A dialgebra $D$ over $K$ is a vector space over $K$ along with two $K$ -linear maps $\dashv:D\otimes D\longrightarrow D$ called left and $\vdash:D\otimes D\longrightarrow D$ called right satisfying the following axioms :

$$\begin{array}[]{rcl}x\dashv(y\dashv z)&\stackrel{{\scriptstyle 1}}{{=}}&(x\dashv y)\dashv z\stackrel{{\scriptstyle 2}}{{=}}x\dashv(y\vdash z)\\ (x\vdash y)\dashv z&\stackrel{{\scriptstyle 3}}{{=}}&x\vdash(y\dashv z)\\ (x\dashv y)\vdash z&\stackrel{{\scriptstyle 4}}{{=}}&x\vdash(y\vdash z)\stackrel{{\scriptstyle 5}}{{=}}(x\vdash y)\vdash z\end{array}$$

(2.1.1)

for all $x,y,z\in D$ .

An algebra is a complex vector space $A$ equipped with an associative multiplication $(a,b)\mapsto ab$ satisfying the following laws:

	$$\displaystyle\lambda(ab)=(\lambda a)b=a(\lambda b)$$
	$$\displaystyle c(a+b)=ca+cb$$
	$$\displaystyle(a+b)c=ac+bc$$

for all $\lambda$ in $\mathds{C}$ and all $a,b,c$ in $A$ .

Let $A$ be an algebra. A unit in $A$ is a non-zero element $e$ of $A$ such that

$$ea=ae=a$$

for all $a$ in $A$ . One says that $A$ is a unital algebra if $A$ has a unit.

A Bol algebra is a vector space $\mathfrak{B}$ over a field $K$ of which is closed with respect a trilinear operation $(x;y,z)$ and with additional bilinear skew-symetric operation $x\cdot y$ satisfying:

1. (i)

   $(x;x,y)=0$

2. (ii)

   $(x;y,z)+(z;x,y)+(y;z,x)=0.$

3. (iii)

   $$((x;y,z);\alpha,\beta)=((x;\alpha,\beta);y,z)+(x;{y;\alpha,\beta},z)+(x;y,(\alpha,\beta,z))$$

4. (iv)

   $$(x\cdot y;\alpha,\beta)=(x;\alpha,\beta)\cdot y+x\cdot(y;\alpha,\beta)+(\alpha\cdot\beta;x,y)+(x\cdot y)\cdot(\alpha\cdot\beta)$$

for all $x$ , $y$ , $z$ , $\alpha$ ,and $\beta$ in $\mathfrak{B}$ .

A vector space $V$ is a Bol module if for all $\alpha,\beta,\gamma,\tau$ in $\mathfrak{B}$ and $v$ in $V$ , the following properties are true:

1. (1)

   $\alpha\cdot v=-v\cdot\alpha$

2. (2)

   $[v;\alpha,\beta]+[v;\beta,\alpha]=0$

3. (3)

   $[v;\alpha,\beta]+[\alpha;v,\beta]+[\alpha;\beta,v]=0$

4. (4)

   $[(\alpha;\beta,\gamma);v,\tau]=[\alpha;\beta,[\gamma;v,\tau]]+[\alpha;[\beta,v,\tau],\gamma]+[[\alpha;v,\tau];\beta,\gamma]$

5. (5)

   $[(\alpha;\beta,\gamma),v\cdot\tau]=v\cdot(\alpha;\beta,\tau)+[v;\alpha,\beta]\cdot\tau+[(\beta;\alpha);\tau,v]+(v\cdot\tau)\cdot(\beta;\alpha)$

Let $\mathbb{C}_{n}$ , $n\in\mathbb{Z}$ , be a filtration of partitions of $\Omega$ .

(i) Let $\tau=\tau(x)$ be a function on $\Omega$ with values in $\{\infty,0,\pm 1,\pm 2,...\}$ . The function $\tau$ is called a stopping time (relative to the filtration) if, for each $n=0,\pm 1,\pm 2,...$ , the set

$$\{x:\tau(x)=n\}$$

is either empty or else is the union of some elements of $\mathbb{C}_{n}$ .

(ii) For a function $f\in\mathcal{L}_{1}(\Omega)$ and $n\in\mathbb{Z}$ , we denote

$$f_{|n}(x)=\,\,\hbox{\rm\bf--}\kern-9.8pt\int_{C_{n}(x)}f(y)\,\mu(dy).$$

We read $f_{|n}$ as “ $f$ given $\mathbb{C}_{n}$ ”, continuing to borrow the terminology from probability theory. If we are also given a stopping time $\tau$ , we let

$$f_{|\tau}(x)=f_{|\tau(x)}(x)$$

for those $x$ for which $\tau(x)<\infty$ and $f_{|\tau}(x)=f(x)$ otherwise.

Given two hyperreal numbers $\xi$ and $\zeta\in{\mathbb{R}}^{\ast}\backslash\left\{0\right\},$ we say that they have the same order if $\xi/\zeta$ and $\zeta/\xi$ are bounded numbers and we will write

$$\xi\approx\zeta$$

(notice the difference between $"\sim"$ and $"\approx"$ since these symbols will be largely used in the rest of this paper). We say that $\xi$ has a larger order than $\zeta$ if $\xi/\zeta$ is an infinite number and we will write

$$\xi\gg\zeta$$

We say that $\xi$ has a smaller order than $\zeta$ if $\xi/\zeta$ is an infinitesimal number and we will write

$$\xi\ll\zeta$$

Define $\beta=q^{2}+q^{-2}$ and

	$\displaystyle\gamma\;=\;-a(q-q^{-1})^{2},\qquad\qquad\varrho\;=\;a^{2}(q-q^{-1})^{2}-bc(q^{2}-q^{-2})^{2},$		(66)
	$\displaystyle\gamma^{*}\;=\;-a^{*}(q-q^{-1})^{2},\quad\qquad\varrho^{*}=a^{*2}(q-q^{-1})^{2}-b^{*}c^{*}(q^{2}-q^{-2})^{2}.$		(67)

Let $M$ be a smooth manifold. A Poisson structure on $M$ is a Lie bracket $\{\cdot,\cdot\}$ on $C^{\infty}(M)$ such that each operator $\{\cdot,h\}$ is a derivation on functions, i.e.

$$\{fg,h\}=\{f,h\}g+\{g,h\}f.$$

A Poisson manifold is a manifold endowed with a Poisson structure.

We call odd composite number $n$ overpseudoprime to base 2 $(n\in\mathbb{S}_{2})$ if

(5)

$$n=r(n)h(n)+1.$$