A cellular automaton consists of a regular system of cells, each containing a symbol from a finite alphabet, together with a uniform rule called a transition function for updating all cells simultaneously based on the values of neighboring cells.
Mathematically, this can be formalized by the notion of a group, a set of elements together with an associative and invertible binary operation.
For instance, a one-dimensional line of cells can be described in this way as the additive group of the integers, and the higher-dimensional integer grids can be described as the free abelian groups.
For such functions, the Curtis–Hedlund–Lyndon theorem ensures that the value of the transition function at each group element depends on the previous state of only a finite set of neighboring elements.
A surjunctive group is a group with the property that, when its elements are used as the cells of cellular automata, every injective transition function of a cellular automaton is also surjective.
[1] The implication from injectivity to surjectivity is a form of the Garden of Eden theorem, and the cellular automata defined from injective and surjective transition functions are reversible.