Let $R$ be a ring, $\sigma$ an endomorphism of $R$ and $\delta$ a $\sigma$ -derivation. The Ore extension $R[x;\sigma,\delta]$ is defined as the ring generated by $R$ and an element $x\notin R$ such that $1,x,x^{2},\ldots$ form a basis for $R[x;\sigma,\delta]$ as a left $R$ -module and all $r\in R$ satisfy

$$xr=\sigma(r)x+\delta(r).$$

(1)

For a metric compatible connection $\nabla$ , recalling the definition of the torsion operator ^{8 8} 8 The torsion operator of a connection $\nabla$ is the mapping $\mathbf{\tau}:\sec TM\times\sec TM\rightarrow\sec TM$ , $(\boldsymbol{u,v})\mapsto\mathbf{\tau}(\boldsymbol{u,v})=\nabla_{\boldsymbol{u}}\boldsymbol{v}-\nabla_{v}\boldsymbol{v}-[\boldsymbol{u,v}]$ . we conveniently write

$$\mathbf{\tau(}u,v)=\mathopen{[\kern-2.2pt[\kern-2.3pt[}u,v\mathclose{]\kern-2.1pt]\kern-2.3pt]},$$

(15)

which we call the ( form ) torsion operator.

Consider two labeled graphs $G$ and $H$ in ${\cal G}_{\pi,\Sigma}$ , they are consistent if and only if for all vertices $u,v,w$ and ports $k,l,p$ we have:

$$\{u:k,v:l\}\in E(G)\wedge\{u:k,w:p\}\in E(H)\Rightarrow v=w\wedge l=p$$

We say that a function $F\colon X^{*}\to Y$ is preassociative if for every $\mathbf{x}\mathbf{y}\mathbf{y}^{\prime}\mathbf{z}\in X^{*}$ we have

$$F(\mathbf{y})~{}=~{}F(\mathbf{y}^{\prime})\quad\Rightarrow\quad F(\mathbf{x}\mathbf{y}\mathbf{z})~{}=~{}F(\mathbf{x}\mathbf{y}^{\prime}\mathbf{z}).$$

We say that a function $F\colon X^{*}\to Y$ is strongly preassociative if for every $\mathbf{x}\mathbf{x}^{\prime}\mathbf{y}\mathbf{z}\mathbf{z}^{\prime}\in X^{*}$ we have

(5)

$$F(\mathbf{x}\mathbf{z})~{}=~{}F(\mathbf{x}^{\prime}\mathbf{z}^{\prime})\quad\Rightarrow\quad F(\mathbf{x}\mathbf{y}\mathbf{z})~{}=~{}F(\mathbf{x}^{\prime}\mathbf{y}\mathbf{z}^{\prime}).$$

Let $F\subset M_{(S,\phi)}$ be a surface admitting an open book foliation. Suppose that $F$ contains a hexagon region $R$ consisting of two bb-tiles of opposite signs meeting along a b-arc as in Sketch (1) of Figure 11 . We name the vertices (elliptic points) $A,B,C,D,E,F$ counterclockwise. We may assume that ${\tt sgn}(A)={\tt sgn}(C)={\tt sgn}(E)=+1$ and ${\tt sgn}(B)={\tt sgn}(D)={\tt sgn}(F)=-1$ . We require that the boundary b-arcs $\overline{AB},\overline{CD},\overline{EF}$ lie on the same page of the open book, and likewise $\overline{BC},\overline{DE},\overline{FA}$ lie on another same page.

Let ${\bf p},{\bf q}$ denote the two hyperbolic points of $R$ . From now on we assume that

$${\tt sgn}({\bf p})=+1,\qquad{\tt sgn}({\bf q})=-1.$$

(If ${\tt sgn}({\bf p})=-1$ , ${\tt sgn}({\bf q})=+1$ similar statements hold.) With this sign assumption there are two possible movie presentations realizing the open book foliation $\mathcal{F}_{ob}(R)$ on $R$ . See Figure 12 . We call them Type1 and Type2 .

[ 28 ] If $A$ is an MV-algebra then a function $s:A\to[0,1]$ is a state if the following properties are satisfied for any $x$ , $y\in A$ :

1. (s1)

   if $x\odot y=0$ then $s(x\oplus y)=s(x)+s(y)$ ,

2. (s2)

   $s(1)=1$ .

A Riesz MV-algebra is a structure

$(R,\cdot,\oplus,^{*},0)$ ,

where $(R,\oplus,^{*},0)$ is an MV-algebra and the operation $\cdot:[0,1]\times R\rightarrow R$ satisfies the following identities for any $r$ , $q\in[0,1]$ and $x$ , $y\in R$ :

1. (RMV1)

   $r\cdot(x\odot y^{*})=(r\cdot x)\odot(r\cdot y)^{*}$ ,

2. (RMV2)

   $(r\odot q^{*})\cdot x=(r\cdot x)\odot(q\cdot x)^{*}$ ,

3. (RMV3)

   $r\cdot(q\cdot x)=(rq)\cdot x$ ,

4. (RMV4)

   $1\cdot x=x$ .

In the following we write $rx$ instead of $r\cdot x$ for $r\in[0,1]$ and $x\in R$ . Note that $rq$ is the real product for any $r$ , $q\in[0,1]$ .

Suppose $\delta\in(0,1]$ , $\rho\in(0,1/2]$ and $\alpha>0$ . Given $\zeta>0$ , let us write

$$a^{\ast}(\delta,\rho;\zeta)\stackrel{{\scriptstyle\rm def}}{{=}}a(\delta,\rho)+\zeta,$$

(5.6)

where $a(\delta,\rho)$ is defined in ( 5.2 ), let us write $\{\Gamma_{i}:i\in S_{n}\}$ for the set of all possible support sets of cardinality $k$ , and let us disjointly partition $S_{n}\stackrel{{\scriptstyle\rm def}}{{=}}\Theta^{1}_{n}\cup\Theta^{2}_{n}$ such that

$$\Theta^{1}_{n}\stackrel{{\scriptstyle\rm def}}{{=}}\left\{i\in S_{n}\;\;:\;\;\|x^{\ast}_{\Lambda\setminus\Gamma_{i}}\|>\sigma\cdot a^{\ast}(\delta,\rho;\zeta)\right\};\;\;\;\;\Theta^{2}_{n}\stackrel{{\scriptstyle\rm def}}{{=}}\left\{i\in S_{n}\;\;:\;\;\|x^{\ast}_{\Lambda\setminus\Gamma_{i}}\|\leq\sigma\cdot a^{\ast}(\delta,\rho;\zeta)\right\}.$$

(5.7)

Let $\SS$ be a semiring. A map $\tau:\SS\to\SS$ is a symmetry if

(2.1a)		$\displaystyle\tau(a+b)=\tau(a)+\tau(b)$
(2.1b)		$\displaystyle\tau(0)=0$
(2.1c)		$\displaystyle\tau(a\cdot b)=a\cdot\tau(b)=\tau(a)\cdot b$
(2.1d)		$\displaystyle\tau(\tau(a))=a.$

Let $\rho:[0,1)^{d}\to\{\pm 1\}$ be a periodic function with

$$\int\rho\,dx=0.$$

Let $\rho_{+}$ and $\rho_{-}$ be the probability densities defined by

$$\rho_{+}=2\max\{\rho,0\}\quad\mbox{and}\quad\rho_{-}=-2\min\{\rho,0\},$$

so that $\rho=\frac{1}{2}(\rho_{+}-\rho_{-})$ . We then define the MKR mixing measure as

$$D(\rho):=\exp\left(d_{\ln}(\rho_{+},\rho_{-})\right).$$

Let $g\in\mathcal{G}^{0}_{2}(M)$ be a generalized metric of Lorentzian signature. Then the generalized space-time $(M,g)$ is said to be geodesically complete if every geodesic can be defined on $\mathbb{R}$ , i.e., every solution of the geodesic equation

$$c^{\prime\prime}=0$$

is in $\mathcal{G}[\mathbb{R},M]$ .

We define the exponential tensor $\_\odot\_:End(\mathbb{N})\times End(\mathbb{N})\rightarrow End(\mathbb{N})$ in terms of the exponential bijection of Definition 4.5 above, as follows:

$$f\odot g\ =\ \psi(f\times g)\psi^{-1}$$

Diagramatically,

$$\xymatrix{\mathbb{N}\times\mathbb{N}\ar[rr]^{f\times g}&&\mathbb{N}\times\mathbb{N}\ar[d]^{\psi}\\ \mathbb{N}\ar[u]^{\psi^{-1}}\ar[rr]_{f\odot g}&&\mathbb{N}}$$

A measure $\mathsf{x}\in\mathcal{X}$ is a DE fixed point for the LDPC( $\lambda,\rho$ ) ensemble if

$\displaystyle\mathsf{x}=\mathsf{c}\varoast\lambda^{\varoast}\left(\rho^{\boxast}(\mathsf{x})\right).$

Consider the following terms.

$${\sf suc}(x)=1;(\breve{p};x;\breve{q})$$

and

$${\sf pred}(x)=\breve{p};{\sf rng}x;q.$$

A bijective, anti-linear operator $\theta:\mathsf{h}\longrightarrow\mathsf{h}$ is called anti-unitary if

$\displaystyle\langle\theta u,\theta v\rangle=\langle v,u\rangle,\ \ \ \text{ for all }x,y\in\mathsf{h}.$

A time bound $\beta$ is said to be of Type 1 iff

$$\beta(n)=n\cdot\alpha(n)$$

with a nondecreasing function $\alpha$ such that $2\mbox{$\leq_{\mbox{\footnotesize\rm a}}$}\alpha(n)\mbox{$\leq_{\mbox{\footnotesize\rm a}}$}n$ and $\alpha$ is f-consistent and constructible in linear time.