For finite inverse semigroups, membership and conjugacy are easy (NC/NP or LOGSPACE) precisely for strict and Clifford varieties, and PSPACE-complete or L-complete otherwise.
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Membership and Conjugacy in Inverse Semigroups
For finite inverse semigroups, membership and conjugacy are easy (NC/NP or LOGSPACE) precisely for strict and Clifford varieties, and PSPACE-complete or L-complete otherwise.