For a finite group G and a finite set A, we study various algebraic aspects of cellular automata over the configuration space A G . In this situation, the set CA(G; A) of all cellular automata over
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 C be a binary linear code of length n. A basis for the code algebra A C consists of n idempotents and a vector for each non-constant codeword of C. 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 s-map, which comes from the code structure. For a general code algebra, we classify the eigenvalues and eigenvectors of the smallest examples of the s-map construction, and hence show that certain code algebras are axial algebras. We give some examples, including that for a Hamming code H 8 where the code algebra A H8 is an axial algebra and embeds in the code VOA V H8 .
Let G be a group and A a set. A cellular automaton (CA) τ over A G is von Neumann regular (vN-regular) if there exists a CA σ over A G such that τ στ = τ , and in such case, σ is called a weak generalised inverse of τ . In this paper, we investigate vN-regularity of various kinds of CA. First, we establish that, over any nontrivial configuration space, there always exist CA that are not vN-regular. Then, we obtain a partial classification of elementary vN-regular CA over {0, 1} Z ; in particular, we show that rules like 128 and 254 are vNregular (and actually generalised inverses of each other), while others, like the well-known rules 90 and 110, are not vN-regular. Next, when A and G are both finite, we obtain a full characterisation of vN-regular CA over A G . Finally, we study vN-regular linear CA when A = V is a vector space over a field F; we show that every vN-regular linear CA is invertible when V = F and G is torsion-free elementary amenable (e.g. when G = Z d , d ∈ N), and that every linear CA is vN-regular when V is finite-dimensional and G is locally finite with char(F) ∤ o(g) for all g ∈ G.
Since first introduced by John von Neumann, the notion of cellular automaton has grown into a key concept in computer science, physics and theoretical biology. In its classical setting, a cellular automaton is a transformation of the set of all configurations of a regular grid such that the image of any particular cell of the grid is determined by a fixed local function that only depends on a fixed finite neighbourhood.
A code algebra AC is a non-associative commutative algebra defined via a binary linear code C. 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 C, we show that small idempotents are primitive and semisimple and we calculate their fusion law. If C is a projective code generated by a conjugacy class of codewords, we show that AC is generated by small idempotents and so is, in fact, an axial algebra. Furthermore, we classify when the fusion law is Z2-graded. In doing so, we exhibit an infinite family of Z2 × Z2-graded axial algebras -these are the first known examples of axial algebras with a non-trivial grading other than a Z2-grading.
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