A simple counterexample is the Brandt semigroup with five elements " B 2 " because the unipotency class of its zero element is not a subsemigroup . " B 2 " is actually the quintessential epigroup which is not unipotently partionable.
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Moreover, each simple group ( " prime " ) or non-group irreducible semigroup ( subsemigroup of the flip-flop monoid ) that divides the transformation semigroup of " A " must divide the transition semigroup of some component of the cascade, and only the primes that must occur as divisors of the components are those that divide " A "'s transition semigroup.
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