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 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 AC 0 . Deciding whether a semigroup has a left (resp. right) zero is shown to be NL-complete, as are the problems of testing whether a transformation semigroup is nilpotent, R-trivial or has central idempotents. We also give 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 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 PSPACE-complete.

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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. 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.

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