1,002 research outputs found
Generating infinite monoids of cellular automata
For a group and a set , let be the monoid of all
cellular automata over , and let be its group of units.
By establishing a characterisation of surjunctuve groups in terms of the monoid
, we prove that the rank of (i.e. the
smallest cardinality of a generating set) is equal to the rank of
plus the relative rank of in
, and that the latter is infinite when has an infinite
decreasing chain of normal subgroups of finite index, condition which is
satisfied, for example, for any infinite residually finite group. Moreover,
when is a vector space over a field , we study the monoid
of all linear cellular automata over and
its group of units . We show that if is an
indicable group and is finite-dimensional, then
is not finitely generated; however, for any
finitely generated indicable group , the group
is finitely generated if and only if
is finite.Comment: 11 page
Code algebras, axial algebras and VOAs
Inspired by code vertex operator algebras (VOAs) and their representation
theory, we define code algebras, a new class of commutative non-associative
algebras constructed from binary linear codes. Let be a binary linear code
of length . A basis for the code algebra consists of idempotents
and a vector for each non-constant codeword of . We show that code algebras
are almost always simple and, under mild conditions on their structure
constants, admit an associating bilinear form. We determine the Peirce
decomposition and the fusion law for the idempotents in the basis, and we give
a construction to find additional idempotents, called the -map, which comes
from the code structure. For a general code algebra, we classify the
eigenvalues and eigenvectors of the smallest examples of the -map
construction, and hence show that certain code algebras are axial algebras. We
give some examples, including that for a Hamming code where the code
algebra is an axial algebra and embeds in the code VOA .Comment: 32 pages, including an appendi
Code algebras which are axial algebras and their -gradings
A code algebra is a non-associative commutative algebra defined via a
binary linear code . We study certain idempotents in code algebras, which we
call small idempotents, that are determined by a single non-zero codeword. For
a general code , we show that small idempotents are primitive and semisimple
and we calculate their fusion law. If is a projective code generated by a
conjugacy class of codewords, we show that is generated by small
idempotents and so is, in fact, an axial algebra. Furthermore, we classify when
the fusion law is -graded. In doing so, we exhibit an infinite
family of -graded axial algebras - these are
the first known examples of axial algebras with a non-trivial grading other
than a -grading.Comment: 29 page
The number of configurations in the full shift with a given least period
For any group and any set , consider the shift action of on the
full shift . A configuration has \emph{least period} if the stabiliser of is precisely . Among other things, the number of
such configurations is interesting as it provides an upper bound for the size
of the corresponding -orbit. In this paper we show that if
is finitely generated and is of finite index, then the number of
configurations in with least period may be computed using the
M\"obius function of the lattice of subgroups of finite index in . Moreover,
when is a normal subgroup, we classify all situations such that the number
of -orbits with least period is at most .Comment: 8 page
Complete Simulation of Automata Networks
Consider a finite set A and . We study complete simulation of transformations of , also known as automata networks. For , a transformation of is n-complete of size m if it may simulate every transformation of by updating one register at a time. Using tools from memoryless computation, we establish that there is no n-complete transformation of size n, but there is one of size . By studying various constructions, we conjecture that the maximal time of simulation of any n-complete transformation is at least 2n. We also investigate the time and size of sequentially n-complete transformations, which may simulate every finite sequence of transformations of . Finally, we show that there is no n-complete transformation updating all registers in parallel, but there exists one updating all but one register in parallel. This illustrates the strengths and weaknesses of sequential and parallel models of computation
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