a $ be an affine object in the topos $\E $. 2.1. [*An order*]{} in $a$ is an object $P$ together with a collection of subobjects $p_{x}:P \mapsto a $, where $x:1 \to  a$ is an arbitrary element, such that:

1\) $x \in  p_{x}$.

2\) If $y \in  p_{x}$, then $z \in p_y $ implies $z \in  p_{x}$.

The order $ \langle P, \{p_{x}\} \rangle $ is denoted as $\O $.

A morphism $f:a \to a $ is called [*affine*]{}, if $\psi \circ f\circ \phi  \in \Gamma $ for any $\phi  \in  \Phi $ and $\psi \in \Psi $. We denote the set of all affine morphisms by $\Aff(a)$.

Let $\A \subset  \Aff(a)$ consist of all commuting morphisms. An order $\O$ is [*invariant with respect to* ]{} if for any $p_{x},p_{y}$ there exists $g_{xy} \in  \A $ such that $ g_{xy}\circ p_{x} =p_{y} $.

A morphism $f:a \to  a$ [*preserves*]{} an order $\O$, if for each $p_{x}$ there exists $p_{y}$ such that $ f\circ p_{x} =p_{y} $. The collection of all morphisms preserving an order $\O$ that is invariant with respect to $ \A $ is denoted by $ Aut(\O)$.

A [*ray*]{} is a morphism $$\lambda : R_{+} \mapsto R_{T} 