Equational theories of semigroups with involution

We employ the techniques developed in an earlier paper to show that involutory semigroups arising in various contexts do not have a finite basis for their identities. Among these are partition semigroups endowed with their natural inverse involution, including the full partition semigroup Cn for n⩾2, the Brauer semigroup Bn for n⩾4 and the annular semigroup An for n⩾4, n even or a prime power. Also, all of these semigroups, as well as the Jones semigroup Jn for n⩾4, turn out to be inherently nonfinitely based when equipped with another involution, the ‘skew’ one. Finally, we show that similar techniques apply to the finite basis problem for existence varieties of locally inverse semigroups.