preinjectivity
Let $X={\prod}_{i\in I}{X}_{i}$ be a Cartesian product. Call two elements $x,y\in X$ almost equal if the set $\mathrm{\Delta}(x,y)=\{i\in I\mid {x}_{i}\ne {y}_{i}\}$ is finite. A function $f:X\to X$ is said to be preinjective if it sends distinct almost equal elements into distinct elements, i.e., if $$ implies $f(x)\ne f(y)$.
If $X$ is finite, preinjectivity is the same as injectivity; in general, the latter implies the former, but not the other way around. Moreover, it is not true in general that a composition^{} of preinjective functions is itself preinjective.
A cellular automaton^{} is said to be preinjective if its global function is. As cellular automata send almost equal configurations^{} into almost equal configurations, the composition of two preinjective cellular automata is preinjective.
Preinjectivity of cellular automata can be characterized via mutually erasable patterns. Given a finite subset $E$ of $G$, two patterns ${p}_{1},{p}_{2}:E\to Q$ are mutually erasable (briefly, m.e.) for a cellular automaton $\mathcal{A}=\u27e8Q,\mathcal{N},f\u27e9$ on $G$ if for any two configurations ${c}_{1},{c}_{2}:G\to Q$ such that ${{c}_{i}}_{E}={p}_{i}$ and ${{c}_{1}}_{G\setminus E}={{c}_{2}}_{G\setminus E}$ one has ${F}_{\mathcal{A}}({c}_{1})={F}_{\mathcal{A}}({c}_{2})$.
Lemma 1
For a cellular automaton $\mathrm{A}\mathrm{=}\mathrm{\u27e8}Q\mathrm{,}\mathrm{N}\mathrm{,}f\mathrm{\u27e9}\mathrm{,}$ the following are equivalent^{}.

1.
$\mathcal{A}$ has no mutually erasable patterns.

2.
$\mathcal{A}$ is preinjective.
Proof. It is immediate that the negation^{} of point 1 implies the negation of point 2. So let ${c}_{1},{c}_{2}:G\to Q$ be two distinct almost equal configurations such that ${F}_{\mathcal{A}}({c}_{1})={F}_{\mathcal{A}}({c}_{2}):$ it is not restrictive to suppose that $\mathcal{N}$ is symmetric^{} (i.e., if $x\in \mathcal{N}$ then ${x}^{1}\in \mathcal{N}$) and $e$, the identity element^{} of $G$, belongs to $\mathcal{N}$. Let $\mathrm{\Delta}$ be a finite subset of $G$ such that ${{c}_{1}}_{G\setminus \mathrm{\Delta}}={{c}_{2}}_{G\setminus \mathrm{\Delta}},$ and let
$$E=\mathrm{\Delta}{\mathcal{N}}^{2}=\{g\in G\mid \exists z\in \mathrm{\Delta},u,v\in \mathcal{N}\mid g=zuv\}:$$  (1) 
we shall prove that ${p}_{1}={{c}_{1}}_{E}$ and ${p}_{2}={{c}_{2}}_{E}$ are mutually erasable. (They surely are distinct, since $\mathrm{\Delta}\subseteq E$.)
So let ${\gamma}_{1},{\gamma}_{2}:G\to Q$ satisfy ${{\gamma}_{i}}_{E}={p}_{i}$ and ${{\gamma}_{2}}_{G\setminus E}={p}_{i}={{\gamma}_{2}}_{G\setminus E}.$ Let $z\in G$. If $z\in \mathrm{\Delta}\mathcal{N}$, then ${F}_{\mathcal{A}}({\gamma}_{1})(z)={F}_{\mathcal{A}}({\gamma}_{2})(z)$, because by construction ${{\gamma}_{i}}_{z\mathcal{N}}={{c}_{i}}_{z\mathcal{N}};$ if $z\in G\setminus \mathrm{\Delta}\mathcal{N},$ then ${F}_{\mathcal{A}}({\gamma}_{1})(z)={F}_{\mathcal{A}}({\gamma}_{2})(z)$ as well, because by construction ${{\gamma}_{1}}_{z\mathcal{N}}={{\gamma}_{2}}_{z\mathcal{N}}.$ Since ${\gamma}_{1}$ and ${\gamma}_{2}$ are arbitrary, ${p}_{1}$ and ${p}_{2}$ are mutually erasable. $\mathrm{\square}$
References
 1 CeccheriniSilberstein, T. and Coornaert, M. (2010) Cellular Automata and Groups. Springer Verlag.
 2
Title  preinjectivity 

Canonical name  Preinjectivity 
Date of creation  20130322 19:22:04 
Last modified on  20130322 19:22:04 
Owner  Ziosilvio (18733) 
Last modified by  Ziosilvio (18733) 
Numerical id  4 
Author  Ziosilvio (18733) 
Entry type  Definition 
Classification  msc 37B15 
Classification  msc 68Q80 
Defines  mutually erasable patterns (cellular automaton) 