Spin structures and baby universes

Definition 6 .

An involutive bisemilattice is an algebra ${\mathbf{B}}=\langle B,\land,\lor,^{{}^{\prime}},0,1\rangle$ of type $(2,2,1,0,0)$ satisfying:

I1 .

$x\lor x\thickapprox x$ ;
I2 .

$x\lor y\thickapprox y\lor x$ ;
I3 .

$x\lor(y\lor z)\thickapprox(x\lor y)\lor z$ ;
I4 .

$(x^{\prime})^{\prime}\thickapprox x$ ;
I5 .

$x\land y\thickapprox(x^{\prime}\lor y^{\prime})^{\prime}$ ;
I6 .

$x\land(x^{\prime}\lor y)\thickapprox x\land y$ ;
I7 .

$0\lor x\thickapprox x$ ;
I8 .

$1\thickapprox 0^{\prime}$ .

Definition 3.1.12 .

{enumerate*}

Suppose $i\in\mathsf{Input}$ and $o\in\mathsf{Output}$ . Then say that $i$ points to $o$ and write $i\mapsto o$ when they share a position:

i\mapsto o\quad\text{when}\quad\mathit{pos}(i)=\mathit{pos}(o).

(The use of ‘point’ here is unrelated to the ‘pointed sets’ from Notation 2.1.3 .)

Recall the notation $\mathit{tx}\and i$ from Notation 3.1.8 ( 3.1.8 ). Suppose that:

\begin{array}[]{l}i=(p,k)\in\mathsf{Input}\\ i\in\mathit{tx}\in\mathsf{Transaction}\ \ \text{and}\\ o=(p,V)\in\mathsf{Output}.\end{array}

Then write

\mathit{validates}(o,\mathit{tx}\and i)\quad\text{when}\quad\mathit{tx}\and i\in V

and say that the output $o$ validates the input-in-context $\mathit{tx}\and i$ .

Definition 2.1 .

The (first-order) differential calculus over an algebra $A$ over a field $k$ is a pair $(\Omega^{1}(A),d)$ , where $\Omega^{1}(A)$ is a bimodule over $A$ , $d$ is a linear map $d:A\rightarrow\Omega^{1}(A)$ , which satisfies the Leibniz rule,

\displaystyle d(ab)=a\,db+(da)\,b,

(2.1)

and $\Omega^{1}(A)$ is generated as a left module by the image of $d$ . We say that $(\Omega^{1}(A),d)$ is connected if $\ker d\cong k$ .

Definition 3.9 .

We say that the connection is star-compatible, if:

\nabla\circ\ast=\sigma\circ\dagger\circ\nabla,

where $(\omega\otimes_{A}\eta)^{\dagger}=\eta^{\ast}\otimes_{A}\omega^{\ast}$ , i.e. $\dagger$ is the induced $\ast$ -structure on higher tensors.

Definition 4.3 .

We say that the metric is compatible with the higher-order differential calculus iff $\mathbf{g}\in\ker\wedge$ , that is

\wedge\,\mathbf{g}=\mathbf{g}^{(1)}\wedge\mathbf{g}^{(2)}=0.

Definition 5.1 .

A linear connection $(\nabla,\sigma)$ is compatible with the metric ${\bf g}$ if

(\nabla\otimes\mathrm{id}){\bf g}+(\sigma\otimes\mathrm{id})(\mathrm{id}% \otimes\nabla){\bf g}=0.

(5.1)

Definition 3 .

Let $S$ be a set with two operations $+$ and $\cdot$ such that $\left(S,+\right)$ is a semigroup (not necessarily commutative) and $\left(S,\cdot\right)$ is an inverse semigroup. Then, we say that $\left(S,+,\cdot\right)$ is a left inverse semi-brace if

\displaystyle a\left(b+c\right)=ab+a\left(a^{-1}+c\right)

(1)

holds, for all $a,b,c\in S$ . We call $(S,+)$ and $(S,\cdot)$ the additive semigroup and the multiplicative semigroup of $S$ , respectively.
A right inverse semi-brace is defined similarly, by replacing condition ( 1 ) with $\;\left(a+b\right)c=\left(a+c^{-1}\right)c+bc,\;$ for all $a,b,c\in S$ .
A two-sided inverse semi-brace $(S,+,\cdot)$ is a left inverse semi-brace that is also a right inverse semi-brace with respect to the same operations $+$ and $\cdot$ .

Definition 2 (Path-differentiable functions) .

A locally Lipschitz function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is path-differentiable if for each Lipschitz ¹ ¹ 1 In other parts of the literature (see e.g. [ 4 ] ), this definition is given with absolutely-continuous curves, and this is equivalent because such curves can be reparameterized (for example, by arclength) to obtain Lipschitz curves, without affecting their role in the definition. curve $\gamma\colon\mathbb{R}\to\mathbb{R}^{n}$ , for almost every $t\in\mathbb{R}$ , the composition $f\circ\gamma$ is differentiable at $t$ and the derivative is given by

(f\circ\gamma)^{\prime}(t)=v\cdot\gamma^{\prime}(t)

for all $v\in\partial^{c}f(\gamma(t))$ .

Definition 2.1 .

Let ” $\cdot$ ” be a bilinear product in a vector space $\mathcal{A}.$ Suppose that it satisfies the following law:

\displaystyle(x\cdot y)\cdot z=-x\cdot(y\cdot z).

(2.3)

Then, we call the pair $(\mathcal{A},\cdot)$ an antiassociative algebra . Combining both associative (q=1) and antiassociative (q=-1) cases, any algebra $\mathcal{A}$ satisfying

\displaystyle(x\cdot y)\cdot z=qx\cdot(y\cdot z),q=1,-1

is called a $q$ -associative algebra .

Definition 2.4 .

[ 12 ] An algebra $(\mathcal{A},\diamond)$ over $K$ is called mock Lie if it is commutative:

\displaystyle x\diamond y=y\diamond x,

(2.4)

and satisfies the Jacobi identity:

\displaystyle(x\diamond y)\diamond z+(z\diamond x)\diamond y+(y\diamond z)% \diamond x=0

(2.5)

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

Definition 5.6 .

Let $\mathcal{A}$ be an antiassociative algebra. We say that $(\mathcal{A},\omega)$ is a sympletic antiassociative algebra if $\omega$ is a non-degenerate skew-symmetric bilinear form on $\mathcal{A}$ such that the following identity satisfied:

\displaystyle\omega(xy,z)+\omega(yz,x)+\omega(zx,y)=0\ \mbox{(invariance % condition),}

(5.2)

for all $x,y,z\in\mathcal{A}.$

Definition 2.1 .

We will say that $\mathbf{f}\in V$ is a $\boxdot$ -null element if for all $\mathbf{x}\in V$ we have

(2.1)

{\bf x}\boxdot{\bf f}={\bf f}.

Definition

Let $P$ be a probability measure on ${\boldsymbol{S}}={\boldsymbol{S}}^{\prime}\times{\boldsymbol{S}}^{\prime\prime}$ with the product metric

\rho((x^{\prime},x^{\prime\prime}),(y^{\prime},y^{\prime\prime}))=\rho^{\prime% }(x^{\prime},y^{\prime})\vee\rho^{\prime\prime}(x^{\prime\prime},y^{\prime% \prime}).

Define the marginal distributions by $P^{\prime}(A^{\prime})=P(A^{\prime}\times{\boldsymbol{S}}^{\prime\prime})$ and $P^{\prime\prime}(A^{\prime\prime})=P({\boldsymbol{S}}^{\prime}\times A^{\prime% \prime})$ . If the marginals are independent, we write $P=P^{\prime}\times P^{\prime\prime}$ . We denote by $\mathcal{S}^{\prime}\times\mathcal{S}^{\prime\prime}$ the product $\sigma$ -algebra generated by the measurable rectangles $A^{\prime}\times A^{\prime\prime}$ for $A^{\prime}\in\mathcal{S}^{\prime}$ and $A^{\prime\prime}\in\mathcal{S}^{\prime\prime}$ . ${}_{\blacksquare}$

Definition 1.2 .

Let $R$ be a commutative unital ring. The set $\{x:x=x^{2}\}$ of idempotents is a Boolean algebra, denoted by $\mathbb{B}$ , with operations

e\wedge f=ef,

\neg e=1-e,

0=0,

1=1,

e\vee f=1-(1-e)(1-f)=e+f-ef.

It carries a partial ordering defined by $e\leq f\Leftrightarrow ef=e$ (which is $\mathcal{L}_{rings}$ -definable). The atoms of $\mathbb{B}$ are the minimal idempotents (with respect to the ordering) that are not equal to $0,1$ . (In fact we assume $0\neq 1$ ).

Definition 3.1

Let $E$ be a Fréchet space and $f\colon W\to E$ be a function on a subset $W\subseteq{\mathbb{R}}\times E$ . We say that the differential equation

y^{\prime}(t)\,=\,f(t,y(t))

(12)

satisfies local uniqueness of Carathéodory solutions if the following holds: For all Carathéodory solutions $\gamma_{1}\colon I_{1}\to E$ and $\gamma_{2}\colon I_{2}\to E$ of ( 12 ) and $t_{0}\in I_{1}\cap I_{2}$ such that $\gamma_{1}(t_{0})=\gamma_{2}(t_{0})$ , there exists an interval $K\subseteq{\mathbb{R}}$ which is an open neighbourhood of $t_{0}$ in $I_{1}\cap I_{2}$ such that

\gamma_{1}|_{K}=\gamma_{2}|_{K}.

Definition 4.1

Let $M$ be a $C^{1}$ -manifold modelled on a Fréchet space $E$ and $f\colon W\to TM$ be a function on a subset $W\subseteq{\mathbb{R}}\times M$ such that $f(t,y)\in T_{y}M$ for all $(t,y)\in W$ . We say that a function $\gamma\colon I\to M$ on a non-degenerate interval $I\subseteq{\mathbb{R}}$ is a Carathéodory solution to the differential equation

\dot{y}(t)=f(t,y(t))

(16)

if $\gamma$ is absolutely continuous, $(t,\gamma(t))\in W$ for all $t\in I$ and

\dot{\gamma}(t)=f(t,\gamma(t))

for $\lambda_{1}$ -almost all $t\in I$ (using notation as in 2.4 ). If $(t_{0},y_{0})\in W$ is given and $\gamma$ satisfies, moreover, the condition $\gamma(t_{0})=y_{0}$ , then $\gamma$ is called a Carathéodory solution to the initial value problem

\dot{y}(t)=f(t,y(t)),\quad y(t_{0})=y_{0}.

(17)

Definition 1 .

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be a smooth function and $M\subset\mathbb{R}^{n}$ a compact, codimension-one manifold with a boundary $\partial M$ . The manifold $M$ is a ridge of the scalar field $f$ if both $M$ and $\partial M$ are normally attracting invariant manifolds for the gradient system

\dot{x}=\nabla f(x).

(VI.4)

Definition 4.4 (Big circle relation) .

Let $\delta$ be the reduced loop class of the circle $D$ in Figure 3.1 . The Big circle relation is

(4.5)

\delta=-q^{2}-q^{-2}.

5.13 Definition .

[ 24 ] An associative dialgebra is a vector space $D$ together with bilinear maps $\dashv,\vdash:D\otimes D\rightarrow D$ satisfying the following identities

	$\displaystyle a\dashv(b\dashv c)=(a\dashv b)\dashv c=a\dashv(b\vdash c),$
	$\displaystyle(a\vdash b)\dashv c=a\vdash(b\dashv c),$
	$\displaystyle(a\dashv b)\vdash c=a\vdash(b\vdash c)=(a\vdash b)\vdash c,~{}~{}% ~{}\text{ for all }a,b,c\in D.$

Definition 2.2

A Nelson algebra is an algebra $\mathbf{A}=\langle A;\land,\lor,\to,\sim,1\rangle$ of type $(2,2,2,1,0)$ such that the following conditions are satisfied for all $x,y,z\in A$ :

(N1)

$x\land(x\lor y)=x$ ,
(N2)

$x\land(y\lor z)=(z\land x)\lor(y\land x)$ ,
(N3)

$\sim\sim x=x$ ,
(N4)

$\sim(x\land y)=\sim x\lor\sim y$ ,
(N5)

$x\land\sim x=(x\land\sim x)\land(y\lor\sim y)$ ,
(N6)

$x\to x=1$ ,
(N7)

$x\to(y\to z)=(x\land y)\to z$ ,
(N8)

$x\land(x\to y)=x\land(\sim x\lor y)$ .

Definition 2.8

An algebra $\mathbf{A}=\langle A;\land,\lor,\to,^{\dagger},0,1\rangle$ of type $(2,2,2,1,0,0)$ is said to be a dually hemimorphic semi-Heyting algebra if $\langle A;\land,\lor,\to,0,1\rangle$ is a semi-Heyting algebra and the following equations are satisfied:

DSM1)

$0^{\dagger}=1$ ,
DSM2)

$1^{\dagger}=0$ ,
DSM3)

$(x\land y)^{\dagger}=x^{\dagger}\lor y^{\dagger}$ .

Definition 2.9

An algebra $\mathbf{A}=\langle A;\land,\lor,\to,\sim,^{\prime},1\rangle$ of type $(2,2,2,1,0)$ is said to be a dually hemimorphic semi-Nelson algebra if $\langle A;\land,\lor,\to,\sim,1\rangle$ is a semi-Nelson algebra and the following equations are satisfied:

DSN1)

$(\sim 1)^{\prime}=1$ ,
DSN2)

$1^{\prime}\to(\sim 1)=1$ ,
DSN3)

$((x\to y)\land(y\to x)\land x^{\prime})\to((x\to y)\land(y\to x)\land y)^{% \prime}=1$ ,
DSN4)

$\sim x^{\prime}\to(\sim x\land(x^{\prime}\to x))=1$ ,
DSN5)

$(\sim x\land(x^{\prime}\to x))\to\sim x^{\prime}=1$ ,
DSN6)

$(x\land y)^{\prime}\to(x^{\prime}\lor y^{\prime})=1$ ,
DSN7)

$(x^{\prime}\lor y^{\prime})\to(x\land y)^{\prime}=1$ .

We use the convention that the unary operation ${}^{\prime}$ has higher priority than $\sim$ , so the expression $\sim x^{\prime}$ means $\sim(x^{\prime})$ .

Definition 2.1 .

[ 8 ] Let $(M,g)$ be a Riemennian manifold. Then a complex-valued function $\phi:M\to{\mathbb{C}}$ is said to be an eigenfunction if it is eigen both with respect to the Laplace-Beltrami operator $\tau$ and the conformality operator $\kappa$ i.e. there exist complex numbers $\lambda,\mu\in{\mathbb{C}}$ such that

\tau(\phi)=\lambda\cdot\phi\ \ \text{and}\ \ \kappa(\phi,\phi)=\mu\cdot\phi^{2}.

A set $\mathcal{E}=\{\phi_{i}:M\to{\mathbb{C}}\ |\ i\in I\}$ of complex-valued functions is said to be an eigenfamily on $M$ if there exist complex numbers $\lambda,\mu\in{\mathbb{C}}$ such that for all $\phi,\psi\in\mathcal{E}$ we have

\tau(\phi)=\lambda\cdot\phi\ \ \text{and}\ \ \kappa(\phi,\psi)=\mu\cdot\phi% \cdot\psi.

Definition 2.1 .

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,

(i)

$\varepsilon(g,g^{\prime})\varepsilon(g^{\prime},g)=1$ ,
(ii)

$\varepsilon(g,g^{\prime}+g^{\prime\prime})=\varepsilon(g,g^{\prime})% \varepsilon(g,g^{\prime\prime})$ ,
(iii)

$\varepsilon(g+g^{\prime},g^{\prime\prime})=\varepsilon(g,g^{\prime\prime})% \varepsilon(g^{\prime},g^{\prime\prime})$ ,

$g,g^{\prime},g^{\prime\prime}\in G$ .

Definition 2.1 .

A (multiplicative) hom-Lie algebra is a triplet $(\mathfrak{g},[~{},~{}],\alpha)$ , where $\mathfrak{g}$ is a vector space equipped with a skew-symmetric bilinear map $[~{},~{}]:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$ , and a linear map $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying $\alpha[x,y]=[\alpha(x),\alpha(y)]$ such that

(2.1)

[\alpha(x),[y,z]]+[\alpha(y),[z,x]]+[\alpha(z),[x,y]]=0,~{}~{}~{}~{}\mbox{for % all}~{}~{}x,y,z\in\mathfrak{g}.

Furthermore, if $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ is a vector space automorphism of $\mathfrak{g}$ , then the hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha)$ is called a regular hom-Lie algebra.

Definition 2.3 .

( [ 21 ] ) A representation of a hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha)$ on a $k$ -vector space $V$ with respect to $\beta\in\mathsf{End}(V)$ is a linear map $\rho:\mathfrak{g}\rightarrow\mathsf{End}(V)$ such that

(2.2)

\rho(\alpha(x))(\beta(v))=\beta(\rho(x)(v)),

(2.3)

\rho([x,y])(\beta(v))=\rho(\alpha(x))\rho(y)(v)-\rho(\alpha(y))\rho(x)(v),

for all $x,y\in\mathfrak{g}$ and $v\in V$ .

Definition 1 .

Let $R$ be a commutative ring, and ${\mathbf{L}}$ a $R$ -linear category. Let $\iota\colon{\mathbf{L}}\rightarrow{\mathbf{C}}$ be a functor that preserves finite products, and $M\in{\textnormal{Obj}}({\mathbf{L}})$ . An endomorphism $f\in{\textnormal{End}}_{\mathbf{C}}(\iota(M))$ is a machine if, for all morphisms $g\colon X\rightarrow\iota(M)$ , there exists a unique $h\colon X\rightarrow\iota(M)$ such that:

h=g+fh.

We call $h$ the stable state of $f$ with initial condition $g$ , and denote by $S_{f}$ the stable state of $f$ with initial condition ${\textnormal{id}}_{\iota(M)}$ .

Definition 3 .

Let $R,{\mathbf{C}},{\mathbf{L}},\iota,M$ be as in definition 1 . Let $f,f^{\prime}\in{\textnormal{End}}_{\mathbf{C}}(\iota(M))$ . We say that $f$ does not depend on $f^{\prime}$ if, for any $X\in{\textnormal{Obj}}({\mathbf{C}})$ , for any pair of maps $b,b^{\prime}\colon X\rightarrow\iota(M)$ , and for all $\lambda\in R$ , the following holds:

f(b+\lambda f^{\prime}b^{\prime})=fb.

(4)

Otherwise, we say that $f$ depends on $f^{\prime}$ .

Definition 4 .

Let $G$ be a $Q$ -group. Let $M$ be, simultaneously, a $G$ -module and a $Q$ -module. We say that $M$ is a $Q$ - $G$ module if

(18)

\displaystyle q(gm)=q(g)qm

for $g\in G,q\in Q,m\in M$ . We denote by $Q$ - $G\,\mathcal{M}od$ the category whose objects are $Q$ - $G$ modules and morphisms are the functions $f:M\rightarrow N$ such that $f$ is both $G$ -linear and $Q$ -linear.

Definition 1 .

Let ${\bf{A}}$ , ${\bf{B}}$ be two bounded distributive lattices. A structure $\langle{\bf{A}},{\bf{B}},f\rangle$ is called a FDL-module, if $f\colon A\times B\to A$ is a function such that for every $x,y\in A$ and every $b,c\in B$ the following conditions hold:

(F1)

$f(x\vee y,b)=f(x,b)\vee f(y,b)$ ,
(F2)

$f(x,b\vee c)=f(x,b)\vee f(x,c)$ ,
(F3)

$f(0,b)=0$ ,
(F4)

$f(x,0)=0$ .

A structure $\langle{\bf{A}},{\bf{B}},i\rangle$ is called an IDL-module, if $i\colon B\times A\to A$ is a function such that for every $x,y\in A$ and every $b,c\in B$ the following conditions hold:

(I1)

$i(b,x\wedge y)=i(b,x)\wedge i(b,y)$ ,
(I2)

$i(b\vee c,x)=i(b,x)\wedge i(c,x)$ ,
(I3)

$i(b,1)=1$ .

Moreover, a structure $\mathcal{M}=\langle{\bf{A}},{\bf{B}},f,i\rangle$ is called a FIDL-module, if $\langle{\bf{A}},{\bf{B}},f\rangle$ is a FDL-module and $\langle{\bf{A}},{\bf{B}},i\rangle$ is an IDL-module.

Definition 1 .

Let $\Sigma$ be an alphabet, $\Gamma=\{\texttt{u},\texttt{d}\}$ , and $f:\Sigma^{*}\times\Sigma\times\Gamma\rightarrow\Sigma^{*}$ a function such that

f(w,a,b)=\left\{\begin{array}[]{ll}aw,&\text{ if }b=\texttt{u},\\ wa,&\text{ if }b=\texttt{d}.\end{array}\right.

(1)

Then, the folding function $h:\Sigma^{*}\times\Gamma^{*}\rightarrow\Sigma^{*}$ is a partial function defined by

h(w,v)=\left\{\begin{array}[]{ll}f(f(\ldots f(\texttt{\textepsilon},a_{1},b_{1% })\ldots,a_{k-1},b_{k-1}),a_{k},b_{k}),&\text{ if }|w|=|v|>0\\ \texttt{\textepsilon},&\text{ if }|w|=|v|=0,\\ \text{undefined,}&\text{ if }|w|\neq|v|.\end{array}\right.

(2)

where $w=a_{1}a_{2}\ldots a_{k}$ , $v=b_{1}b_{2}\ldots b_{k}$ , $a_{i}\in\Sigma$ , with $b_{i}\in\Gamma$ for $i=1,2,\cdots,k$ , and \textepsilon is the empty string.

Definition 4 .

$(x,y)\in T$ is an extreme point of $T$ if and only if there are no points $(x^{\prime},y^{\prime}),~{}(x^{\prime\prime},y^{\prime\prime})\in T$ and no $\alpha\in(0,1)$ such that

(x,y)=\alpha(x^{\prime},y^{\prime})+(1-\alpha)(x^{\prime\prime},y^{\prime% \prime}).

(3)

2.3 Definition .

A Hom-coassociative coalgebra is a Hom-vector space $(C,\alpha)$ together with a linear map $\triangle:C\rightarrow C\otimes C$ satisfying

(1)

\displaystyle(\triangle\otimes\alpha)\circ\triangle=(\alpha\otimes\triangle)% \circ\triangle.

A Hom-coassociative coalgebra as above may be denoted by $(C,\alpha,\triangle)$ or simply by $C$ . A Hom-coassociative coalgebra $(C,\alpha,\triangle)$ is called multiplicative if $(\alpha\otimes\alpha)\circ\triangle=\triangle\circ\alpha$ .

2.4 Definition .

An infinitesimal Hom-bialgebra is a quadruple $(A,\alpha,\mu,\triangle)$ in which $(A,\alpha,\mu)$ is a Hom-associative algebra, $(A,\alpha,\triangle)$ is a Hom-coassociative coalgebra and satisfying the following compatibility

(2)

\displaystyle\triangle\circ\mu=(\mu\otimes\alpha)\circ(\alpha\otimes\triangle)% +(\alpha\otimes\mu)\circ(\triangle\otimes\alpha).

Definition 4.12 .

Define the static part of $u$ to be the part of $u$ that has not been pushed yet, i.e. the subword $s(u)$ satisfying

u=s(u)\cdot p(u).

Definition 2.4 .

Let $(A,\cdot)$ be a ${\bf k}$ -algebra with multiplication $\cdot$ and let $(R,\circ)$ be a ${\bf k}$ -algebra with multiplication $\circ$ . Let $\ell,r:A\rightarrow{\rm End}_{\bf k}(R)$ be linear maps. We call $R$ (or the quadruple $(R,\circ,\ell,r)$ ) an $A$ -bimodule ${\bf k}$ -algebra if $(R,\ell,r)$ is an $A$ -bimodule that is compatible with the multiplication $\circ$ on $R$ in the sense that

\ell(x)(v\circ w)=(\ell(x)v)\circ w,\;(v\circ w)r(x)=v\circ(wr(x)),\;(vr(x))% \circ w=v\circ(\ell(x)w),\;

for all $x,y\in A,v,w\in R$ .

Definition 1.4

( [ 19 ] , [ 20 ] ) A Novikov-Poisson algebra is a triple $(A,\mu,*)$ such that $(A,\mu)$ is a commutative associative algebra, $(A,*)$ is a Novikov algebra and the following compatibility conditions hold, for all $x,y,z\in A$ :

	$\displaystyle(xy)z-x(yz)=(yx)z-y(xz),$		(1.3)
	$\displaystyle(xy)z=(xz)y.$		(1.4)

A morphism of Novikov-Poisson algebras from $(A,\mu,*)$ to $(A^{\prime},\mu^{\prime},*^{\prime})$ is a linear map $f:A\rightarrow A^{\prime}$ satisfying $f\circ\mu=\mu^{\prime}\circ(f\otimes f)$ and $f(x*y)=f(x)*^{\prime}f(y)$ , for all $x,y\in A$ .

Definition 1.5

( [ 15 ] ) A BiHom-Novikov algebra is a 4-tuple $(A,\ast,\alpha,\beta)$ , where $A$ is a linear space, $\ast:A\otimes A\rightarrow A$ is a linear map and $\alpha,\beta:A\rightarrow A$ are commuting linear maps (called the structure maps of $A$ ), satisfying the following conditions, for all $x,y,z\in A$ :

	$\displaystyle\alpha(x\ast y)=\alpha(x)\ast\alpha(y),~{}~{}\beta(x\ast y)=\beta% (x)\ast\beta(y),$		(1.6)
	$\displaystyle(\beta(x)\ast\alpha(y))\ast\beta(z)-\alpha\beta(x)\ast(\alpha(y)% \ast z)$
	$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;=(\beta(y)\ast\alpha(x))\ast\beta(z)-% \alpha\beta(y)\ast(\alpha(x)\ast z),$		(1.7)
	$\displaystyle(x\ast\beta(y))\ast\alpha\beta(z)=(x\ast\beta(z))\ast\alpha\beta(% y).$		(1.8)

A morphism $f:(A,\ast_{A},\alpha_{A},\beta_{A})\rightarrow(B,\ast_{B},\alpha_{B},\beta_{B})$ of BiHom-Novikov algebras is a linear map $f:A\rightarrow B$ such that $\alpha_{B}\circ f=f\circ\alpha_{A}$ , $\beta_{B}\circ f=f\circ\beta_{A}$ and $f(x\ast_{A}y)=f(x)\ast_{B}f(y)$ , for all $x,y\in A$ .

Definition 1.6

( [ 15 ] ) A BiHom-Novikov-Poisson algebra is a 5-tuple $(A,\mu,\ast,\alpha,\beta)$ such that:

(1) $(A,\mu,\alpha,\beta)$ is a BiHom-commutative algebra;

(2) $(A,\ast,\alpha,\beta)$ is a BiHom-Novikov algebra;

(3) the following compatibility conditions hold for all $x,y,z\in A$ :

	$\displaystyle(\beta(x)\ast\alpha(y))\beta(z)-\alpha\beta(x)\ast(\alpha(y)z)=(% \beta(y)\ast\alpha(x))\beta(z)-\alpha\beta(y)\ast(\alpha(x)z),$		(1.9)
	$\displaystyle(x\beta(y))\ast\alpha\beta(z)=(x\ast\beta(z))\alpha\beta(y),$		(1.10)
	$\displaystyle\alpha(x)(y\ast z)=(xy)\ast\beta(z).$		(1.11)

The maps $\alpha$ and $\beta$ (in this order) are called the structure maps of $A$ .

A morphism $f:(A,\mu,\ast,\alpha,\beta)\rightarrow(A^{\prime},\mu^{\prime},\ast^{\prime},% \alpha^{\prime},\beta^{\prime})$ of BiHom-Novikov-Poisson algebras is a map that is a morphism of BiHom-associative algebras from $(A,\mu,\alpha,\beta)$ to $(A^{\prime},\mu^{\prime},\alpha^{\prime},\beta^{\prime})$ and a morphism of BiHom-Novikov algebras from $(A,\ast,\alpha,\beta)$ to $(A^{\prime},\ast^{\prime},\alpha^{\prime},\beta^{\prime})$ .

Definition 4.1

( [ 16 ] ) A BiHom-Lie algebra $\left(L,\left[\cdot,\cdot\right],\alpha,\beta\right)$ is a 4-tuple in which $L$ is a linear space, $\alpha,\beta:L\rightarrow L$ are linear maps and $\left[\cdot,\cdot\right]:L\times L\rightarrow L$ is a bilinear map, such that

	$\displaystyle\alpha\circ\beta=\beta\circ\alpha,$		(4.1)
	$\displaystyle\alpha(\left[x,y\right])=\left[\alpha\left(x\right),\alpha(y)% \right]\;\;\text{ and }\;\;\beta(\left[x,y\right])=\left[\beta\left(x\right),% \beta\left(y\right)\right],$		(4.2)
	$\displaystyle\left[\beta\left(x\right),\alpha\left(y\right)\right]=-\left[% \beta\left(y\right),\alpha\left(x\right)\right],\;\;\;\;\text{ (BiHom-skew-% symmetry)}$		(4.3)
	$\displaystyle\left[\beta^{2}\left(x\right),\left[\beta\left(y\right),\alpha% \left(z\right)\right]\right]+\left[\beta^{2}\left(y\right),\left[\beta\left(z% \right),\alpha\left(x\right)\right]\right]+\left[\beta^{2}\left(z\right),\left% [\beta\left(x\right),\alpha\left(y\right)\right]\right]=0,$		(4.4)
	(BiHom-Jacobi condition)

for all $x,y,z\in L$ . The maps $\alpha$ and $\beta$ (in this order) are called the structure maps of $L$ .

Definition 4.2

A BiHom-Poisson algebra is a 5-tuple $(A,\mu,\left[\cdot,\cdot\right],\alpha,\beta)$ , with the property that

(1) $(A,\mu,\alpha,\beta)$ is a BiHom-commutative algebra;

(2) $(A,\left[\cdot,\cdot\right],\alpha,\beta)$ is a BiHom-Lie algebra;

(3) the following BiHom-Leibniz identity holds for all $x,y,z\in A$ :

\displaystyle\left[\alpha\beta(x),yz\right]=\left[\beta(x),y\right]\beta(z)+% \beta(y)\left[\alpha(x),z\right].

(4.5)

Definition 4.3

Let $(A,\mu,\ast,\alpha,\beta)$ be a BiHom-Novikov-Poisson algebra. Then $A$ is called left BiHom-associative if the following condition holds for all $x,y,z\in A$ :

\displaystyle\alpha(x)\ast(yz)=(xy)\ast\beta(z).

(4.6)

Definition 1 (Degree of Algebraic Number) .

Let $L$ be an algebraic number field of degree $n$ , and let $\{1,\alpha,\cdots,\alpha^{n-1}\}$ be a $\mathbb{Q}$ -basis of $L$ for some primitive element $\alpha\in L$ . Given any element $v\in L$ , we can write $v$ uniquely as

v=f(\alpha)

for some $f\in\mathbb{Q}[x]$ with $\deg f\leq n-1$ . Then the degree of $v$ with respect to $\alpha$ , written as $\deg_{\alpha}(v)$ , is the degree of $f(x)$ .

Definition 2.1 .

[ 10 ] Let ” $\cdot$ ” be a bilinear product in a vector space $\mathcal{A}.$ Suppose that it satisfies the following law:

\displaystyle(x\cdot y)\cdot z=-x\cdot(y\cdot z).

(2.1)

Then, we call the pair $(\mathcal{A},\cdot)$ an antiassociative algebra .

Definition 2.2 .

[ 13 ] An algebra $(\mathcal{A},\diamond)$ over $K$ is called JJ if it is commutative:

\displaystyle x\diamond y=y\diamond x,

(2.2)

and satisfies the Jacobi identity:

\displaystyle(x\diamond y)\diamond z+(z\diamond x)\diamond y+(y\diamond z)% \diamond x=0

(2.3)

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

Definition 2.4 .

[ 13 ] A vector space $V$ is a module over a JJ algebra $\mathcal{A}$ , if there is a linear map (a representation) $\rho:\mathcal{A}\to End(V)$ such that

\displaystyle\rho(x\diamond y)(v)=-\rho(x)(\rho(y)v)-\rho(y)(\rho(x)v)

(2.4)

for any $x,y\in\mathcal{A}$ and $v\in V$ .

Definition 2.6 .

Let $(\mathfrak{g},\diamond)$ be a JJ algebra. Two representations $(V,\rho)$ and $(V^{\prime},\rho^{\prime})$ of $\mathfrak{g}$ are said to be isomorphic if there exists a linear map $\phi\ :V\rightarrow V^{\prime}$ such that

\forall x\in\mathfrak{g},\ \ \rho^{\prime}(x)\circ\phi=\phi\circ\rho(x).

Definition 4 (Superposition measures and solutions) .

We say that $\boldsymbol{\eta}\in\mathscr{P}(\mathbb{R}^{d}\times\Sigma_{T})$ is a superposition measure generated by $v(\cdot,\cdot)$ if it is concentrated on the pairs $(x,\sigma)\in\mathbb{R}^{d}\times\textnormal{AC}([0,T],\mathbb{R}^{d})$ such that

\sigma(0)=x\qquad\text{and}\qquad\dot{\sigma}(t)=v(t,\sigma(t)),

(10)

for $\mathscr{L}^{1}$ -almost every $t\in[0,T]$ . We further say that a distributional solution $\mu(\cdot)\in C^{0}([0,T],\mathscr{P}(\mathbb{R}^{d}))$ of ( 8 ) is a superposition solution if there exists a superposition measure $\boldsymbol{\eta}\in\mathscr{P}(\mathbb{R}^{d}\times\Sigma_{T})$ generated by $v(\cdot,\cdot)$ such that $\mu(t)=(e_{t})_{\#}\boldsymbol{\eta}$ for all times $t\in[0,T]$ .