A semitorsor is a set $G$ together with a map $G^{3}\to G$ , $(x,y,z)\mapsto(xyz)$ such that the following identity, called the para-associative law , holds:

$$(xy(zuv))=(x(uzy)v)=((xyz)uv)\,.$$

A torsor is a semitorsor in which, moreover, the following idempotent law holds:

$$(xxy)=y=(yxx)\,.$$

The commutator in an algebra is the bilinear operation

$$[a,b]=ab-ba.$$

This operation is anticommutative : it satisfies $[a,b]+[b,a]\equiv 0$ .

The anticommutator in an algebra is the bilinear operation

$$a\circ b=ab+ba;$$

we omit the scalar $\frac{1}{2}$ . This operation is commutative : it satisfies $a\circ b-b\circ a\equiv 0$ .

The dicommutator in a dialgebra is the bilinear operation

$$\langle a,b\rangle=a\dashv b-b\vdash a.$$

In general, this operation is not anticommutative.

The antidicommutator in a dialgebra is the bilinear operation

$$a\star b=a\dashv b+b\vdash a.$$

In general, this operation is not commutative.

For two vectors $u=[m,n]^{T}$ and $w=[m^{\prime},n^{\prime}]^{T}$ , the value of the symplectic form $\omega$ is defined as

$$\omega(u,w)=mn^{\prime}-nm^{\prime}.$$

(4.3)

Conjugation $A^{\ast}$ of a matrix $A$ representing an endomorphism in $\mathbf{E}$ is the adjugate matrix, namely

$$\hbox{if }A=\left[{{\begin{array}[]{*{20}c}a&b\\ c&d\\ \end{array}}}\right]\hbox{ then }A^{\ast}=\left[{{\begin{array}[]{*{20}c}d&{-b}\\ {-c}&a\\ \end{array}}}\right]$$

(4.4)

Conjugation of vectors in $\mathbf{E}$ is a map into the dual space, expressed in terms of matrices as

$$\left[{{\begin{array}[]{*{20}c}m\\ n\\ \end{array}}}\right]^{\ast}\quad=\quad\left[{{\begin{array}[]{*{20}c}{-n}&m\\ \end{array}}}\right]\,.$$

(4.5)

Now, the symplectic product may be performed via matrix multiplication: $\omega(u,w)=u^{\ast}w$ . The map defined by ( 4.5 ) is the symplectic conjugation of the spinor. Also, note that $AA^{*}=A^{*}A=\det(A)I$ .

$\phi$ is zero-preserving if

$$\phi(\emptyset)=0\,.$$

(4)

$\phi$ is unital if

$$\phi({{\mathbf{1}}})=1\,.$$

(5)

$\phi$ is MP if

$$\phi(A\to B)=1\,,\ \phi(A)=1\;\Rightarrow\;\phi(B)=1,\ \ \ \forall A,B\in\mathfrak{A}\,.$$

(9)

$\phi$ is C1 if

$$\phi(A)=1\;\Rightarrow\;\phi({\mathbf{1}}+A)=0,\ \ \ \forall A\in\mathfrak{A}.$$

(10)

$\phi$ is C2 if

$$\phi(A)=0\;\Rightarrow\;\phi({\mathbf{1}}+A)=1,\ \ \ \forall A\in\mathfrak{A}.$$

(11)

A co-event $\phi$ is preclusive if

$$\mu(A)=0\Rightarrow\phi(A)=0,\ \ \ \forall A\in\mathfrak{A}\,.$$

(13)

Let $A$ and $B$ be subgroups of $S_{X}$ and $S_{Y}$ , respectively, where $X$ and $Y$ are finite disjoint sets. Then we define an action of the group $A\times B$ on the set $Y^{X}$ of functions from $X$ to $Y$ in the following way: the image of $f\in Y^{X}$ under $(\alpha,\beta)\in A\times B$ is given by

$$f^{(\alpha,\beta)}(x)=\left(f\left(x^{\alpha}\right)\right)^{\beta}$$

for all $x\in X$ . We will refer to $A\times B$ with this action as a power group .

Let $u$ and $v$ be two distinct points, and the line $(u,v)$ join the points $p$ and $q$ . We call $q($ or $p)$ the characteristic mapping point of $p($ or $q)$ with respect to the basic points $u$ and $v$ if

$$[u,v;p,q]=1,$$

and denote $q=\chi_{(u,v)}(p)$ (or $p=\chi_{(u,v)}(q)$ ).

Given two words $w$ and $w^{\prime}$ in ${{\mathbb{P}}}^{*}$ , $w$ is said to be below $w^{\prime}$ if every letter in $w$ is smaller than every letter in $w^{\prime}$ . The partial order $\ll$ on ${{\mathbb{P}}}^{*}$ is defined as follows:

$\displaystyle w\ll w^{\prime}\iff\text{$w=w^{\prime}$ or $w$ is below $w^{\prime}$.}$

(1.1)

Two words $w$ and $w^{\prime}$ are said to be comparable if $w\ll w^{\prime}$ or $w\gg w^{\prime}$ , otherwise they are incomparable.

Let $X$ and $X^{\prime}$ be two torsors under the action of $G$ . A map $f:X\to X^{\prime}$ is a $G$ -morphism of torsors if it is $G$ -equivariant, i.e. for any $x\in X$ and any $g\in g$ ,

$$f(g\cdot x)=g\cdot f(x)\;.$$

In other words, we ask for the following diagram to be commutative :

$$\xymatrix{G\times X\ar[rr]^{\gamma}\ar[d]_{{\rm id}_{G}\times f}&&X\ar[d]^{f}\\ G\times X\ar[rr]_{\gamma^{\prime}}&&X^{\prime}}$$

By $\Gamma^{\sharp}$ we denote the alphabet $\Gamma\cup\{\sharp\}$ where $\sharp$ is a special symbol not in $\Gamma$ . Let $p$ be any $d$ -picture of domain $[n]^{d}$ on $\Gamma$ . The bordered $d$ -picture of $p$ , denoted by $p^{\sharp}$ , is the $d$ -picture $p^{\sharp}:[0,n+1]^{d}\to\Gamma^{\sharp}$ defined by

$$p^{\sharp}(a)=\left\{\begin{array}[]{l}p(a)\textrm{ if }a\in dom(p);\\ \sharp\textrm{ otherwise}.\end{array}\right.$$

Here, "otherwise" means that $a$ is on the border of $p^{\sharp}$ , that is, some component $a_{i}$ of $a$ is $\mathrm{0}$ or $n+1$ .

To each $d$ -picture $p$ of length $n$ , one associates its squared $d$ -picture denoted $p^{=}$ of domain $[n]^{d}$ obtained by putting the new special symbol $\square$ in each additional cell. Formally,

$$p^{=}(a)=\left\{\begin{array}[]{l}p(a)\textrm{ if }a\in dom(p);\\ \square\textrm{ otherwise}.\end{array}\right.$$

To any $d$ -picture language $L$ one associates its squared $d$ -picture language

$$L^{=}=\{p^{=}:p\in L\}.$$

A structure $(X,+,0,\mathop{\leftrightarrow},\cdot,1)$ is a disring when $(X,+,0,\mathop{\leftrightarrow})$ is a disgroup, $(X,+,0,\cdot,1)$ a unital commutative semiring, and for $\mathop{\leftrightarrow}$ the additional property

• •

  $(a\mathop{\leftrightarrow}b)\cdot x=(a\cdot x)\mathop{\leftrightarrow}(b\cdot x)$ (distributivity, or homogeneity)

holds for all $a,b,x\in X$ . The order of operations is to first evaluate $\cdot$ , then $\mathop{\leftrightarrow}$ , then $+$ .

A structure $(X,+\colon X\times X\to X,\mathop{\leftrightarrow}\colon X\times X\to X,0,\cdot\colon\mathbb{D}\times X\to X$ is a module over a disring $\mathbb{D}$ when $(X,+,\mathop{\leftrightarrow},0)$ is a disgroup and the following holds for all $x,y\in X$ , $\lambda,\mu\in\mathbb{D}$ :

• •

  $\mu\cdot(\lambda\cdot x)=(\lambda\mu)\cdot x$ ,

• •

  $\lambda\cdot(x\mathop{\leftrightarrow}y)=(\lambda\cdot x)\mathop{\leftrightarrow}(\lambda\cdot y)$ ,

• •

  $1\cdot x=x$ ,

• •

  $0\cdot x=0$ .

Furthermore, if the disring $\mathbb{D}$ is ordered, we define that the module $(X,+,\mathop{\leftrightarrow},0,\cdot,\|{\text{{---}}}\|)$ is normed when the operation $\|{\text{{---}}}\|\colon X\to\mathbb{D}$ has the properties

• •

  $\|{x}\|=0\implies x=0$ ,

• •

  $\|{\lambda\cdot x}\|=\lambda\|{x}\|$ (in particular $\|{0}\|=0$ ),

• •

  $\|{a}\|\mathop{\leftrightarrow}\|{b}\|\leq\|{a\mathop{\leftrightarrow}b}\|$ .