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0-simple 5-congruence 9l-class 9l-simple archimedean semigroup automorphism binary relation bitranslations cancellative semigroup commutative semigroup completely prime ideal completely regular semigroups completely semiprime ideal completely simple semigroup congruence Conversely COROLLARY cyclic semigroup define DEFINITION denoted dense extension dually element equivalent Exercise exists finite following conditions function globally idempotent Hence homomorphic image identity implies intersection inverse semigroup isomorphism left ideal left translation left zero semigroup LEMMA mapping maximal multiplication nonempty set nonempty subset nonzero normal band notation obtain one-to-one partial homomorphism partially ordered set permutable Petrich product of semigroups Proof PROPOSITION Prove rectangular band rectangular group Rees matrix semigroup retract extension right ideal right zero semigroup semi semigroup of transformations semigroup Q semigroups of order semilattice congruence semilattice decomposition semilattice of groups semilattice of semigroups Show strict extension strong semilattice subdirect product subdirectly irreducible subgroup subsemigroup THEOREM translational hull unique weakly reductive semigroup