Some algorithmic problems in monoids of Boolean matrices

  • Peter Fenner

Student thesis: Phd


A Boolean matrix is a matrix with elements from the Boolean semiring ({0, 1}, +, x), where the addition and multiplication are as usual with the exception that 1 + 1 = 1. In this thesis we study eight classes of monoids whose elements are Boolean matrices. Green's relations are five equivalence relations and three pre-orders which are defined on an arbitrary monoid M and describe much of its structure. In the monoids we consider the equivalence relations are uninteresting - and in most cases completely trivial - but the pre-orders are not and play a vital part in understanding the structure of the monoids. Each of the three pre-orders in each of the eight classes of monoids can be viewed as a computational decision problem: given two elements of the monoid, are they related by the pre-order? The main focus of this thesis is determining the computational complexity of each of these twenty-four decision problems, which we successfully do for all but one.
Date of Award31 Dec 2018
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorMarianne Johnson (Supervisor) & Mark Kambites (Supervisor)


  • semigroup theory
  • computational complexity
  • Boolean matrices

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