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John Rhodes

Publications and source records attributed to John Rhodes.

At least 19 recordsLinked to original sources
arXiv2026

Defining the Crick-Franklin-Watson genes of any robot (equals a finite state machine) and defining the Shannon genetic code attached to each gene. We also define the Krohn-Rhodes complexity of any regular maximal prefix code

This paper will discuss the relationships among three pillars of mathematics: important research in codes and automata by Marcel-Paul Schutzenberger and others; random walks on finite semigroups by Persi Diaconis and others; and advanced techniques from finite semigroup theory used in proving Krohn-Rhodes complexity c is decidable by Stuart Margolis, Anne Schilling, and myself. Very surprising connections exist between the Fundamental Lemma of Complexity c (epimorphisms between finite semigroups that are one-to-one on subgroups preserve c) and coupling from the past in Markov chains, Diaconis' strong stationary time, and the Crick-Franklin-Watson genes of a finite automaton (which will be defined in the paper).

cs.FL↗
arXiv2026

Aperiodic Flows on Finite Semigroups II: Smallish Monoids Suffice for Complexity 1

A smallish monoid M is a monoid that has a unique 0-minimal ideal I(M) that is a 0-simple subsemigroup and such that its regular J -classes are the group of units and the two in I(M). We show constructively how to embed an arbitrary finite semigroup S into the evaluation semigroup of a smallish monoid S^{Ev} . We use the theory of flows to show that a group mapping semigroup S admits an aperiodic flow if and only if S^{Ev} admits one. This reduces the computation of Krohn-Rhodes complexity 1 to the class of smallish monoids.

math.GR↗
arXiv2026

Bases of Permutation Groups and Boolean Representable Simplicial Complexes

A base of a permutation group (X,G) is a subset B of X such that its pointwise stabilizer is the trivial group. A list (x1,x2, ... ,xk) of elements of X is irredundant if each element is not in the pointwise stabilizer of its predecessors. We define a Boolean representable simplicial complex B(X,G) such that a subset Y of X is independent if and only if some enumeration of its elements is irredundant. In addition Y is a base if and only if its closure is X. We give a number of examples and close with a conjecture whose solution leads to a new proof of the Feit-Thompson Theorem.

math.GR↗
arXiv2025

Aperiodic Flows on Finite Semigroups: Foundations and First Examples

The theory of flows was used as a crucial tool in the recent proof by Margolis, Rhodes and Schilling that Krohn-Rhodes complexity is decidable. In this paper we begin a systematic study of aperiodic flows. We give the foundations of the theory of flows and give a unified approach to the Presentation Lemma and its relations to flows and the Slice Theorem. We completely describe semigroups having a flow over the trivial semigroup and connect this to classical results in inverse semigroup theory. We reinterpret Tilson's Theorem on the complexity of small monoids in terms of flows. We conclude with examples of semigroups built from the character table of Abelian Groups that have an aperiodic flows.

math.DS↗
arXiv2025

Complexity of Finite Semigroups: History and Decidability

In recent papers, Margolis, Rhodes and Schilling proved that the complexity of a finite semigroup is computable. This solved a problem that had been open for more than 50 years. The purpose of this paper is to survey the basic results of Krohn-Rhodes complexity of finite semigroups and to outline the proof of its computability.

math.GR↗
arXiv2024

Decidability of Krohn-Rhodes complexity for all finite semigroups and automata

The Krohn-Rhodes Theorem proves that a finite semigroup divides a wreath product of groups and aperiodic semigroups. Krohn-Rhodes complexity equals the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for more than 50 years. The main result of this paper proves that it is decidable whether a semigroup has complexity k for any k greater than or equal to 0. This builds on our previous work for complexity 1. In that paper we proved using profinite methods and results on free Burnside semigroups by McCammond and others that the lower bound from a 2012 paper by Henckell, Rhodes and Steinberg is precise for complexity 1. In this paper we define an improved version of the lower bound from the 2012 paper and prove that it is exact for arbitrary complexity.

math.GR↗
arXiv2023

Topics in Boolean Representable Simplicial Complexes

We study a number of topics in the theory of Boolean Representable Simplicial Complexes (BRSC). These include various operators on BRSC. We look at shellability in higher dimensions and propose a number of new conjectures.

math.CO↗
arXiv2022

Upper bounds on mixing time of finite Markov chains

We provide a general framework for computing upper bounds on mixing times of finite Markov chains when its minimal ideal is left zero. Our analysis is based on combining results by Brown and Diaconis with our previous work on stationary distributions of finite Markov chains. Stationary distributions can be computed from the Karnofsky--Rhodes and McCammond expansion of the right Cayley graph of the finite semigroup underlying the Markov chain. Using loop graphs, which are planar graphs consisting of a straight line with attached loops, there are rational expressions for the stationary distribution in the probabilities. From these we obtain bounds on the mixing time. In addition, we provide a new Markov chain on linear extension of a poset with $n$ vertices, inspired by but different from the promotion Markov chain of Ayyer, Klee and the last author. The mixing time of this Markov chain is $O(n \log n)$.

math.PR↗
arXiv2021

Decidability of Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata

When decomposing a finite semigroup into a wreath product of groups and aperiodic semigroups, complexity measures the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for almost 60 years. The main result of this paper proves decidability of Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata. This is achieved by showing the lower bounds in work by Henckell, Rhodes and Steinberg from 2012 is sharp using profinite methods and results of McCammond from 1991 and 2001.

math.GR↗
arXiv2021

Holonomy theorem for finite semigroups

We provide a simple proof of the Holonomy Theorem using a new Lyndon-Chiswell length function on the Karnofsky-Rhodes expansion of a semigroup. Unexpectedly, we have both a left and a right action on the Chiswell tree by elliptic maps.

math.GR↗
arXiv2020

Degree 2 Transformation Semigroups as Continuous Maps on Graphs: Foundations and Structure

We develop the theory of transformation semigroups that have degree 2, that is, act by partial functions on a finite set such that the inverse image of points have at most two elements. We show that the graph of fibers of such an action gives a deep connection between semigroup theory and graph theory. It is known that the Krohn-Rhodes complexity of a degree 2 action is at most 2. We show that the monoid of continuous maps on a graph is the translational hull of an appropriate 0-simple semigroup. We show how group mapping semigroups can be considered as regular covers of their right letter mapping image and relate this to their graph of fibers.

math.GR↗
arXiv2019

Join irreducible semigroups

We begin a systematic study of finite semigroups that generate join irreducible members of the lattice of pseudovarieties of finite semigroups, which are important for the spectral theory of this lattice. Finite semigroups $S$ that generate join irreducible pseudovarieties are characterized as follows: whenever $S$ divides a direct product $A \times B$ of finite semigroups, then $S$ divides either $A^n$ or $B^n$ for some $n \geq 1$. We present a new operator ${ \mathbf{V} \mapsto \mathbf{V}^\mathsf{bar} }$ that preserves the property of join irreducibility, as does the dual operator, and show that iteration of these operators on any nontrivial join irreducible pseudovariety leads to an infinite hierarchy of join irreducible pseudovarieties. We also describe all join irreducible pseudovarieties generated by a semigroup of order up to five. It turns out that there are $30$ such pseudovarieties, and there is a relatively easy way to remember them. In addition, we survey most results known about join irreducible pseudovarieties to date and generalize a number of results in Sec. 7.3 of The $q$-theory of Finite Semigroups, Springer Monographs in Mathematics (Springer, Berlin, 2009).

math.GR↗
arXiv2019

NANUQ: A method for inferring species networks from gene trees under the coalescent model

Species networks generalize the notion of species trees to allow for hybridization or other lateral gene transfer. Under the Network Multispecies Coalescent Model, individual gene trees arising from a network can have any topology, but arise with frequencies dependent on the network structure and numerical parameters. We propose a new algorithm for statistical inference of a level-1 species network under this model, from data consisting of gene tree topologies, and provide the theoretical justification for it. The algorithm is based in an analysis of quartets displayed on gene trees, combining several statistical hypothesis tests with combinatorial ideas such as a quartet-based intertaxon distance appropriate to networks, the NeighborNet algorithm for circular split systems, and the Circular Network algorithm for constructing a splits graph.

q-bio.PE↗
arXiv2019

On the Wilson Monoid of a Pairwise Balanced Design

We give a new perspective of the relationship between simple matroids of rank 3 and pairwise balanced designs, connecting Wilson's theorems and tools with the theory of truncated boolean representable simplicial complexes. We also introduce the concept of Wilson monoid W(X) of a pairwise balanced design X. We present some general algebraic properties and study in detail the cases of Steiner triple systems up to 19 points, as well as the case where a single block has more than 2 elements

math.CO↗
arXiv2019

Truncated Boolean Representable Simplicial Complexes

We extend, in significant ways, the theory of truncated boolean representable simplicial complexes introduced in 2015. This theory, which includes all matroids, represents the largest class of finite simplicial complexes for which combinatorial geometry can be meaningfully applied

math.CO↗
arXiv2019

Unified theory for finite Markov chains

We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup $S$. Our methods use geometric finite semigroup theory via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to $P(n)$, the set of all subsets of an $n$ element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or $\mathscr{R}$-trivial bands) by Brown and Diaconis, extensions to $\mathscr{R}$-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of $S$ in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded $S$. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded $S$ and the original $S$. Analyzing the normal forms also provides an estimate on the mixing time.

math.PR↗