On actions on cubic stochastic matrices

TL;DR

This paper explores algebraic structures and actions on cubic stochastic matrices, providing a framework for modeling changes in biological transition probabilities.

Contribution

It introduces new multiplication rules and actions on cubic stochastic matrices, establishing an associative semigroup structure and analyzing their effects on marginal distributions.

Findings

Defined new multiplication rules preserving stochastic properties

Established an associative semigroup structure for cubic matrices

Analyzed how actions affect transition probabilities in biological models

Abstract

We consider the set of ($n\times n\times n$) cubic stochastic matrices of type (1,2) together with different multiplication rules that not only retain their stochastic properties but also endow this set with an associative semigroup structure. Then we introduce different actions of the semigroup of nonnegative column stochastic $n\times n$ matrices on the set of cubic stochastic matrices of type (1,2) and study how these actions translate to the cubic matrix slices and marginal distributions. Actions introduced here provide an algebraic framework where considering different changes affecting the transition probabilities ruling certain biological populations.