Ternary cellular automata induced by semigroups of order 3 are solvable

TL;DR

This paper proves that all cellular automata rules derived from semigroups of order 3 are solvable, providing explicit formulas for their evolution based on algebraic properties.

Contribution

It demonstrates that cellular automata induced by all semigroups of order 3 are solvable and derives explicit formulas for their states after n iterations.

Findings

All 18 semigroup-based rules are solvable.

Explicit formulas are derived for each rule.

Properties like commutativity and idempotence aid in derivations.

Abstract

The minimal number of inputs in the local function of a non-trivial cellular automaton is two. Such a function can be viewed as as a kind of binary operation. If this operation is associative, it forms, together with the set of states, a semigroup. There are 18 semigroups of order 3 up to equivalence, and they define 18 cellular automata rules with three states. We investigate these rules with respect to solvability and show that all of them are solvable, meaning that the state of a given cell after $n$ iterations can be expressed by an explicit formula. We derive the relevant formulae for all 18 rules using some additional properties possessed by particular semigroups of order 3, such as commutativity and idempotence.