A right moving frame is a $G$ -equivariant map $\rho\colon\mathcal{M}\to G$ . The $G$ -equivariance means that

$$\rho(g\cdot z)=\rho(z)g^{-1}.$$

(Reversal convention applies) An edge-weight assignment $\overrightarrow{w}$ on a tournament $\overrightarrow{\mathcal{H}}=(V,E)$ satisfies exact quantitative transitivity if

$$\overrightarrow{w}(x,y)+\overrightarrow{w}(y,z)=\overrightarrow{w}(x,z)$$

(7)

holds for every three distinct vertices $x,y,z\in V$ .

A totally associative $3$ -ary algebra is a $\mathcal{K}$ -vector space $\mathcal{T}$ equipped with a trilinear operation $\mu$ satisfying

$\displaystyle\mu\circ(\mu\otimes\operatorname{id}\otimes\operatorname{id})=\mu\circ(\operatorname{id}\otimes\mu\otimes\operatorname{id})=\mu\circ(\operatorname{id}\otimes\operatorname{id}\otimes\mu).$

(2.1)

A weak totally associative $3$ -ary algebra is a $\mathcal{K}$ -vector space $\mathcal{W}$ with a trilinear operation $\mu$ satisfying

$\displaystyle\mu\circ(\mu\otimes\operatorname{id}\otimes\operatorname{id})=\mu\circ(\operatorname{id}\otimes\operatorname{id}\otimes\mu).$

(2.2)

A partially associative $3$ -ary algebra is a $\mathcal{K}$ -vector space $\mathcal{P}$ endowed with a trilinear operation $\mu$ satisfying

$\displaystyle\mu\circ(\mu\otimes\operatorname{id}\otimes\operatorname{id}+\operatorname{id}\otimes\mu\otimes\operatorname{id}+\operatorname{id}\otimes\operatorname{id}\otimes\mu)=0.$

(2.3)

A commutative monoid is a set $M$ with a distinguished element $e$ and a binary operation on $M$ denoted by $\cdot$ such that for all $x,y,z\in M$ :

• •

  $x\cdot(y\cdot z)=(x\cdot y)\cdot z$ (associativity);

• •

  $x\cdot y=y\cdot x$ (commutativity);

• •

  $x\cdot e=e\cdot x=x$ (identity).

Let $H$ be a left quaternion module, that is, $H$ consists of an abelian group with a left scalar multiplication $(q,u)\mapsto qu$ from $\mathbb{H}\times H$ into $H$ , such that for all $u,v\in H$ and $p,q\in\mathbb{H}$

$$(p+q)u=pu+qu,~{}~{}p(u+v)=pu+qu,~{}~{}(pq)u=p(qu).$$

An element $\omega\in{\cal F}({\mathbb{R}}^{2})$ is $\Gamma_{\eta}$ -equivariant if, for all $\gamma\in\Gamma$ ,

$$\omega(\gamma\cdot(x,y,dx,dy))\ =\ \eta(\gamma)\omega(x,y,dx,dy).$$

(2.2)

A modified double Poisson bracket on $\mathcal{A}$ is a map $\mathcal{A}\otimes\mathcal{A}\rightarrow\mathcal{A}\otimes\mathcal{A}$ s.t. for all $a,b,c\in\mathcal{A}$

(1a)		$\displaystyle\mathopen{\{\!\!\{}a\otimes bc\mathopen{\}\!\!\}}=(b\otimes 1)\mathopen{\{\!\!\{}a\otimes c\mathopen{\}\!\!\}}+\mathopen{\{\!\!\{}a\otimes b\mathopen{\}\!\!\}}(1\otimes c)$
(1b)		$\displaystyle\mathopen{\{\!\!\{}ab\otimes c\mathopen{\}\!\!\}}=(1\otimes a)\mathopen{\{\!\!\{}b\otimes c\mathopen{\}\!\!\}}+\mathopen{\{\!\!\{}a\otimes c\mathopen{\}\!\!\}}(b\otimes 1)$
(1c)		$\displaystyle\{a\otimes\{b\otimes c\}\}-\{b\otimes\{a\otimes c\}\}=\{\{a\otimes b\}\otimes c\}\quad\mathrm{where}\quad\{\_\}:=\mu\circ\mathopen{\{\!\!\{}\_\mathopen{\}\!\!\}}$
(1d)		$\displaystyle\{a,b\}+\{b,a\}=0\,\bmod[\mathcal{A},\mathcal{A}]$

For a finitely generated associative algebra $\mathcal{A}$ , let $\delta:\mathcal{A}\rightarrow\mathcal{A}\otimes\mathcal{A}$ be a linear map satisfying the following form of Leibnitz identity

$\displaystyle\delta(ab)=(a\otimes 1)\,\delta(b)+\delta(a)\,(1\otimes b).$

Then we call $\delta$ a noncommutative vector field . The space of all noncommutative vector fields for a given algebra $\mathcal{A}$ we denote by $\mathcal{D}_{\mathcal{A}}$ in what follows.

Let $G$ be an abelian group. A map $\varepsilon:G\times G\rightarrow{\bf\mathbb{K}^{*}}$ is called a skew-symmetric bicharacter on $G$ if the following identities hold,

1. (i)

   $\varepsilon(a,b)\varepsilon(b,a)=1$ ,

2. (ii)

   $\varepsilon(a,b+c)=\varepsilon(a,b)\varepsilon(a,c)$ ,

3. (iii)

   $\varepsilon(a+b,c)=\varepsilon(a,c)\varepsilon(b,c)$ ,

$a,b,c\in G$ ,

A Lie color algebra is a triple $(A,[\cdot,\cdot],\varepsilon)$ in which $(A,[\cdot,\cdot])$ is a $G$ -graded algebra and $\varepsilon:G\times G\rightarrow{\bf\mathbb{K}^{*}}$ is a bicharacter such that

	$\displaystyle[x,y]=-\varepsilon(x,y)[y,x],\qquad\qquad\qquad\qquad(\varepsilon\mbox{-skew-symmetry})$		(2.2)
	$\displaystyle\varepsilon(z,x)[x,[y,z]]+\varepsilon(x,y)[y,[z,x]]+\varepsilon(y,z)[z,[x,y]]=0,\;(\varepsilon\mbox{-Jacobi identity})$		(2.3)

for any $x,y,z\in\mathcal{H}(A)$ .

1) An averaging operator over an associative color algebra $(A,\cdot,\varepsilon)$ is an even linear map $\alpha:A\rightarrow A$ such that

$$\alpha(\alpha(x)\cdot y)=\alpha(x)\cdot\alpha(y)=\alpha(x\cdot\alpha(y)),$$

for all $x,y\in\mathcal{H}(A)$ .
2) An averaging operator over a Lie color algebra $(A,[\cdot,\cdot],\varepsilon)$ is an even linear map $\alpha:A\rightarrow A$ such that

$$\alpha([\alpha(x),y])=[\alpha(x),\alpha(y)]=\alpha([x,\alpha(y)]),$$

for all $x,y\in\mathcal{H}(A)$ .

[ 4 ] A Leibniz color algebra is a $G$ -graded vector space $L$ together with an even bilinear map $[-,-]:L\otimes L\rightarrow L$ and a bicharacter $\varepsilon:G\otimes G\rightarrow\mathbb{K}^{*}$ such that

$\displaystyle[[x,y],z]=[x,[y,z]]+\varepsilon(y,z)[[x,z],y]$

(2.5)

holds, for all $x,y,z\in\mathcal{H}(L)$ .

[ 5 ] A color algebra $(A,\cdot,\varepsilon)$ is said to be a Lie admissible color algebra if, for any hogeneous elements $x,y\in A$ , the bracket $[-,-]$ defined by

$$[x,y]=x\cdot y-\varepsilon(x,y)y\cdot x$$

satisfies the $\varepsilon$ -Jacobi identity.

[ 5 ] A post-Lie color algebra $(L,[-,-],\cdot,\varepsilon)$ is a Lie color algebra $(L,[-,-],\varepsilon)$ together with an even bilinear map $\cdot:L\otimes L\rightarrow L$ such that

	$\displaystyle z\cdot[x,y]-[z\cdot x,y]-\varepsilon(z,x)[x,z\cdot y]=0,$		(2.9)
	$\displaystyle z\cdot(y\cdot x)-\varepsilon(z,y)y\cdot(z\cdot x)+\varepsilon(z,y)(y\cdot z)\cdot x-(z\cdot y)\cdot x+\varepsilon(z,y)[y,z]\cdot x=0,$		(2.10)

for any $x,y,z\in\mathcal{H}(L)$ .

A ternary Leibniz color algebra is a $G$ -graded vector space $A$ over a field $\mathbb{K}$ equipped with a bicharacter $\varepsilon:G\otimes G\rightarrow\mathbb{K}^{*}$ and an even trilinear operation $[-,-,-]:A\otimes A\otimes A\rightarrow A$ (i.e. $[x,y,z]\subseteq A_{x+y+z}$ whenever $x,y,z\in\mathcal{H}(A)$ ) satisfying the following ternary $\varepsilon$ -Nambu identity :

$\displaystyle[[x,y,z],t,u]=[x,y,[z,t,u]]+\varepsilon(z,t+u)[x,[y,t,u],z]+\varepsilon(y+z,t+u)[[x,t,u],y,z]$

(3.1)

for any $x,y,z,t,u\in\mathcal{H}(A)$ .
If the trilinear map $[-,-,-]$ is $\varepsilon$ -skew-symmetric for any pair of variables, then $(A,[-,-,-],\varepsilon)$ is said to be a ternary Lie color algebra [ 18 ] .

A color Lie triple system is a $G$ -graded vector space $A$ over a field $\mathbb{K}$ equipped with a bicharacter $\varepsilon:G\times G\rightarrow\mathbb{K}^{*}$ and an even trilinear bracket which satisfies the identity ( 3.1 ), instead of skew-symmetry, satisfies the conditions

	$\displaystyle\qquad\qquad\qquad[x,y,z]=-\varepsilon(y,z)[x,z,y],\quad(\mbox{\it right}\;\varepsilon\mbox{\it-skew-symmetry})$		(3.16)
	$\displaystyle\varepsilon(z,x)[x,y,z]+\varepsilon(x,y)[y,z,x]+\varepsilon(y,z)[z,x,y]=0,\;\;(\mbox{\it ternary}\;\;\varepsilon\mbox{\it-Jacobi identity})$		(3.17)

for each $x,y,z\in\mathcal{H}(A)$ .

Let $(L,\cdot,[-,-,-],\varepsilon)$ and $(L^{\prime},\cdot^{\prime},[-,-,-]^{\prime},\varepsilon)$ be two non-commutative ternary Leibniz-Nambu-Poisson color algebras. Let $\alpha:L\rightarrow L^{\prime}$ be an even linear mapping such that, for any $x,y,z\in\mathcal{H}(L)$ ,

$$\alpha(x\cdot y)=\alpha(x)\cdot\alpha(y)\quad\mbox{and}\quad\alpha([x,y,z])=[\alpha(x),\alpha(y),\alpha(z)]^{\prime}.$$

Then $\alpha$ is called a morphism of non-commutative ternary Leibniz-Nambu-Poisson color algebras.

An associate of $F_{{\rm add}}(x,y)$ is a formal series $\phi(x,z)\in\mathbb{C}((x))[[z]]$ satisfying the conditions

(2.1)

$\displaystyle\phi(x,0)=x,\ \ \ \ \phi(\phi(x,y),z)=\phi(x,y+z).$

A Hom-associative algebra $A$ is called unital if there exists a linear map $\eta:\Bbbk\rightarrow A$ $\$ such that $\alpha\circ\eta=\eta$ and

$$\mathit{\ }\mu\circ\left(\eta\otimes id_{A}\right)=\mu\circ\left(id_{A}\otimes\eta\right)=\alpha.$$

(2)

The algebra $\mathcal{W}\mathbin{\otimes}\Lambda^{\bullet}=\mathopen{}\mathclose{{}\left(\prod_{k=0}^{\infty}\mathrm{S}^{k}\mathfrak{g}^{*}\mathbin{\otimes}\Lambda^{\bullet}\mathfrak{g}^{*}}\right)[[t]]$ is called the formal Weyl algebra where the product $\mu$ is defined by

$$(f\mathbin{\otimes}\alpha)\cdot(g\mathbin{\otimes}\beta)=\mu(f\mathbin{\otimes}\alpha,g\mathbin{\otimes}\beta)=f\vee g\mathbin{\otimes}\alpha\wedge\beta.$$

(2.2)

for any factorizing tensors $f\mathbin{\otimes}\alpha,g\mathbin{\otimes}\beta\in\mathcal{W}\mathbin{\otimes}\Lambda^{\bullet}$ and extended $\mathsf{R}[[t]]$ -bilinearly. ${}_{\blacksquare}$

A Kan complex $K_{\cdot}$ is called minimal, if for any $x,y\in K_{n}$ with $\partial x=\partial y$ homotopy implies equality:

$$x\sim y\Rightarrow x=y.$$

A Kan complex $(K_{\cdot},\star)$ is called $n$ -connected, if we have $\pi_{k}(K_{\cdot},\star)=0$ for all $0\leq k\leq n$ .

Let $A$ be a nonempty set. A majority operation on $A$ is a ternary operation $m:A^{3}\to A$ such that $m(b,a,a)=m(a,b,a)=m(a,a,b)=a$ for all $a$ , $b\in A$ . A Maltsev operation on $A$ is a ternary operation $p:A^{3}\to A$ such that $p(b,a,a)=p(a,a,b)=b$ for all $a$ , $b\in A$ . For $k\geq 2$ , an operation $n:A^{(k+1)}\to A$ is a $(k+1)$ -ary near unanimity operation on $A$ if for all $a$ , $b\in A$ ,

$$n(b,a,a,\ldots,a)=n(a,b,a,\ldots,a)=\cdots=n(a,a,\ldots,a,b)=a.$$

(Note that a majority operation is a $3$ -ary near-unanimity operation.)