arXiv subjects

Trevor Jack

Publications and source records attributed to Trevor Jack.

4 recordsLinked to original sources

arXiv2024

Answering Five Open Problems Involving Semigroup Conjugacy

A semigroup conjugacy is an equivalence relation that equals group conjugacy when the semigroup is a group. In this note, we answer five open problems related to semigroup conjugacy. (Problem One) We say a conjugacy ~ is partition-covering if for every set X and every partition of the set, there exists a semigroup with universe X such that the partition gives the ~-conjugacy classes of the semigroup. We prove that six well-studied conjugacy relations -- ~o, ~c, ~n, ~p, ~p*, and ~tr -- are all partition-covering. (Problem Two) For two semigroup elements a and b in S, we say a ~p b if there exists u and v in S such that a=uv and b=vu. We give an example of a semigroup that is embeddable in a group for which ~p is not transitive. (Problem Three) We construct an infinite chain of first-order definable semigroup conjugacies. (Problem Four) We construct a semigroup for which ~o is a congruence and S\~o is not cancellative. (Problem Five) We construct a semigroup for which ~p is not transitive while, for each of the semigroup's variants, ~p is transitive.

math.GR↗

arXiv2021

On the complexity of inverse semigroup conjugacy

We investigate the computational complexity of various decision problems related to conjugacy in finite inverse semigroups. We describe polynomial-time algorithms for checking if two elements in such a semigroup are ~p conjugate and whether an inverse monoid is factorizable. We describe a connection between checking ~i conjugacy and checking membership in inverse semigroups. We prove that ~o and ~c are partition covering for any countable set and that ~p, ~p* , and ~tr are partition covering for any finite set. Finally, we prove that checking for nilpotency, R-triviality, and central idempotents in partial bijection semigroups are NL-complete problems and we extend several complexity results for partial bijection semigroups to inverse semigroups.

math.GR↗

arXiv2021

On the Complexity of Properties of Partial Bijection Semigroups

We examine the computational complexity of problems in which we are given generators for a partial bijection semigroup and asked to check properties of the generated semigroup. We prove that the following problems are in AC$^0$: (1) enumerating left and right identities and (2) checking if the semigroup is completely regular. We also describe a nondeterministic logspace algorithm for checking if an inverse semigroup given by generators satisfies a fixed semigroup identity that may involve a unary inverse operation. We conclude with an alternative proof that checking membership of a given idempotent in a partial bijection semigroup is a PSPACE-complete problem. The proof reduces from the well-known PSPACE-complete Rectangle Tiling Problem, thereby illustrating a connection between Wang tilings and partial bijection semigroups.

math.GR↗

arXiv2018

On the Complexity of Properties of Transformation Semigroups

We investigate the computational complexity for determining various properties of a finite transformation semigroup given by generators. We introduce a simple framework to describe transformation semigroup properties that are decidable in $\mathsf{AC^0}$. This framework is then used to show that the problems of deciding whether a transformation semigroup is a group, commutative or a semilattice are in $\mathsf{AC^0}$. Deciding whether a semigroup has a left (resp.right) zero is shown to be $\mathsf{NL}$-complete, as are the problems of testing whether a transformation semigroup is nilpotent, $\mathcal{R}$-trivial or has central idempotents. We also give $\mathsf{NL}$ algorithms for testing whether a transformation semigroup is idempotent, orthodox, completely regular, Clifford or has commuting idempotents. Some of these algorithms are direct consequences of the more general result that arbitrary fixed semigroup equations can be tested in~$\mathsf{NL}$. Moreover, we show how to compute left and right identities of a transformation semigroup in polynomial time. Finally, we show that checking whether an element is regular is $\mathsf{PSPACE}$-complete. \

math.GR↗