A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension.

Definition 5.9 .

Let $V$ be a free graded module of finite rank and let $\omega\in DR^{2}(\Sigma V)$ be a constant symplectic form. Let $m$ and $m^{\prime}$ be two minimal unital symplectic $C_{\infty}$ -structures on $V$ with the same underlying Frobenius algebra. We say that two unital symplectic $C_{\infty}$ -morphisms $\phi$ and $\phi^{\prime}$ from $m$ to $m^{\prime}$ are homotopic if there exists a normalised symplectic vector field $\eta$ of degree $-1$ such that

\phi=\phi^{\prime}\circ\exp([m,\eta]).

We denote the moduli space of unital symplectic $C_{\infty}$ -morphisms from $m$ to $m^{\prime}$ by $\mathcal{USM}_{\infty}(m;m^{\prime})$ and define it as the quotient of the set

\{\phi:\widehat{L}\Sigma V^{*}\to\widehat{L}\Sigma V^{*}:\phi\text{ is a \emph% {pointed unital symplectic} $C_{\infty}$-morphism from $m$ to $m^{\prime}$}\}

by the homotopy equivalence relation defined above.

Definition 8.8 .

Let ${\mathcal{A}}$ be a $*$ -algebra and $\sigma:{\mathcal{A}}\to{\mathcal{A}}$ an algebra automorphism such that

\sigma(a)^{*}=\sigma^{-1}(a^{*}).

then we say that $\sigma$ is a regular automorphism, [ 23 ] .

Definition B.1.1 .

Let $\mathscr{J}$ be a ring which is not necessarily commutative with respect to multiplication. Then an Abelian (commutative) group $S$ is called a left $\mathscr{J}$ -module or a left module over $\mathscr{J}$ with respect to a mapping (scalar multiplication on the left which is simply denoted by juxtaposition) $\mathscr{J}\times S\to S$ such that for all $a,b\in\mathscr{J}$ and $g,h\in S$ ,

1)

$a(g+h)=ag+ah$ ,
2)

$(a+b)g=ag+bg$ ,
3)

$(ab)g=a(bg)$ .

Definition 2.3 .

( [ 12 ] ) Let $\mathfrak{a}$ be a quiver. An $A_{[0,\infty[}$ -structure on $\mathfrak{a}$ is an element $b\in\mathbf{C}_{br}^{1}(\mathfrak{a})$ satisfying

b\{b\}=0

(7)

The morphisms

b_{n}:\Sigma\mathfrak{a}^{\otimes n}\longrightarrow\Sigma\mathfrak{a}

defining $b$ are sometimes called (Taylor) coefficients of $b$ . The couple $(\mathfrak{a},b)$ is called an $A_{[0,\infty]}$ -category . If $b_{0}=0$ , it is called an $A_{\infty}$ -category . If $b_{n}=0$ for $n\geq 3$ , it is called a cdg category . If $b_{0}=0$ and $b_{n}=0$ for $n\geq 3$ , it is called a dg category .

Definition 2.1 .

A mapping $f:S\to X$ is said to be a kannappan mapping if it satisfies equation

(2.1)

f(xyz)+f(x)+f(y)+f(z)-f(xy)-f(xz)-f(yz)=0.

Definition 1 .

Consider a curve $\gamma(t):[a,b]\rightarrow Q$ . A proper variation of $\gamma(t)$ is a differentiable function $q(s,t):[-\varepsilon,\varepsilon]\times[a,b]\rightarrow Q$ that satisfies the following conditions:

(i)

$q(0,t)=\gamma(t),\ \ \forall t\in[a,b]$
(ii)

$q(s,a)=\gamma(a)$ and $q(s,b)=\gamma(b),\ \forall s\in[-\varepsilon,\varepsilon]$ .

Definition 3 .

A lattice is a poset $L$ such that any elements $x$ , $y$ $\in$ $L$ have a unique greatest lower bound, denoted $x\wedge y$ , and a unique least upper bound $x\vee y$ , i.e.,

x\wedge y=\inf\{x,y\},\qquad x\vee y=\sup\{x,y\}.

(4)

The operation $\wedge$ is also called meet , and the operation $\vee$ is called join . A lattice $L$ is complete , when for any set $D\subseteq L$ the bounds $\sup D$ and $\inf D$ exist; it is $\sigma$ -complete , when for any countable set $D\subseteq L$ the bounds $\sup D$ and $\inf D$ exist. A lattice is atomic if every element is a join of atoms. $\lozenge$

Definition 30 .

[ 26 , 30 , 25 ] A t-norm (“triangular norm”) is a binary operation $*:[0,1]^{2}\to[0,1]$ satisfying the following conditions for all $x$ , $y$ , $z\in[0,1]$ :

(commutativity)	$\displaystyle xy=yx$	(41)
(associativity)	$\displaystyle(xy)z=x(yz)$	(42)
(monotony)	$\displaystyle xz\leqq yz\ \mbox{ if }x\leqq y$	(43)
(boundary condition)	$\displaystyle 1*x=x,$	(44)

$\lozenge$

Definition 3.1 .

A perversity $p$ is said to be standard if

(5)

p(x)=p^{-}(x)=p^{+}(x)=0\qquad\text{if $\operatorname{codim}\bar{x}=0$.}

Definition 3.1 .

A random variable $Y$ is called a GGC variable if it is infinitely divisible with Lévy measure $\nu$ of the form:

\nu\left({\mathrm{d}}x\right)=\dfrac{{\mathrm{d}}x}{x}\int\mu\left({\mathrm{d}% }\xi\right)\exp\left(-\xi x\right),

(3.1)

where $\mu\left({\mathrm{d}}\xi\right)$ is a Radon measure on $\mathbb{R}_{+}$ , called the Thorin measure of $Y$ .

Definition 7 .

A property $P$ of colorings on a graph $G$ is a frame property just in case

Frame(C)=Frame(C^{\prime})\Rightarrow[C\in P\Rightarrow C^{\prime}\in P],

for any colorings $C$ , $C^{\prime}$ of $G$ .