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- W2572604825 abstract "For any group G and any set A, a cellular automaton (CA) is a transformation of the configuration space (A^G) defined via a finite memory set and a local function. Let (mathrm {CA}(G;A)) be the monoid of all CA over (A^G). In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element (tau in mathrm {CA}(G;A)) is von Neumann regular (or simply regular) if there exists (sigma in mathrm {CA}(G;A)) such that (tau circ sigma circ tau = tau ) and (sigma circ tau circ sigma = sigma ), where (circ ) is the composition of functions. Such an element (sigma ) is called a generalised inverse of (tau ). The monoid (mathrm {CA}(G;A)) itself is regular if all its elements are regular. We establish that (mathrm {CA}(G;A)) is regular if and only if (vert G vert = 1) or (vert A vert = 1), and we characterise all regular elements in (mathrm {CA}(G;A)) when G and A are both finite. Furthermore, we study regular linear CA when (A= V) is a vector space over a field (mathbb {F}); in particular, we show that every regular linear CA is invertible when G is torsion-free (e.g. when (G=mathbb {Z}^d, d ge 1)), and that every linear CA is regular when V is finite-dimensional and G is locally finite with (mathrm {char}(mathbb {F}) not mid o(g)) for all (g in G)." @default.
- W2572604825 created "2017-01-26" @default.
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- W2572604825 date "2017-01-01" @default.
- W2572604825 modified "2023-09-27" @default.
- W2572604825 title "Von Neumann Regular Cellular Automata" @default.
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- W2572604825 doi "https://doi.org/10.1007/978-3-319-58631-1_4" @default.
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