(median algebra, first definition) A median algebra is a set $X$ endowed with a ternary operation $(a,b,c)\mapsto m(a,b,c)$ such that:

• (1)

  $m(a,a,b)=a$ ;

• (2)

  $m(a,b,c)=m(b,a,c)=m(b,c,a)$ ;

• (3)

  $m(m(a,b,c),d,e)=m(a,m(b,d,e),m(c,d,e))$ .

Property (3) can be replaced by $(3^{\prime})$ $m(a,m(a,c,d),m(b,c,d))=m(a,c,d)$ .

The element $m(a,b,c)$ is the median of the points $a,b,c$ . In a median algebra $(X,m)$ , given any two points $a,b$ the set $I(a,b)=\{x\;;\;x=m(a,b,x)\}$ is called the interval of endpoints $a,b$ . This defines a map $I:X\times X\to{\mathcal{P}}(X)$ . We say that a point $x\in I(a,b)$ is between $a$ and $b$ .

A homomorphism of median algebras is a map $f:(X,m_{X})\to(Y,m_{Y})$ such that $m_{Y}(f(x),f(y),f(z))=f(m_{X}(x,y,z))$ . Equivalently, $f$ is a homomorphism if and only if it preserves the betweenness relation. If moreover $f$ is injective (bijective) then $f$ is called embedding or monomorphism (respectively isomorphism ) of median algebras.

The integral functional ( P ) it said to be invariant under the $\varepsilon$ -parameter infinitesimal transformations

$$\begin{cases}\bar{t}=t+\varepsilon\tau(t,q)+o(\varepsilon)\,,\\ \bar{q}(t)=q(t)+\varepsilon\xi(t,q)+o(\varepsilon)\,,\\ \end{cases}$$

(3)

where $\tau$ and $\xi$ are piecewise-smooth, if

$$\int_{t_{a}}^{t_{b}}L\left(t,q(t),\dot{q}(t)\right)dt=\int_{\bar{t}(t_{a})}^{\bar{t}(t_{b})}L\left(\bar{t},\bar{q}(\bar{t}),\dot{\bar{q}}(\bar{t})\right)d\bar{t}$$

(4)

for any subinterval $[{t_{a}},{t_{b}}]\subseteq[a,b]$ .

The parameters $(a,\alpha,b,\beta,c,\gamma,m,\mu)$ are called admissible for a process with state space $\mathbb{R}_{\geqslant 0}$ if

	$$\displaystyle a=0,$$
	$$\displaystyle\alpha,b,c,\gamma\in\mathbb{R}_{\geqslant 0}\,,$$
	$$\displaystyle\beta\in\mathbb{R}\,,$$
	$$\displaystyle\begin{split}\displaystyle m,\mu&\displaystyle\text{are L\'{e}vy measures on $(0,\infty)$, where $m$ satisfies}\\ &\displaystyle\int_{(0,\infty)}{(\xi\land 1)\,m(d\xi)}<\infty\;,\end{split}$$

and admissible for a process with state space $\mathbb{R}$ if

	$$\displaystyle a,c\in\mathbb{R}_{\geqslant 0}\,,$$
	$$\displaystyle b,\beta\in\mathbb{R}\,,$$
	$$\displaystyle m\;\text{is a L\'{e}vy measure on}\;\mathbb{R}\setminus\left\{0\right\}\;,$$
	$$\displaystyle\alpha=0,\gamma=0,\mu\equiv 0\;.$$

Moreover define the truncation functions

$$h_{F}(\xi)=\begin{cases}0&\text{if}\quad D=\mathbb{R}_{\geqslant 0}\\ \frac{\xi}{1+\xi^{2}}&\text{if}\quad D=\mathbb{R}\end{cases}\qquad\text{and}\qquad h_{R}(\xi)=\begin{cases}\frac{\xi}{1+\xi^{2}}&\text{if}\quad D=\mathbb{R}_{\geqslant 0}\\ 0&\text{if}\quad D=\mathbb{R}\end{cases},$$

and finally the functions $F(u)$ , $R(u)$ for $u\in\mathbb{C}$ as

(2.2)		$$\displaystyle F(u)=au^{2}+bu-c+\int_{D\setminus\left\{0\right\}}{\left(e^{u\xi}-1-uh_{F}(\xi)\right)\,m(d\xi)}\;,$$
(2.3)		$$\displaystyle R(u)=\alpha u^{2}+\beta u-\gamma+\int_{D\setminus\left\{0\right\}}{\left(e^{u\xi}-1-uh_{R}(\xi)\right)\,\mu(d\xi)}\;.$$

Let $j_{\varsigma}$ be the least nucleus $j$ on $\operatorname{\mathfrak{T}}(L)$ such that

$$j(a)=j(\varsigma aa)\;.$$

We define $\operatorname{\mathfrak{T}}_{\varsigma}(L,\lozenge,\blacklozenge)$ to be $\operatorname{\mathfrak{T}}(L)_{j_{\varsigma}}$ . We also write $\operatorname{\mathfrak{T}}_{\varsigma}(L)$ if $\lozenge$ and $\blacklozenge$ are clear from the context.