Generalizedω−ℒ-unipotent bisimple semigroups
R. J. Warne · Pacific Journal of Mathematics · 1974
Let S be a bisimple semigroup and let E(S) be the set of idempotents of S. If E(S) is an ω-chain of rectangular bands (E n : neN, the nonnegative integers) and £?, Green's equivalence relation, is a left congruence on E(S) 9 we term Sa generalized ω-^-unipotent bisimple semigroup.We characterize S in terms of (J, o), an ω-chain of left zero semigroups (I k : keN); (/, *) an ω-chain of right groups (J k : k e N); a homomorphism (n, r) ->α (n , r) of C, the bicyclic semigroup, into End (J, o), the semigroup of endomorphisms of (/, o) (iteration); a homomorphism (n, r) -> β (n , r) of C into End (J, *); and an (upper) anti-homomorphism j -> Aj of (J, *) into T I9 the full transformation semigroup on I (A ά is "almost" an endomorphism).In fact, S ^ ((i, (n, k), j): i 6 /", j e J k , n,keN) under the multiplication (i, (n, k),j)(u, (r, s), / y)=(^°(^A 7 α (fc , 7l) )), (π+r-min (fc, r), /c+s-min (k, r)), jβ(r, S )*v) (Theorem 4.1).We then characterize (J, *) as a semi-direct product of an ω-chain of right zero semigroups by an ω-chain of groups.Finally, we specialize Theorem 4.1 to obtain our previous characterization of ω-^-unipotent bisimple semigroups S(E(S) is an ω-chain of right zero semigroups).