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  • 1School of Physical Sciences, University of Tasmania, Hobart, Australia. jsumner@utas.edu.au.

Bulletin of Mathematical Biology
|August 4, 2018
PubMed
Summary
This summary is machine-generated.

We developed a method to create Lie-Markov models from finite semigroups. This results in a k-state continuous-time Markov chain with multiplicative closure, generalizing group-based models.

Keywords:
Continuous-time Markov chainsGroup-based modelsLie algebrasPhylogenetics

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Area of Science:

  • Mathematical modeling
  • Stochastic processes
  • Algebraic theory

Background:

  • Group-based models are established in stochastic modeling.
  • Finite semigroups offer a rich algebraic structure.
  • The need for models with multiplicative closure is recognized.

Purpose of the Study:

  • To present a general method for deriving Lie-Markov models.
  • To explore the properties of models derived from finite semigroups.
  • To generalize existing group-based modeling concepts.

Main Methods:

  • Derivation of a Lie-Markov model from a finite semigroup.
  • Utilizing the semigroup's degree (k) to define the Markov chain states.
  • Leveraging the semigroup's product rule to ensure multiplicative closure.

Main Results:

  • A general method for constructing Lie-Markov models from finite semigroups is established.
  • The resulting models are continuous-time Markov chains on k-states.
  • The property of multiplicative closure is demonstrated for the derived models.

Conclusions:

  • The proposed construction provides a natural generalization of group-based models.
  • Lie-Markov models derived from finite semigroups possess multiplicative closure.
  • This method offers a novel approach to building stochastic models with specific algebraic properties.