An implication zroupoid $\mathbf{A}=\langle A,\to,0\rangle$ is a De Morgan algebra ( $\mathbf{DM}$ -algebra, for short ) if $\mathbf{A}$ satisfies the axiom:

• (DM)

  $(x\to y)\to x\approx x.$

A $\mathbf{DM}$ -algebra $\mathbf{A}=\langle A,\to,0\rangle$ is a Kleene algebra ( $\mathbf{KL}$ -algebra, for short ) if $\mathbf{A}$ satisfies the axiom:

• (KL ${}_{1}$ )

  $(x\to x)\to(y\to y)^{\prime}\approx x\to x$

or, equivalently,

• (KL ${}_{2}$ )

  $(y\to y)\to(x\to x)\approx x\to x$ .

A $\mathbf{DM}$ -algebra $\mathbf{A}=\langle A,\to,0\rangle$ is a Boolean algebra ( $\mathbf{BA}$ -algebra, for short ) if $\mathbf{A}$ satisfies the axiom:

• (BA)

  $x\to x\approx 0^{\prime}$ .

An implication zroupoid $\mathbf{A}=\langle A,\to,0\rangle$ is a semilattice with $\mathrm{0}$ ( $\mathbf{SL}$ -algebra, for short ) if $\mathbf{A}$ satisfies the axioms:

• (SM1)

  $x^{\prime}\approx x$

• (SM2)

  $x\to y\approx y\to x$ . (Commutativity) .

We denote by $\mathbf{DM}$ , $\mathbf{KL}$ , $\mathbf{BA}$ and $\mathbf{SL}$ , respectively, the variety of $\mathbf{DM}$ -algebras, $\mathbf{KL}$ -algebras, $\mathbf{BA}$ -algebras, and $\mathbf{SL}$ -algebras.

$\mathbf{I}_{2,0}$ denotes the subvariety of $\mathbf{I}$ defined by the identity:

$$x^{\prime\prime}\approx x.$$

A $\pi$ - $\varepsilon$ -cocycle on $(A,\varepsilon)$ is called Gaussian if it is a derivation, i.e. if

$$\eta(ab)=\varepsilon(a)\eta(b)+\eta(b)\varepsilon(b)$$

for $a,b\in A$ .

A generating functional $\psi$ on $(A,\varepsilon)$ is called Gaussian , if $\psi|_{K_{3}}=0$ .

If $(\pi,\eta,\psi)$ is a Schürmann triple over $(A,\varepsilon)$ , then $\eta$ is Gaussian if and only if $\psi$ is Gaussian, in which case we call the Schürmann triple Gaussian .

Let $f$ be a $\mathbb{D}$ -valued 2-functional on $\mathcal{M}\times\mathcal{N}$ . If $f$ is such that for each $\alpha,\beta\in\mathbb{D}$ and for all $x,y\in\mathcal{M}$ and $z,w\in\mathcal{N}$ we have:

1. (i)

   $f(x+y,z+w)=f(x,z)+f(y,z)+f(x,w)+f(y,w)$ ,

2. (ii)

   $f(\alpha x,\beta z)=\alpha\beta f(x,z)$ ,

then $f$ is called a $\mathbb{D}$ -linear 2-functional with domain $\mathcal{M}\times\mathcal{N}$ . Further, it is easy to show that if $x$ and $y$ are linear dependent in $X$ , then $f(x,y)=0$ for $(x,y)\in\mathcal{M}\times\mathcal{N}$ .
Let $f:\mathcal{M}\times\mathcal{N}\rightarrow\mathbb{D}$ be a $\mathbb{D}$ -linear 2-functional. For any $x,z\in\mathcal{M}\times\mathcal{N}$ , one can write

$\displaystyle f(x,z)=\phi(x,z)+\textbf{k}\psi(x,z)=e_{1}f_{1}(x,z)+e_{2}f_{2}(x,z),$

(4.1)

where $\phi(x,z)\in\mathbb{R}$ , $\psi(x,z)\in\mathbb{R}$ , $f_{1}(x,z)\in\mathbb{R}$ and $f_{2}(x,z)\in\mathbb{R}$ with $f_{1}=\phi+\psi$ and $f_{2}=\phi-\psi$ .

Let us first show that $f_{1},f_{2}:\mathcal{M}_{\mathbb{R}}\times\mathcal{N}_{\mathbb{R}}\rightarrow\mathbb{R}$ are real linear 2-functionals, where $\mathcal{M}_{\mathbb{R}}$ and $\mathcal{N}_{\mathbb{R}}$ are real linear subspaces of $X$ .
Given $\alpha=\alpha_{1}+\textbf{k}\alpha_{2}$ , $\beta=\beta_{1}+\textbf{k}\beta_{2}\in\mathbb{D}$ with $x,y\in\mathcal{M}$ and $z,w\in\mathcal{N}$ , one has:

	$\displaystyle f(x+y,z+w)=e_{1}f_{1}(x+y,z+w)+e_{2}f_{2}(x+y,z+w)\;\;\text{implies}$
	$\displaystyle f(x,z)+f(y,z)+f(x,w)+f(y,w)=e_{1}f_{1}(x+y,z+w)+e_{2}f_{2}(x+y,z+w).$

Then $e_{1}(f_{1}(x,z)+f_{1}(y,z)+f_{1}(x,w)+f_{1}(y,w))+e_{2}(f_{2}(x,z)+f_{2}(y,z)$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+f_{2}(x,w)+f_{2}(y,w))=e_{1}f_{1}(x+y,z+w)+e_{2}f_{2}(x+y,z+w).$

Hence $f_{1}(x,z)+f_{1}(y,z)+f_{1}(x,w)+f_{1}(y,w)=f_{1}(x+y,z+w)$ and

$$f_{2}(x,z)+f_{2}(y,z)+f_{2}(x,w)+f_{2}(y,w)=f_{2}(x+y,z+w).$$

Further, $f(\alpha x,\beta z)=e_{1}f_{1}(\alpha x,\beta z)+e_{2}f_{2}(\alpha x,\beta z)$ implies
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\alpha\beta f(x,z)=e_{1}f_{1}(\alpha x,\beta z)+e_{2}f_{2}(\alpha x,\beta z).$

That is, $e_{1}((\alpha_{1}+\alpha_{2})(\beta_{1}+\beta_{2})f_{1}(x,z))+e_{2}((\alpha_{1}-\alpha_{2})(\beta_{1}-\beta_{2})f_{2}(x,z))$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=e_{1}f_{1}((\alpha_{1}+\textbf{k}\alpha_{2})x,(\beta_{1}+\textbf{k}\beta_{2})z)+e_{2}f_{2}((\alpha_{1}+\textbf{k}\alpha_{2})x,(\beta_{1}+\textbf{k}\beta_{2})z)$ .

This implies $(\alpha_{1}+\alpha_{2})(\beta_{1}+\beta_{2})f_{1}(x,z)=f_{1}((\alpha_{1}+\textbf{k}\;\alpha_{2})x,(\beta_{1}+\textbf{k}\;\beta_{2})z)$ and

$$(\alpha_{1}-\alpha_{2})(\beta_{1}-\beta_{2})f_{2}(x,z)=f_{2}((\alpha_{1}+\textbf{k}\;\alpha_{2})x,(\beta_{1}+\textbf{k}\;\beta_{2})z).$$

In particular, setting $\alpha_{2}=0$ and $\beta_{2}=0$ , we have:

$$\alpha_{1}\beta_{1}f_{1}(x,z)=f_{1}(\alpha_{1}x,\beta_{1}z)\;\;\text{and}\;\;\alpha_{1}\beta_{1}f_{2}(x,z)=f_{2}(\alpha_{1}x,\beta_{1}z).$$

Thus, the mappings $f_{1}$ and $f_{2}$ are real linear 2-functionals on $\mathcal{M}_{\mathbb{R}}\times\mathcal{N}_{\mathbb{R}}$ . Similarly one can show that $\phi$ and $\psi$ are also real linear 2-functionals on $\mathcal{M}_{\mathbb{R}}\times\mathcal{N}_{\mathbb{R}}$ .

Given $m$ , $s$ , and $M$ such that $m<s<M$ , let $\kappa=\frac{1}{2}(s-2m-M)$ , then the cuspon with exponential decay and speed $s+\kappa$ is defined through

$$u(t,x)=\phi(x-(s+\kappa)t)+\kappa$$

where $\phi(x)$ is implicitly given through

(4.1)

$$\phi_{x}^{2}=\frac{(M-\phi)(\phi-m)^{2}}{(s-\phi)}$$

and satisfies

(4.2a)		$\displaystyle\phi(-x)=\phi(x)$
(4.2b)		$\displaystyle\phi(0)=s$
(4.2c)		$\displaystyle\phi_{x}(x)<0\quad\text{for}\quad x>0$
(4.2d)		$\displaystyle\lim_{x\to\infty}\phi(x)=m.$

A groupoid over a set $M$ is a set $G$ equipped with source and target mappings $\alpha,\beta:G\to M$ , a multiplication map $m$ from $G_{2}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}\{(g,h)\in G\times G|\ \beta(g)=\alpha(h)\}$ to $G$ , an injective units mapping $\epsilon:M\rightarrow G$ , and an inversion mapping $\iota:G\rightarrow G$ , satisfying the following properties (where we write $gh$ for $m(g,h)$ and $g^{-1}$ for $\iota(g)$ ):

• •

  (associativity) $g(hk)=(gh)k$ in the sense that, if one side of the equation is defined, so is the other, and then they are equal;

• •

  (identities) $\epsilon(\alpha(g))g=g=g\epsilon(\beta(g))$ ;

• •

  (inverses) $gg^{-1}=\epsilon(\alpha(g))$ and $g^{-1}g=\epsilon(\beta(g))$ .

A semiloopoid will be called a left inverse semiloopoid if there is a left inversion map $\mathchar 28947\relax_{l}:G\to G$ such that for each $(g,h)\in G_{2}$ also $(\mathchar 28947\relax_{l}(g),gh)\in G_{2}$ and $\mathchar 28947\relax_{l}(g)(gh)=h$ . A right inverse semiloopoid can be defined analogously.

A semiloopoid will be called an inverse semiloopoid if there is an inversion map $\mathchar 28947\relax:G\to G$ , to be denoted simply by $\mathchar 28947\relax(g)=g^{-1}$ , such that, for each $(g,h),(u,g)\in G_{2}$ , also $(g^{-1},gh),(ug,g^{-1})\in G_{2}$ and

$$g^{-1}(gh)=h\,,\quad(ug)g^{-1}=u\,.$$

Let $A,B$ be two $C^{*}$ -algebras. Let $\alpha:\text{K}_{0}A\rightarrow K_{0}B$ be a scaled ordered homomorphism, and $\xi:\text{T}B\rightarrow\text{T}A$ be an affine map. We say that $\alpha$ and $\xi$ are compatible if

$$\tau(\alpha(x))=(\xi(\tau))(x)$$

for all $x\in\text{K}_{0}A$ and $\tau\in\text{T}B$ .

For a process transition label $\lambda$ , define $\mathop{\mathsf{sel}}(\lambda)$ by

	$$\displaystyle\mathop{\mathsf{sel}}(k!v)=\mathop{\mathsf{sel}}(k?v)=\mathop{\mathsf{sel}}(τ)=\emptyset$$
	$$\displaystyle\mathop{\mathsf{sel}}(k\&l)=\mathop{\mathsf{sel}}(k⊕l)=\mathop{\mathsf{sel}}(\tau:l)=l$$

Given a trace $\alpha$ we lift $\mathop{\mathsf{sel}}(-)$ pointwise, that is, $\mathop{\mathsf{sel}}(\alpha)=\{\mathop{\mathsf{sel}}(\lambda)|\alpha=\phi\lambda\alpha^{\prime}\}$ .