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Abstract
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Let
be an amenable
group and let
be an irreducible complete algebraic variety over an algebraically closed field
. Let
denote the
set of
-points
of
and let
be an algebraic cellular
automaton over
,
that is, a cellular automaton over the group
and the
alphabet
whose local defining map is induced by a morphism of
-algebraic
varieties. We introduce a weak notion of preinjectivity for algebraic cellular automata, namely
-preinjectivity, and prove
that
is surjective if and
only if it is
-preinjective.
In particular,
has the Myhill property, i.e., is surjective whenever it is preinjective. Our result gives
a positive answer to a question raised by Gromov (J. Eur. Math. Soc. 1:2 (1999),
109–197) and yields an analogue of the classical Moore–Myhill Garden of Eden
theorem.
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Keywords
algebraic cellular automaton, complete algebraic variety,
projective algebraic variety, amenable group, Krull
dimension, algebraic mean dimension, preinjectivity, Garden
of Eden theorem
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Mathematical Subject Classification 2010
Primary: 14A10, 14A15, 37B10, 37B15, 43A07, 68Q80
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Milestones
Received: 29 March 2018
Accepted: 20 December 2019
Published: 14 June 2020
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