Epimorphisms, permutation identities and finite semigroups (Q798444)

In a joint result with N. M. Khan the author proves that a semigroup variety that admits the identity \(x_ 1x_ 2...x_ n=x_{1\pi}x_{2\pi}...x_{n\pi},\) where \(\pi\) is a permutation on 1,2,...,n is closed under taking epimorphisms iff either 1\(\pi \neq 1\) or \(n\pi \neq n\). A semigroup U is said to be saturated if the dominion of U in S is \(\neq S\) for every properly containing semigroup S. The author proves that any finite permutative semigroup is saturated.