A trialgebra $(A,*,\Delta,\cdot)$ with $*$ and $\cdot$ associative products on a vector space $A$ (where $*$ may be partially defined, only) and $\Delta$ a coassociative coproduct on $A$ is given if both $(A,*,\Delta)$ and $(A,\cdot,\Delta)$ are bialgebras and the following compatibility condition between the products is satisfied for arbitrary elements $a,b,c,d\in A$ :

$$(a*b)\cdot(c*d)=(a\cdot c)*(b\cdot d)$$

whenever both sides are defined.

Let the function $\phi:(0,\infty)^{2}\to\mathbb{R}$ be given by

$$\phi(\alpha,\beta)=(\beta+1)\log(\beta+1)-\beta\log\beta+\beta\log\alpha-(\beta+1)\log(\alpha+1).$$

Let $u,x\in S$ and $d\geq 0$ . The equation (see 1.4 ( 3 ))

$$x\left(t\right)=u(t-d)$$

is called the fixed delay condition (FDC). The delay defined by this equation is also called pure, ideal or non-inertial. A delay different from FDC is called inertial.

[ M , Definition 2.14] . A matched pair of groupoids is a pair of groupoids $({\mathcal{V}},{\mathcal{H}})$ over ${\mathcal{P}}$ with ${\mathcal{V}}$ denoted vertically and ${\mathcal{H}}$ horizontally, endowed with a left action $\,{\rightharpoonup}\,:{\mathcal{H}}{}_{r}\hskip-2.845276pt\times_{t}{\mathcal{V}}\to{\mathcal{V}}$ of ${\mathcal{H}}$ on $t:{\mathcal{V}}\to{\mathcal{P}}$ , and a right action $\,{\leftharpoonup}\,:{\mathcal{H}}{}_{r}\hskip-2.845276pt\times_{t}{\mathcal{V}}\to{\mathcal{H}}$ of ${\mathcal{V}}$ on $r:{\mathcal{H}}\to{\mathcal{P}}$ , satisfying

(1.7)		$\displaystyle b(x\,{\rightharpoonup}\,g)=l(x\,{\leftharpoonup}\,g),$
(1.8)		$\displaystyle x\,{\rightharpoonup}\,fg=(x\,{\rightharpoonup}\,f)((x\,{\leftharpoonup}\,f)\,{\rightharpoonup}\,g),$
(1.9)		$\displaystyle xy\,{\leftharpoonup}\,g=(x\,{\leftharpoonup}\,(y\,{\rightharpoonup}\,g))(y\,{\leftharpoonup}\,g),$

for all $f,g\in{\mathcal{V}}$ , $x,y\in{\mathcal{H}}$ such that the compositions are possible.

Let us first say that a quiver in a category ${\mathcal{C}}$ is a pair of arrows $\mathfrak{s},\mathfrak{e}:{\mathfrak{A}}\to{\mathfrak{P}}$ in ${\mathcal{C}}$ .

A double quiver is a quiver in the category $\operatorname{Quiv}$ of all quivers. That is, in the “vertical and horizontal” notation, a double quiver is a pair of morphisms of quivers $t,b:{\mathfrak{B}}\to{\mathfrak{H}}$ , where ${\mathfrak{B}}$ and ${\mathfrak{H}}$ are quivers in the usual sense: $l,r:{\mathfrak{B}}\to{\mathfrak{V}}$ , $l,r:{\mathfrak{H}}\to{\mathcal{P}}$ , and $t,b$ should preserve $l,r$ :

(4.5)

$$tr=rt,\qquad tl=lt,\qquad br=rb,\qquad bl=lb.$$

In short, a double quiver is a collection of sets and maps

$$\begin{matrix}\qquad{\mathfrak{B}}&\overset{t,b}{\rightrightarrows}&{\mathfrak{H}}\\ l,r\downdownarrows&&\downdownarrows l,r\\ \qquad{\mathfrak{V}}&\underset{t,b}{\rightrightarrows}&{\mathcal{P}}\end{matrix}$$

satisfying ( 4.5 ). By abuse of notation we shall say that $({\mathfrak{B}},{\mathfrak{V}},{\mathfrak{H}})$ is a double quiver over ${\mathcal{P}}$ ; or alternatively that ${\mathfrak{B}}$ is a double quiver with sides in ${\mathfrak{V}}$ and ${\mathfrak{H}}$ ; or that ${\mathfrak{B}}$ is a double quiver with sides in ${\mathcal{A}}$ in case ${\mathfrak{V}}={\mathfrak{H}}={\mathcal{A}}$ . An element $B$ of ${\mathfrak{B}}$ is called an oriented box and depicted as a box

$$B=\begin{matrix}\quad t\\ l\,\,\begin{tabular}[]{|p{0,1cm}|}\hline\hbox{}\\ \hline\hbox{}\end{tabular}\,\,r\\ \quad b\end{matrix}$$

where $t=t(B)$ , $b=b(B)$ , $r=r(B)$ , $l=l(B)$ , and the four corners are $tl(B)$ , $tr(B)$ , $bl(B)$ , $br(B)$ . In this picture, we keep in mind the orientations top-to-bottom and left-to-right. Morphisms of double quivers, or of double quivers over ${\mathcal{P}}$ , are defined in the standard way.

Let ${\mathcal{P}}$ be a set and ${\mathcal{A}}$ , ${\mathcal{B}}$ be quivers over ${\mathcal{P}}$ denoted vertically and horizontally, respectively. The coarse double quiver with sides in ${\mathcal{A}}$ and ${\mathcal{B}}$ is the collection $({\mathfrak{V}}\boxplus{\mathfrak{H}},{\mathfrak{V}},{\mathfrak{H}})$ where ${\mathfrak{V}}\boxplus{\mathfrak{H}}$ is the set of all quadruples $\begin{pmatrix}\quad x\\ f\quad g\\ \quad y\end{pmatrix}$ with $x,y\in{\mathfrak{H}}$ , $f,g\in{\mathfrak{V}}$ such that

(4.6)

$$l(x)=t(f),\quad r(x)=t(g),\quad l(y)=b(f),\quad r(y)=b(g).$$

Such a quadruple is called a face . We omit the obvious description of the arrows.

If $({\mathfrak{B}},{\mathfrak{V}},{\mathfrak{H}})$ is a double quiver over ${\mathcal{P}}$ then there are maps $\Theta:{\mathfrak{B}}\to{\mathfrak{V}}\boxplus{\mathfrak{H}}$ , $\Xi:{\mathfrak{B}}\to{\mathfrak{H}}{}_{r}\hskip-2.845276pt\times_{l}{\mathfrak{V}}$ given by

$$\Theta\left(\begin{matrix}\quad x\\ f\,\,\begin{tabular}[]{|p{0,1cm}|}\hline\hbox{}\\ \hline\hbox{}\end{tabular}\,\,g\\ \quad y\end{matrix}\right)=\begin{pmatrix}\quad x\\ f\quad g\\ \quad y\end{pmatrix},\qquad\Xi\left(\begin{matrix}\quad x\\ f\,\,\begin{tabular}[]{|p{0,1cm}|}\hline\hbox{}\\ \hline\hbox{}\end{tabular}\,\,g\\ \quad y\end{matrix}\right)=(x,g),\qquad\begin{matrix}\quad x\\ f\,\,\begin{tabular}[]{|p{0,1cm}|}\hline\hbox{}\\ \hline\hbox{}\end{tabular}\,\,g\\ \quad y\end{matrix}\in{\mathfrak{B}}.$$

Clearly $\Theta$ is a morphism of double quivers.

We shall say that $({\mathfrak{B}},{\mathfrak{V}},{\mathfrak{H}})$ is thin if $\Theta$ is injective (any box is determined by its sides) and $\Xi$ is surjective.

We shall say that $({\mathfrak{B}},{\mathfrak{V}},{\mathfrak{H}})$ is vacant if $\Xi$ is bijective.

Let $\xi:(0,1)\times(0,\infty)\to\mathbb{R}$ be given by

$$\xi(p,\beta)=\beta\log\beta-(\beta+1)\log(\beta+1)-\log{p}-\beta\log(1-p).$$

Let $w\in B_{n+1}^{+}$ , $w=u\sigma v$ . We note

$$w=u\sigma v\stackrel{{\scriptstyle 1}}{{\rightarrow}}uv$$

to say that we pass from $w$ to $uv$ by eliminating one generator.This last is said to be a 1-sequence. So for any $X$ in $B_{n+1}^{+}$ ,the k-sequence

$$X\stackrel{{\scriptstyle k}}{{\rightarrow}}Y$$

means that we pass from $X$ to $Y$ by eliminating $k$ generators,i.e there exists $k+1$ positive words $L_{j}$ with $0\leq{j\leq{k}}$ such as

$$L_{k}=X$$

$$L_{j+1}\stackrel{{\scriptstyle 1}}{{\rightarrow}}L_{j}$$

$$L_{0}=Y$$