lambda = ⋅ italic_λ

  4. (iv)

    - = ⋅ ⋅ \raisebox{-12.9pt}{\includegraphics[height=30.1pt]{figures/skein3a}}-\raisebox% {-12.9pt}{\includegraphics[height=30.1pt]{figures/skein3b}}=\raisebox{-12.9pt}% {\includegraphics[height=30.1pt]{figures/skein3c}}\cdot\raisebox{-12.9pt}{% \includegraphics[height=30.1pt]{figures/skein3d}} - = ⋅ .

Here K 𝐾 K italic_K is depicted in black and K ′ superscript 𝐾 ′ K^{\prime} italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT parallel to K 𝐾 K italic_K in gray, and cords are drawn in red.

Definition 2.10 (Bi-gyration inversion law) .

A bi-gyrogroupoid B 𝐵 B italic_B satisfies the bi-gyration inversion law if

lgyr - 1 ⁢ [ a , b ] = lgyr ⁢ [ b , a ] and rgyr - 1 ⁢ [ a , b ] = rgyr ⁢ [ b , a ] formulae-sequence superscript lgyr 1 𝑎 𝑏 lgyr 𝑏 𝑎 and superscript rgyr 1 𝑎 𝑏 rgyr 𝑏 𝑎 {\mathrm{lgyr^{-1}}[{a,b}]}{}={\mathrm{lgyr}[{b,a}]}{}\quad\textrm{and}\quad{% \mathrm{rgyr^{-1}}[{a,b}]}{}={\mathrm{rgyr}[{b,a}]}{} roman_lgyr start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT [ italic_a , italic_b ] = roman_lgyr [ italic_b , italic_a ] and roman_rgyr start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT [ italic_a , italic_b ] = roman_rgyr [ italic_b , italic_a ]

for all a , b ∈ B 𝑎 𝑏 𝐵 a,b\in B italic_a , italic_b ∈ italic_B .

Definition 4.27 (Gyrocommutative gyrogroup, [ 29 ] ) .

A gyrogroup ( G , ⊕ ) 𝐺 direct-sum (G,\oplus) ( italic_G , ⊕ ) is gyrocommutative if it satisfies the gyrocommutative law

a ⊕ b = gyr ⁢ [ a , b ] ⁢ ( b ⊕ a ) direct-sum 𝑎 𝑏 gyr 𝑎 𝑏 direct-sum 𝑏 𝑎 a\oplus b={\mathrm{gyr}[{a,b}]}{(}{b\oplus a}) italic_a ⊕ italic_b = roman_gyr [ italic_a , italic_b ] ( italic_b ⊕ italic_a )

for all a , b ∈ G 𝑎 𝑏 𝐺 a,b\in G italic_a , italic_b ∈ italic_G .

Definition 3.10

[ 29 ] A crossed module in 𝖢 𝖢 \mathsf{C} sansserif_C is a triple ( A , B , α ) 𝐴 𝐵 𝛼 (A,B,\alpha) ( italic_A , italic_B , italic_α ) , where A 𝐴 A italic_A and B 𝐵 B italic_B are the objects of 𝖢 𝖢 \mathsf{C} sansserif_C , B 𝐵 B italic_B acts on A 𝐴 A italic_A , i.e., we have a derived action in 𝖢 𝖢 \mathsf{C} sansserif_C , and α : A → B normal-: 𝛼 normal-→ 𝐴 𝐵 \alpha\colon A\rightarrow B italic_α : italic_A → italic_B is a morphism in 𝖢 𝖢 \mathsf{C} sansserif_C with the conditions:

  1. CM1.

    α ⁢ ( b ⋅ a ) = b + α ⁢ ( a ) - b 𝛼 ⋅ 𝑏 𝑎 𝑏 𝛼 𝑎 𝑏 \alpha(b\cdot a)=b+\alpha(a)-b italic_α ( italic_b ⋅ italic_a ) = italic_b + italic_α ( italic_a ) - italic_b ;

  2. CM2.

    α ⁢ ( a ) ⋅ a ′ = a + a ′ - a ⋅ 𝛼 𝑎 superscript 𝑎 ′ 𝑎 superscript 𝑎 ′ 𝑎 \alpha(a)\cdot a^{\prime}=a+a^{\prime}-a italic_α ( italic_a ) ⋅ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_a + italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_a ;

  3. CM3.

    α ⁢ ( a ) ⋆ a ′ = a ⋆ a ′ ⋆ 𝛼 𝑎 superscript 𝑎 ′ ⋆ 𝑎 superscript 𝑎 ′ \alpha(a)\star a^{\prime}=a\star a^{\prime} italic_α ( italic_a ) ⋆ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_a ⋆ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ;

  4. CM4.

    α ⁢ ( b ⋆ a ) = b ⋆ α ⁢ ( a ) 𝛼 ⋆ 𝑏 𝑎 ⋆ 𝑏 𝛼 𝑎 \alpha(b\star a)=b\star\alpha(a) italic_α ( italic_b ⋆ italic_a ) = italic_b ⋆ italic_α ( italic_a ) , α ⁢ ( a ⋆ b ) = α ⁢ ( a ) ⋆ b 𝛼 ⋆ 𝑎 𝑏 ⋆ 𝛼 𝑎 𝑏 \alpha(a\star b)=\alpha(a)\star b italic_α ( italic_a ⋆ italic_b ) = italic_α ( italic_a ) ⋆ italic_b

for any b ∈ B 𝑏 𝐵 b\in B italic_b ∈ italic_B , a , a ′ ∈ A 𝑎 superscript 𝑎 normal-′ 𝐴 a,a^{\prime}\in A italic_a , italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A , and ⋆ ∈ Ω 2 ′ fragments normal-⋆ superscript subscript normal-Ω 2 normal-′ \star\in\Omega_{2}^{\prime} ⋆ ∈ roman_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

Definition 2.2 .

Let p : E → B : 𝑝 → 𝐸 𝐵 p:E\to B italic_p : italic_E → italic_B be a map and let α ~ ⁢ a ⁢ n ⁢ d ⁢ β ~ ~ 𝛼 𝑎 𝑛 𝑑 ~ 𝛽 \widetilde{\alpha}\ and\ \widetilde{\beta} ~ start_ARG italic_α end_ARG italic_a italic_n italic_d ~ start_ARG italic_β end_ARG be two arbitrary paths in E. Then we say that
(i) p has unique path lifting property (upl) if

α ~ ⁢ ( 0 ) = β ~ ⁢ ( 0 ) , p ∘ α ~ = p ∘ β ~ ⇒ α ~ = β ~ . formulae-sequence ~ 𝛼 0 ~ 𝛽 0 𝑝 ~ 𝛼 𝑝 ~ 𝛽 ⇒ ~ 𝛼 ~ 𝛽 \widetilde{\alpha}(0)=\widetilde{\beta}(0),\ p\circ\widetilde{\alpha}=p\circ% \widetilde{\beta}\Rightarrow\widetilde{\alpha}=\widetilde{\beta}. ~ start_ARG italic_α end_ARG ( 0 ) = ~ start_ARG italic_β end_ARG ( 0 ) , italic_p ∘ ~ start_ARG italic_α end_ARG = italic_p ∘ ~ start_ARG italic_β end_ARG ⇒ ~ start_ARG italic_α end_ARG = ~ start_ARG italic_β end_ARG .

(ii) p has homotopically unique path lifting property (hupl) if

α ~ ⁢ ( 0 ) = β ~ ⁢ ( 0 ) , p ∘ α ~ = p ∘ β ~ ⇒ α ~ ≃ β ~ ⁢ rel ⁢ I ˙ . formulae-sequence ~ 𝛼 0 ~ 𝛽 0 𝑝 ~ 𝛼 𝑝 ~ 𝛽 ⇒ ~ 𝛼 similar-to-or-equals ~ 𝛽 rel ˙ 𝐼 \widetilde{\alpha}(0)=\widetilde{\beta}(0),\ p\circ\widetilde{\alpha}=p\circ% \widetilde{\beta}\Rightarrow\widetilde{\alpha}\simeq\widetilde{\beta}\ \mathrm% {rel}\ \dot{I}. ~ start_ARG italic_α end_ARG ( 0 ) = ~ start_ARG italic_β end_ARG ( 0 ) , italic_p ∘ ~ start_ARG italic_α end_ARG = italic_p ∘ ~ start_ARG italic_β end_ARG ⇒ ~ start_ARG italic_α end_ARG ≃ ~ start_ARG italic_β end_ARG roman_rel ˙ start_ARG italic_I end_ARG .

(iii) p has weakly homotopically unique path lifting property (whupl) if

α ~ ⁢ ( 0 ) = β ~ ⁢ ( 0 ) , α ~ ⁢ ( 1 ) = β ~ ⁢ ( 1 ) , p ∘ α ~ = p ∘ β ~ ⇒ α ~ ≃ β ~ ⁢ rel ⁢ I ˙ . formulae-sequence ~ 𝛼 0 ~ 𝛽 0 formulae-sequence ~ 𝛼 1 ~ 𝛽 1 𝑝 ~ 𝛼 𝑝 ~ 𝛽 ⇒ ~ 𝛼 similar-to-or-equals ~ 𝛽 rel ˙ 𝐼 \widetilde{\alpha}(0)=\widetilde{\beta}(0),\ \widetilde{\alpha}(1)=\widetilde{% \beta}(1),\ p\circ\widetilde{\alpha}=p\circ\widetilde{\beta}\Rightarrow% \widetilde{\alpha}\simeq\widetilde{\beta}\ \mathrm{rel}\ \dot{I}. ~ start_ARG italic_α end_ARG ( 0 ) = ~ start_ARG italic_β end_ARG ( 0 ) , ~ start_ARG italic_α end_ARG ( 1 ) = ~ start_ARG italic_β end_ARG ( 1 ) , italic_p ∘ ~ start_ARG italic_α end_ARG = italic_p ∘ ~ start_ARG italic_β end_ARG ⇒ ~ start_ARG italic_α end_ARG ≃ ~ start_ARG italic_β end_ARG roman_rel ˙ start_ARG italic_I end_ARG .

(iv) p has unique path homotopically lifting property (uphl) if

α ~ ⁢ ( 0 ) = β ~ ⁢ ( 0 ) , p ∘ α ~ ≃ p ∘ β ~ ⁢ rel ⁢ I ˙ ⇒ α ~ ≃ β ~ ⁢ rel ⁢ I ˙ . formulae-sequence ~ 𝛼 0 ~ 𝛽 0 similar-to-or-equals 𝑝 ~ 𝛼 𝑝 ~ 𝛽 rel ˙ 𝐼 ⇒ ~ 𝛼 similar-to-or-equals ~ 𝛽 rel ˙ 𝐼 \widetilde{\alpha}(0)=\widetilde{\beta}(0),\ p\circ\widetilde{\alpha}\simeq p% \circ\widetilde{\beta}\ \mathrm{rel}\ \dot{I}\Rightarrow\widetilde{\alpha}% \simeq\widetilde{\beta}\ \mathrm{rel}\ \dot{I}. ~ start_ARG italic_α end_ARG ( 0 ) = ~ start_ARG italic_β end_ARG ( 0 ) , italic_p ∘ ~ start_ARG italic_α end_ARG ≃ italic_p ∘ ~ start_ARG italic_β end_ARG roman_rel ˙ start_ARG italic_I end_ARG ⇒ ~ start_ARG italic_α end_ARG ≃ ~ start_ARG italic_β end_ARG roman_rel ˙ start_ARG italic_I end_ARG .

(v) p has weakly unique path homotopically lifting property (wuphl) if

α ~ ⁢ ( 0 ) = β ~ ⁢ ( 0 ) , α ~ ⁢ ( 1 ) = β ~ ⁢ ( 1 ) , p ∘ α ~ ≃ p ∘ β ~ ⁢ rel ⁢ I ˙ ⇒ α ~ ≃ β ~ ⁢ rel ⁢ I ˙ . formulae-sequence ~ 𝛼 0 ~ 𝛽 0 formulae-sequence ~ 𝛼 1 ~ 𝛽 1 similar-to-or-equals 𝑝 ~ 𝛼 𝑝 ~ 𝛽 rel ˙ 𝐼 ⇒ ~ 𝛼 similar-to-or-equals ~ 𝛽 rel ˙ 𝐼 \widetilde{\alpha}(0)=\widetilde{\beta}(0),\ \widetilde{\alpha}(1)=\widetilde{% \beta}(1),\ p\circ\widetilde{\alpha}\simeq p\circ\widetilde{\beta}\ \mathrm{% rel}\ \dot{I}\Rightarrow\widetilde{\alpha}\simeq\widetilde{\beta}\ \mathrm{rel% }\ \dot{I}. ~ start_ARG italic_α end_ARG ( 0 ) = ~ start_ARG italic_β end_ARG ( 0 ) , ~ start_ARG italic_α end_ARG ( 1 ) = ~ start_ARG italic_β end_ARG ( 1 ) , italic_p ∘ ~ start_ARG italic_α end_ARG ≃ italic_p ∘ ~ start_ARG italic_β end_ARG roman_rel ˙ start_ARG italic_I end_ARG ⇒ ~ start_ARG italic_α end_ARG ≃ ~ start_ARG italic_β end_ARG roman_rel ˙ start_ARG italic_I end_ARG .
Definition 2 .

Let 𝛎 ( p ) superscript 𝛎 𝑝 \bm{\mathbf{\nu}}^{(p)} bold_italic_ν start_POSTSUPERSCRIPT ( italic_p ) end_POSTSUPERSCRIPT denote the one-parametric portfolio family that interpolates 𝛎 ( 0 ) superscript 𝛎 0 \bm{\mathbf{\nu}}^{(0)} bold_italic_ν start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT and 𝛎 ( 1 ) superscript 𝛎 1 \bm{\mathbf{\nu}}^{(1)} bold_italic_ν start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT :

𝝂 ( p ) = ( 1 - p ) ⁢ 𝝂 ( 0 ) + p ⁢ 𝝂 ( 1 ) . superscript 𝝂 𝑝 1 𝑝 superscript 𝝂 0 𝑝 superscript 𝝂 1 \bm{\mathbf{\nu}}^{(p)}=\left(1-p\right)\,\bm{\mathbf{\nu}}^{(0)}+p\,\bm{% \mathbf{\nu}}^{(1)}\,. bold_italic_ν start_POSTSUPERSCRIPT ( italic_p ) end_POSTSUPERSCRIPT = ( 1 - italic_p ) bold_italic_ν start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT + italic_p bold_italic_ν start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT . (23)
Definition 2.5.4 .

The commutator

[ a , b ] = a ⁢ b - b ⁢ a 𝑎 𝑏 𝑎 𝑏 𝑏 𝑎 {\color[rgb]{.4,0,.9}[a,b]}=ab-ba [ italic_a , italic_b ] = italic_a italic_b - italic_b italic_a

measures commutativity in D 𝐷 D italic_D -algebra A 1 subscript 𝐴 1 A_{1} italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT . D 𝐷 D italic_D -algebra A 1 subscript 𝐴 1 A_{1} italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is called commutative , if

[ a , b ] = 0 𝑎 𝑏 0 [a,b]=0 [ italic_a , italic_b ] = 0

∎

Definition 2.5.5 .

The associator

(2.5.2) ( a , b , c ) = ( a ⁢ b ) ⁢ c - a ⁢ ( b ⁢ c ) 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 {\color[rgb]{.4,0,.9}(a,b,c)}=(ab)c-a(bc) ( italic_a , italic_b , italic_c ) = ( italic_a italic_b ) italic_c - italic_a ( italic_b italic_c )

measures associativity in D 𝐷 D italic_D -algebra A 1 subscript 𝐴 1 A_{1} italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT . D 𝐷 D italic_D -algebra A 1 subscript 𝐴 1 A_{1} italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is called associative , if

( a , b , c ) = 0 𝑎 𝑏 𝑐 0 (a,b,c)=0 ( italic_a , italic_b , italic_c ) = 0

∎

Definition 3.1.3 .

Let D 𝐷 D italic_D be normed ring. Element a ∈ D 𝑎 𝐷 a\in D italic_a ∈ italic_D is called limit of a sequence { a n } subscript 𝑎 𝑛 \{a_{n}\} { italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }

a = 𝑎 absent a= italic_a =

if for every ϵ ∈ R italic-ϵ 𝑅 \epsilon\in R italic_ϵ ∈ italic_R , ϵ > 0 italic-ϵ 0 \epsilon>0 italic_ϵ > 0 , there exists positive integer n 0 subscript 𝑛 0 n_{0} italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT depending on ϵ italic-ϵ \epsilon italic_ϵ and such, that

| a n - a | < ϵ subscript 𝑎 𝑛 𝑎 italic-ϵ |a_{n}-a|<\epsilon | italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_a | < italic_ϵ

for every n > n 0 𝑛 subscript 𝑛 0 n>n_{0} italic_n > italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT . ∎

Definition 4.4 .

Let 𝐏𝐕 2 , 1 ⁢ ( ℂ n + 1 ) subscript 𝐏𝐕 2 1 superscript ℂ 𝑛 1 {{\mathbf{PV}}_{2,1}({\mathbb{C}}^{n+1})} bold_PV start_POSTSUBSCRIPT 2 , 1 end_POSTSUBSCRIPT ( blackboard_C start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT ) denote 𝐏𝐕 2 ⁢ ( ℂ n + 1 ) subscript 𝐏𝐕 2 superscript ℂ 𝑛 1 {{\mathbf{PV}}_{2}({\mathbb{C}}^{n+1})} bold_PV start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_C start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT ) equipped with the left 𝕋 𝕋 {\mathbb{T}} blackboard_T -action

z * [ u , v ] = [ ( z ) - 1 ⁢ u , z ⁢ v ] , 𝑧 𝑢 𝑣 superscript 𝑧 1 𝑢 𝑧 𝑣 z*[u,v]=[(\sqrt{z})^{-1}u,\sqrt{z}v], italic_z * [ italic_u , italic_v ] = [ ( square-root start_ARG italic_z end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_u , square-root start_ARG italic_z end_ARG italic_v ] ,

where z 𝑧 \sqrt{z} square-root start_ARG italic_z end_ARG is a square root of z ∈ 𝕋 ⊆ ℂ 𝑧 𝕋 ℂ z\in{\mathbb{T}}\subseteq{\mathbb{C}} italic_z ∈ blackboard_T ⊆ blackboard_C . Note that the action is well-defined. We write 𝐏𝐕 2 , q ⁢ ( ℂ n + 1 ) subscript 𝐏𝐕 2 𝑞 superscript ℂ 𝑛 1 {{\mathbf{PV}}_{2,q}({\mathbb{C}}^{n+1})} bold_PV start_POSTSUBSCRIPT 2 , italic_q end_POSTSUBSCRIPT ( blackboard_C start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT ) for the associated 𝕋 𝕋 {\mathbb{T}} blackboard_T -space, where 𝕋 𝕋 {\mathbb{T}} blackboard_T acts via z ↦ z q maps-to 𝑧 superscript 𝑧 𝑞 z\mapsto z^{q} italic_z ↦ italic_z start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT .

Definition 4.5 .

A crossed module in 𝖳𝖢 𝖳𝖢 \mathsf{TC} sansserif_TC is a morphism α : A → B normal-: 𝛼 normal-→ 𝐴 𝐵 \alpha\colon A\rightarrow B italic_α : italic_A → italic_B in 𝖳𝖢 𝖳𝖢 \mathsf{TC} sansserif_TC , where B 𝐵 B italic_B acts topologically on A 𝐴 A italic_A (i.e. we have a continuous derived action in 𝖳𝖢 𝖳𝖢 \mathsf{TC} sansserif_TC ) with the conditions for any b ∈ B 𝑏 𝐵 b\in B italic_b ∈ italic_B , a , a ′ ∈ A 𝑎 superscript 𝑎 normal-′ 𝐴 a,a^{\prime}\in A italic_a , italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A , and ⋆ ∈ Ω 2 ′ fragments normal-⋆ superscript subscript normal-Ω 2 normal-′ \star\in\Omega_{2}^{\prime} ⋆ ∈ roman_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT :

  1. CM1

    α ⁢ ( b ⋅ a ) = b + α ⁢ ( a ) - b 𝛼 ⋅ 𝑏 𝑎 𝑏 𝛼 𝑎 𝑏 \alpha(b\cdot a)=b+\alpha(a)-b italic_α ( italic_b ⋅ italic_a ) = italic_b + italic_α ( italic_a ) - italic_b ;

  2. CM2

    α ⁢ ( a ) ⋅ a ′ = a + a ′ - a ⋅ 𝛼 𝑎 superscript 𝑎 ′ 𝑎 superscript 𝑎 ′ 𝑎 \alpha(a)\cdot a^{\prime}=a+a^{\prime}-a italic_α ( italic_a ) ⋅ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_a + italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_a ;

  3. CM3

    α ⁢ ( a ) ⋆ a ′ = a ⋆ a ′ ⋆ 𝛼 𝑎 superscript 𝑎 ′ ⋆ 𝑎 superscript 𝑎 ′ \alpha(a)\star a^{\prime}=a\star a^{\prime} italic_α ( italic_a ) ⋆ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_a ⋆ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ;

  4. CM4

    α ⁢ ( b ⋆ a ) = b ⋆ α ⁢ ( a ) 𝛼 ⋆ 𝑏 𝑎 ⋆ 𝑏 𝛼 𝑎 \alpha(b\star a)=b\star\alpha(a) italic_α ( italic_b ⋆ italic_a ) = italic_b ⋆ italic_α ( italic_a ) and α ⁢ ( a ⋆ b ) = α ⁢ ( a ) ⋆ b 𝛼 ⋆ 𝑎 𝑏 ⋆ 𝛼 𝑎 𝑏 \alpha(a\star b)=\alpha(a)\star b italic_α ( italic_a ⋆ italic_b ) = italic_α ( italic_a ) ⋆ italic_b .

□ □ \Box □

Definition 2.15 .

Consider a subanalytic set X 𝑋 X italic_X . Call f : X → ℂ : 𝑓 → 𝑋 ℂ f:X\to\mathbb{C} italic_f : italic_X → blackboard_C a generator for 𝒞 exp ⁢ ( X ) superscript 𝒞 normal-X \mathcal{C}^{\exp}(X) caligraphic_C start_POSTSUPERSCRIPT roman_exp end_POSTSUPERSCRIPT ( italic_X ) if f 𝑓 f italic_f is of the form

(9) f ⁢ ( x ) = g ⁢ ( x ) ⁢ e i ⁢ ϕ ⁢ ( x ) ⁢ γ ⁢ ( x ) , 𝑓 𝑥 𝑔 𝑥 superscript e i italic-ϕ 𝑥 𝛾 𝑥 f(x)=g(x)\text{e}^{\text{i}\phi(x)}\gamma(x), italic_f ( italic_x ) = italic_g ( italic_x ) e start_POSTSUPERSCRIPT i italic_ϕ ( italic_x ) end_POSTSUPERSCRIPT italic_γ ( italic_x ) ,

where g ∈ 𝒞 ⁢ ( X ) 𝑔 𝒞 𝑋 g\in\mathcal{C}(X) italic_g ∈ caligraphic_C ( italic_X ) , ϕ ∈ 𝒮 ⁢ ( X ) italic-ϕ 𝒮 𝑋 \phi\in\mathcal{S}(X) italic_ϕ ∈ caligraphic_S ( italic_X ) , and γ = γ h , ℓ 𝛾 subscript 𝛾 ℎ ℓ \gamma=\gamma_{h,\ell} italic_γ = italic_γ start_POSTSUBSCRIPT italic_h , roman_ℓ end_POSTSUBSCRIPT for some ℓ ∈ ℕ ℓ ℕ \ell\in\mathbb{N} roman_ℓ ∈ blackboard_N and h ∈ 𝒮 ⁢ ( X × ℝ ) ℎ 𝒮 𝑋 ℝ h\in\mathcal{S}(X\times\mathbb{R}) italic_h ∈ caligraphic_S ( italic_X × blackboard_R ) with t ↦ h ⁢ ( x , t ) maps-to 𝑡 ℎ 𝑥 𝑡 t\mapsto h(x,t) italic_t ↦ italic_h ( italic_x , italic_t ) in L 1 ⁢ ( ℝ ) superscript 𝐿 1 ℝ L^{1}(\mathbb{R}) italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_R ) . When γ = 1 𝛾 1 \gamma=1 italic_γ = 1 , we shall also call f 𝑓 f italic_f a generator for 𝒞 naive exp ⁢ ( X ) superscript subscript 𝒞 normal-naive normal-X \mathcal{C}_{\operatorname{naive}}^{\exp}(X) caligraphic_C start_POSTSUBSCRIPT roman_naive end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_exp end_POSTSUPERSCRIPT ( italic_X ) . Note that a function is in 𝒞 exp ⁢ ( X ) superscript 𝒞 𝑋 \mathcal{C}^{\exp}(X) caligraphic_C start_POSTSUPERSCRIPT roman_exp end_POSTSUPERSCRIPT ( italic_X ) if and only if the function can be expressed as a finite sum of generators for 𝒞 exp ⁢ ( X ) superscript 𝒞 𝑋 \mathcal{C}^{\exp}(X) caligraphic_C start_POSTSUPERSCRIPT roman_exp end_POSTSUPERSCRIPT ( italic_X ) , and likewise for 𝒞 naive exp ⁢ ( X ) superscript subscript 𝒞 naive 𝑋 \mathcal{C}_{\operatorname{naive}}^{\exp}(X) caligraphic_C start_POSTSUBSCRIPT roman_naive end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_exp end_POSTSUPERSCRIPT ( italic_X ) .

Definition 1 .

[ 18 ] Let ( Q ; ⋅ ) 𝑄 ⋅ (Q;\cdot) ( italic_Q ; ⋅ ) be a group isotope and 0 0 be an arbitrary element of Q 𝑄 Q italic_Q , then the right part of the formula

x ⋅ y = α ⁢ x + a + β ⁢ y ⋅ 𝑥 𝑦 𝛼 𝑥 𝑎 𝛽 𝑦 x\cdot y=\alpha x+a+\beta y italic_x ⋅ italic_y = italic_α italic_x + italic_a + italic_β italic_y (1)

is called a 0 0 -canonical decomposition , if ( Q ; + ) 𝑄 (Q;+) ( italic_Q ; + ) is a group, 0 0 is its neutral element and α 𝛼 \alpha italic_α , β 𝛽 \beta italic_β are unitary permutations of ( Q ; + ) 𝑄 (Q;+) ( italic_Q ; + ) .

Definition 11 .

A Bohrium array operation, f 𝑓 f italic_f , is data parallel, i.e., each output element can be calculated independently, when the following holds:

∀ i ∈ 𝗂𝗇 ⁢ [ f ] , ∀ o , o ′ ∈ 𝗈𝗎𝗍 ⁢ [ f ] : ( i ∩ o = ∅ ∨ i = o ) ∧ ( o ∩ o ′ = ∅ ∨ o = o ′ ) fragments for-all 𝑖 𝗂𝗇 fragments [ 𝑓 ] , for-all 𝑜 , superscript 𝑜 ′ 𝗈𝗎𝗍 fragments [ 𝑓 ] : fragments ( 𝑖 𝑜 𝑖 𝑜 ) fragments ( 𝑜 superscript 𝑜 ′ 𝑜 superscript 𝑜 ′ ) \displaystyle\forall{i\in\mathsf{in}[f]},\forall{o,o^{\prime}\in\mathsf{out}[f% ]}:\\ \displaystyle(i\cap o=\emptyset\lor i=o)\land(o\cap o^{\prime}=\emptyset\lor o% =o^{\prime}) start_ROW start_CELL ∀ italic_i ∈ sansserif_in [ italic_f ] , ∀ italic_o , italic_o start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ sansserif_out [ italic_f ] : end_CELL end_ROW start_ROW start_CELL ( italic_i ∩ italic_o = ∅ ∨ italic_i = italic_o ) ∧ ( italic_o ∩ italic_o start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∅ ∨ italic_o = italic_o start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_CELL end_ROW (2)

In other words, if an input and an output or two output arrays overlaps, they must be identical.

Definition 1.4 .

A borelian probability measure μ 𝜇 \mu italic_μ is said to be absolutely continuous with respect to the Lebesgue measure λ 𝜆 \lambda italic_λ (resp. equivalent to λ 𝜆 \lambda italic_λ ) if for any borelian A ⊂ ℝ 𝐴 ℝ A\subset\mathbb{R} italic_A ⊂ blackboard_R ,

λ ( A ) = 0 ⇒ μ ( A ) = 0 , ( resp. λ ( A ) = 0 ⇔ μ ( A ) = 0 ) . fragments λ fragments ( A ) 0 ⇒ μ fragments ( A ) 0 , fragments ( resp. λ fragments ( A ) 0 ⇔ μ fragments ( A ) 0 ) . \lambda(A)=0\Rightarrow\mu(A)=0,\quad(\text{resp. }\lambda(A)=0\Leftrightarrow% \mu(A)=0). italic_λ ( italic_A ) = 0 ⇒ italic_μ ( italic_A ) = 0 , ( resp. italic_λ ( italic_A ) = 0 ⇔ italic_μ ( italic_A ) = 0 ) .
Definition 5.1 .

A crossed module in 𝖢 𝖢 \mathsf{C} sansserif_C is α : A → B : 𝛼 → 𝐴 𝐵 \alpha\colon A\rightarrow B italic_α : italic_A → italic_B is a morphism in 𝖢 𝖢 \mathsf{C} sansserif_C , where B 𝐵 B italic_B acts on A 𝐴 A italic_A (i.e. we have a derived action in 𝖢 𝖢 \mathsf{C} sansserif_C ) with the conditions for any b ∈ B 𝑏 𝐵 b\in B italic_b ∈ italic_B , a , a ′ ∈ A 𝑎 superscript 𝑎 ′ 𝐴 a,a^{\prime}\in A italic_a , italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A , and ⋆ ∈ Ω 2 ′ fragments ⋆ superscript subscript Ω 2 ′ \star\in\Omega_{2}^{\prime} ⋆ ∈ roman_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT :

    1. CM1

      α ⁢ ( b ⋅ a ) = b + α ⁢ ( a ) - b 𝛼 ⋅ 𝑏 𝑎 𝑏 𝛼 𝑎 𝑏 \alpha(b\cdot a)=b+\alpha(a)-b italic_α ( italic_b ⋅ italic_a ) = italic_b + italic_α ( italic_a ) - italic_b ;

    2. CM2

      α ⁢ ( a ) ⋅ a ′ = a + a ′ - a ⋅ 𝛼 𝑎 superscript 𝑎 ′ 𝑎 superscript 𝑎 ′ 𝑎 \alpha(a)\cdot a^{\prime}=a+a^{\prime}-a italic_α ( italic_a ) ⋅ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_a + italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_a ;

    3. CM3

      α ⁢ ( a ) ⋆ a ′ = a ⋆ a ′ ⋆ 𝛼 𝑎 superscript 𝑎 ′ ⋆ 𝑎 superscript 𝑎 ′ \alpha(a)\star a^{\prime}=a\star a^{\prime} italic_α ( italic_a ) ⋆ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_a ⋆ italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ;

    4. CM4

      α ⁢ ( b ⋆ a ) = b ⋆ α ⁢ ( a ) 𝛼 ⋆ 𝑏 𝑎 ⋆ 𝑏 𝛼 𝑎 \alpha(b\star a)=b\star\alpha(a) italic_α ( italic_b ⋆ italic_a ) = italic_b ⋆ italic_α ( italic_a ) and α ⁢ ( a ⋆ b ) = α ⁢ ( a ) ⋆ b 𝛼 ⋆ 𝑎 𝑏 ⋆ 𝛼 𝑎 𝑏 \alpha(a\star b)=\alpha(a)\star b italic_α ( italic_a ⋆ italic_b ) = italic_α ( italic_a ) ⋆ italic_b .

Definition 1 .

A bilinear map { . , . } : 𝒞 ( ℳ ) × 𝒞 ( ℳ ) → 𝒞 ( ℳ ) fragments fragments { . , . } : C fragments ( M ) C fragments ( M ) → C fragments ( M ) \{.,.\}:\ \mathscr{C}(\mathscr{M})\times\mathscr{C}(\mathscr{M})\to\mathscr{C}% (\mathscr{M}) { . , . } : script_C ( script_M ) × script_C ( script_M ) → script_C ( script_M ) which satisfies

for all f , g , h ∈ 𝒞 ⁢ ( ℳ ) 𝑓 𝑔 ℎ 𝒞 ℳ f,g,h\ \in\mathscr{C}(\mathscr{M}) italic_f , italic_g , italic_h ∈ script_C ( script_M ) is called Poisson structure or Poisson bracket on ℳ ℳ \mathscr{M} script_M . A manifold ℳ ℳ \mathscr{M} script_M equipped with a Poisson structure is called a Poisson manifold ( ℳ , { . , . } ) fragments normal-( M normal-, fragments normal-{ normal-. normal-, normal-. normal-} normal-) (\mathscr{M},\{.,.\}) ( script_M , { . , . } ) .

Definition 5.8 .

Let ( 𝔤 , [ ⋅ , ⋅ , ⋅ ] ) 𝔤 normal-⋅ normal-⋅ normal-⋅ (\mathfrak{g},[\cdot,\cdot,\cdot]) ( fraktur_g , [ ⋅ , ⋅ , ⋅ ] ) be a 3 3 3 3 -Lie algebra. A symplectic structure on ( 𝔤 , [ ⋅ , ⋅ , ⋅ ] ) 𝔤 normal-⋅ normal-⋅ normal-⋅ (\mathfrak{g},[\cdot,\cdot,\cdot]) ( fraktur_g , [ ⋅ , ⋅ , ⋅ ] ) is a nondegenerate skew-symmetric bilinear form ω : ∧ 2 𝔤 ⟶ ℝ normal-: 𝜔 normal-⟶ superscript 2 𝔤 ℝ \omega:\wedge^{2}\mathfrak{g}\longrightarrow\mathbb{R} italic_ω : ∧ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT fraktur_g ⟶ blackboard_R , such that for all x , y , z , w ∈ 𝔤 𝑥 𝑦 𝑧 𝑤 𝔤 x,y,z,w\in\mathfrak{g} italic_x , italic_y , italic_z , italic_w ∈ fraktur_g , the following identity hold:

ω ⁢ ( [ x , y , z ] , w ) - ω ⁢ ( [ x , y , w ] , z ) + ω ⁢ ( [ x , z , w ] , y ) - ω ⁢ ( [ y , z , w ] , x ) = 0 . 𝜔 𝑥 𝑦 𝑧 𝑤 𝜔 𝑥 𝑦 𝑤 𝑧 𝜔 𝑥 𝑧 𝑤 𝑦 𝜔 𝑦 𝑧 𝑤 𝑥 0 \omega([x,y,z],w)-\omega([x,y,w],z)+\omega([x,z,w],y)-\omega([y,z,w],x)=0. italic_ω ( [ italic_x , italic_y , italic_z ] , italic_w ) - italic_ω ( [ italic_x , italic_y , italic_w ] , italic_z ) + italic_ω ( [ italic_x , italic_z , italic_w ] , italic_y ) - italic_ω ( [ italic_y , italic_z , italic_w ] , italic_x ) = 0 . (24)
Definition 13 .

Let g ∈ G 𝑔 𝐺 g\in G italic_g ∈ italic_G be a permutation, and let 𝐜 ⁢ ( g ) 𝐜 𝑔 \boldsymbol{c}(g) bold_italic_c ( italic_g ) be the signature of g 𝑔 g italic_g . Then the cycle index of g 𝑔 g italic_g is the formal monomial

z ⁢ ( g ; 𝒔 ) = 𝒔 𝒄 ⁢ ( g ) . 𝑧 𝑔 𝒔 superscript 𝒔 𝒄 𝑔 z(g;\boldsymbol{s})=\boldsymbol{s}^{\boldsymbol{c}(g)}. italic_z ( italic_g ; bold_italic_s ) = bold_italic_s start_POSTSUPERSCRIPT bold_italic_c ( italic_g ) end_POSTSUPERSCRIPT .

The cycle index of the whole permutation group G 𝐺 G italic_G is the terminating formal power series

Z ⁢ ( G ; 𝒔 ) = 1 | G | ⁢ ∑ g ∈ G z ⁢ ( g ; 𝒔 ) . 𝑍 𝐺 𝒔 1 𝐺 subscript 𝑔 𝐺 𝑧 𝑔 𝒔 Z(G;\boldsymbol{s})=\frac{1}{|G|}\sum_{g\in G}z(g;\boldsymbol{s}). italic_Z ( italic_G ; bold_italic_s ) = divide start_ARG 1 end_ARG start_ARG | italic_G | end_ARG ∑ start_POSTSUBSCRIPT italic_g ∈ italic_G end_POSTSUBSCRIPT italic_z ( italic_g ; bold_italic_s ) .
Definition 2 (discrete symmetric channel with q 𝑞 q italic_q -ary inputs) .

A DMC with q 𝑞 q italic_q -ary inputs is said to be symmetric if and only if for any α 𝛼 \alpha italic_α in 𝔽 q subscript 𝔽 𝑞 \mathbb{F}_{q} blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT we have

p ( α ) 𝗉𝗋𝗈𝖻 ( π = 𝐩 ) = p ( 0 ) 𝗉𝗋𝗈𝖻 ( π = 𝐩 + α ) . fragments p fragments ( α ) 𝗉𝗋𝗈𝖻 fragments ( π p ) p fragments ( 0 ) 𝗉𝗋𝗈𝖻 fragments ( π superscript 𝐩 𝛼 ) . p(\alpha)\textsf{prob}(\pi=\mathbf{p})=p(0)\textsf{prob}(\pi=\mathbf{p}^{+% \alpha}). italic_p ( italic_α ) prob ( italic_π = bold_p ) = italic_p ( 0 ) prob ( italic_π = bold_p start_POSTSUPERSCRIPT + italic_α end_POSTSUPERSCRIPT ) . (1)
Definition 2.3 .

Let M 𝑀 M italic_M be a von Neumann algebra and let A ⊂ 1 A ⁢ M ⁢ 1 A 𝐴 subscript 1 𝐴 𝑀 subscript 1 𝐴 A\subset 1_{A}M1_{A} italic_A ⊂ 1 start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_M 1 start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and B ⊂ 1 B ⁢ M ⁢ 1 B 𝐵 subscript 1 𝐵 𝑀 subscript 1 𝐵 B\subset 1_{B}M1_{B} italic_B ⊂ 1 start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_M 1 start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT be two subalgebras with expectations. We say that A 𝐴 A italic_A and B 𝐵 B italic_B are semi-conjugated by a partial isometry v ∈ 1 B ⁢ M ⁢ 1 A 𝑣 subscript 1 𝐵 𝑀 subscript 1 𝐴 v\in 1_{B}M1_{A} italic_v ∈ 1 start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_M 1 start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT if

∀ x ∈ A , v ⁢ x = 0 ⇒ x = 0 formulae-sequence for-all 𝑥 𝐴 𝑣 𝑥 0 ⇒ 𝑥 0 \forall x\in A,\;vx=0\Rightarrow x=0 ∀ italic_x ∈ italic_A , italic_v italic_x = 0 ⇒ italic_x = 0

and there exists an (onto) *-isomorphism ψ : A → B : 𝜓 → 𝐴 𝐵 \psi:A\rightarrow B italic_ψ : italic_A → italic_B such that

∀ x ∈ A , ψ ⁢ ( x ) ⁢ v = v ⁢ x . formulae-sequence for-all 𝑥 𝐴 𝜓 𝑥 𝑣 𝑣 𝑥 \forall x\in A,\;\psi(x)v=vx. ∀ italic_x ∈ italic_A , italic_ψ ( italic_x ) italic_v = italic_v italic_x .

We denote this relation by A ∼ v B subscript similar-to 𝑣 𝐴 𝐵 A\sim_{v}B italic_A ∼ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT italic_B . We will also write A ∼ M B subscript similar-to 𝑀 𝐴 𝐵 A\sim_{M}B italic_A ∼ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT italic_B if there exists a partial isometry v ∈ M 𝑣 𝑀 v\in M italic_v ∈ italic_M such that A ∼ v B subscript similar-to 𝑣 𝐴 𝐵 A\sim_{v}B italic_A ∼ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT italic_B .

Definition 1 .

A faithful action of a group G 𝐺 G italic_G on the set X * ∪ X ω superscript 𝑋 superscript 𝑋 𝜔 X^{*}\cup X^{\omega} italic_X start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ∪ italic_X start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT is called self-similar if for every g ∈ G 𝑔 𝐺 g\in G italic_g ∈ italic_G and x ∈ X 𝑥 𝑋 x\in X italic_x ∈ italic_X there exist h ∈ G ℎ 𝐺 h\in G italic_h ∈ italic_G and y ∈ X 𝑦 𝑋 y\in X italic_y ∈ italic_X such that

g ⁢ ( x ⁢ w ) = y ⁢ h ⁢ ( w ) 𝑔 𝑥 𝑤 𝑦 ℎ 𝑤 g(xw)=yh(w) italic_g ( italic_x italic_w ) = italic_y italic_h ( italic_w )

for all w ∈ X * ∪ X ω 𝑤 superscript 𝑋 superscript 𝑋 𝜔 w\in X^{*}\cup X^{\omega} italic_w ∈ italic_X start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ∪ italic_X start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT . The element h ℎ h italic_h is called the restriction of g 𝑔 g italic_g at x 𝑥 x italic_x and is denoted by h = g | x ℎ evaluated-at 𝑔 𝑥 h=g|_{x} italic_h = italic_g | start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT .

Definition 5 (Quasi-shuffle product)

For a given alphabet 𝔸 𝔸 \mathbb{A} blackboard_A , suppose [ ⋅ , ⋅ ] : ℝ ⁢ 𝔸 ⊗ ℝ ⁢ 𝔸 → ℝ ⁢ 𝔸 : ⋅ ⋅ → tensor-product ℝ 𝔸 ℝ 𝔸 ℝ 𝔸 [\,\cdot\,,\,\cdot\,]\colon\mathbb{RA}\otimes\mathbb{RA}\rightarrow\mathbb{RA} [ ⋅ , ⋅ ] : blackboard_R blackboard_A ⊗ blackboard_R blackboard_A → blackboard_R blackboard_A is a commutative, associative product on ℝ ⁢ 𝔸 ℝ 𝔸 \mathbb{RA} blackboard_R blackboard_A . The quasi-shuffle product on ℝ ⁢ ⟨ 𝔸 ⟩ ℝ delimited-⟨⟩ 𝔸 \mathbb{R}\langle\mathbb{A}\rangle blackboard_R ⟨ blackboard_A ⟩ , which is commutative, is generated inductively as follows: if 1 1 \mathbbold{1} 1 is the empty word then u * 1 = 1 * u = u 𝑢 1 1 𝑢 𝑢 u*\mathbbold{1}=\mathbbold{1}*u=u italic_u * 1 = 1 * italic_u = italic_u and

u ⁢ a * v ⁢ b = ( u * v ⁢ b ) ⁢ a + ( u ⁢ a * v ) ⁢ b + ( u * v ) ⁢ [ a , b ] , 𝑢 𝑎 𝑣 𝑏 𝑢 𝑣 𝑏 𝑎 𝑢 𝑎 𝑣 𝑏 𝑢 𝑣 𝑎 𝑏 ua*vb=(u*vb)a+(ua*v)b+(u*v)[a,b], italic_u italic_a * italic_v italic_b = ( italic_u * italic_v italic_b ) italic_a + ( italic_u italic_a * italic_v ) italic_b + ( italic_u * italic_v ) [ italic_a , italic_b ] ,

for all words u , v ∈ 𝔸 * 𝑢 𝑣 superscript 𝔸 u,v\in\mathbb{A}^{*} italic_u , italic_v ∈ blackboard_A start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT and letters a , b ∈ 𝔸 𝑎 𝑏 𝔸 a,b\in\mathbb{A} italic_a , italic_b ∈ blackboard_A . Here u ⁢ a 𝑢 𝑎 ua italic_u italic_a denotes the concatenation of u 𝑢 u italic_u and a 𝑎 a italic_a .

Definition 2.3 .

A homomorphism of Hom-Leibniz algebras f : ( L , α L ) → ( L ′ , α L ′ ) normal-: 𝑓 normal-→ 𝐿 subscript 𝛼 𝐿 superscript 𝐿 normal-′ subscript 𝛼 superscript 𝐿 normal-′ f:(L,\alpha_{L})\to(L^{\prime},\alpha_{L^{\prime}}) italic_f : ( italic_L , italic_α start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) → ( italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_α start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) is a linear map f : L → L ′ normal-: 𝑓 normal-→ 𝐿 superscript 𝐿 normal-′ f:L\to L^{\prime} italic_f : italic_L → italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT such that

f ⁢ ( [ x , y ] ) = [ f ⁢ ( x ) , f ⁢ ( y ) ] , 𝑓 𝑥 𝑦 𝑓 𝑥 𝑓 𝑦 \displaystyle f([x,y])=[f(x),f(y)], italic_f ( [ italic_x , italic_y ] ) = [ italic_f ( italic_x ) , italic_f ( italic_y ) ] ,
f ∘ α L = α L ′ ∘ f , 𝑓 subscript 𝛼 𝐿 subscript 𝛼 superscript 𝐿 ′ 𝑓 \displaystyle f\circ\alpha_{L}=\alpha_{L^{\prime}}\circ f, italic_f ∘ italic_α start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = italic_α start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∘ italic_f ,

for all x , y ∈ L 𝑥 𝑦 𝐿 x,y\in L italic_x , italic_y ∈ italic_L .

Definition 6.1 .

Let R 𝑅 R italic_R be a commutative ring with unit. We say R 𝑅 R italic_R is absolutely flat (also known as von Neumann regular ) if for every r ∈ R 𝑟 𝑅 r\in R italic_r ∈ italic_R there exists some x ∈ R 𝑥 𝑅 x\in R italic_x ∈ italic_R satisfying

r = r 2 ⁢ x . 𝑟 superscript 𝑟 2 𝑥 r=r^{2}x. italic_r = italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x .