Monoid of graphical patterns in $lca(2,1)_{110}$

Graphical patterns that arise from evolutions of rule 110, obey other rules that we can study in this paper, but in general, patterns like triangles can be used for covering the discrete 2-space. The covers themself define an additive discrete monoid. Let us remember that a monoid is defined by a semigroup with identity element over its operation.

We will define the necessary elements for cover the 2-space, and a function that allow us put patterns in both ways, space and time; with the intention that we can cover the 2-space in a infinite way, starting with a definition that can help us to understand the basic idea of full covering space by triangles generated by $lca(2,1)_{110}$ .

Definition 1   If $T$ is a set of triangles, then $\mathcal{X}_T$ is a $T$ -full covering and is a space completely covered by triangles elements of $T$ . That covering has neither overlapping nor holes. A $\tau$ -full covering is a full covering space by tiles elements of $\tau\subset T$ . When is completely clear what is the set of graphical patters used, we can omit the subindex $T$ . From now on we will use $T$ as set of triangles.

Definition 2   We call Patch $p$ , a set of cells such that, for any two cells $c, c'$ in the interior of $p$ , we can always find at least one $path$ from $c$ to $c'$ . A sub-patch, is a graphical region that is adjacent to a patch.

A subspace in the plane that is covered by a patch is called ``cover''. Elements used for build covers are ``patches''. In general, a cover is a patch too, from this moment onwards we call path to this kind of covers that are not full-covers, and we left the covering term to name the topological property of a metrical space.

These covers not necessarily obey rules established in local evolution rule on $lca(2,1)_{110}$ . After, we introduce some restrictions with the purpose that make valid constructions into evolution space of $lca(2,1)_{110}$ . A patch that require $n$ sub-patches is a element of a class of patches that requires $n$ sub-patches, we name this $\mathcal{X}^{(n)}$ . In particular $\mathcal{X}^{(0)}$ is the class of patches that have no patches, the only element of this class is $p^{\lambda}$ . $p^{\lambda}$ cover an area 0 region and we call it the null element; $p^{\lambda}\in\mathcal{X}^\lambda$ . $\mathcal{X}^{(\ast)}\rightleftharpoons \mathcal{X}^{\lambda} \cup \mathcal{X}^{(1)} \cup \mathcal{X}^{(2)} \cup \dots \cup \mathcal{X}^{(n)}$ , is the set of all patches of any number of patches, for some arbitrarily $n<\infty$ .

We can define a binary operation over patches, the graphical-concatenation. Two patches can be graphic-concatenated by a path between two cells, both of them in the border of each patch. In symbolic form, $\bowtie : \mathcal{X}^{(\ast)} \times \mathcal{X}^{(\ast)} \rightarrow \mathcal{X}^{(\ast)}$ , and for $x\in\mathcal{X}^{(i)}$ , $y\in\mathcal{X}^{(j)}$ , $y\in\mathcal{X}^{(j)}$ and $z\in\mathcal{X}^{(\ast)}$ , we have $x \bowtie y = z$ , where $z\in\mathcal{X}^{(i+j)}$ ; so $\bowtie$ is closed into $\mathcal{X}^{(\ast)}$ . The set $\mathcal{X}^{(\ast)}$ with graphical-concatenation operation, define an additive monoid $(\mathcal{X}^{(\ast)};\bowtie)$ of patches in $acl(2,1)_{110}$ , with only one identity element, and the associative. The definition of the operation $\bowtie$ is in table 3.

Table 3: Graphical-concatenation of the additive monoid of patches in $lca(2,1)_{110}$

$\bowtie$	$\mathcal{X}^\lambda$	$\mathcal{X}^{(1)}$	$\mathcal{X}^{(2)}$	$\mathcal{X}^{(3)}$	$\dots$	$\mathcal{X}^{(n)}$
$\mathcal{X}^\lambda$	$\mathcal{X}^\lambda$	$\mathcal{X}^{(1)}$	$\mathcal{X}^{(2)}$	$\mathcal{X}^{(3)}$	...	$\mathcal{X}^{(n)}$
$\mathcal{X}^{(1)}$	$\mathcal{X}^{(1)}$	$\mathcal{X}^{(2)}$	$\mathcal{X}^{(3)}$	$\mathcal{X}^{(4)}$	...	$\mathcal{X}^{(n+1)}$
$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	...	$\vdots$
$\mathcal{X}^{(n)}$	$\mathcal{X}^{(n)}$	$\mathcal{X}^{(n+1)}$	$\mathcal{X}^{(n+2)}$	$\mathcal{X}^{(n+3)}$	...	$\mathcal{X}^{(2n)}$