A semigroup is permutative if it
satisfies an identity of the form x1x2⋯xn= xσ1xσ2⋯xσn where σ is a non-identical
permutation of {1,2,…,n}. The finite permutative semigroups form a pseudovariety
and permutative pseudovarieties enjoy many properties first obtained for
commutative pseudovarieties. Several types of permutation identities are considered,
and all pseudovarieties minimal with respect to the property of containing a
finite semigroup which fails an identity of a given type are determined. This
includes the cases of the commutativity identity, the general permutation
identities, and the “strong (left) permutation identities”. As a preliminary, all
minimal non-commutative pseudovarieties of groups and monoids are also
determined.
