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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 ( En : n ∈ N, the nonnegative integers) and ℒ, Green’s equivalence
relation, is a left congruence on E(S), we term S a generalized ω-ℒ-unipotent
bisimple semigroup. We characterize S in terms of (I,0), an co-chain of left zero
semigroups (Ik : k ∈ N); (J∗) an ω-chain of right groups (Jk : k ∈ N); a
homomorphism (n,r) → α(n,r) of C, the bicyclic semigroup, into End (I,o), the
semigroup of endomorphisms of (I,o) (iteration); a homomorphism (n,r) → β(n,r) of
C into End (J,∗); and an (upper) anti-homomorphism j → Af of (J,∗) into TI,
the full transformation semigroup on I (Aj is “almost” an endomorphism).
In fact, S≅((i,(n,k),j) : i ∈ In,j ∈ Jk,n,k ∈ N) under the multiplication
(i,(n,k),j)(u,(r,s),v) = (i∘ (uAjα(k,n))),(n + r − min(k,r),k + 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-ℒ-unipotent bisimple semigroups SE(S) is
an ω-chain of right zero semigroups).
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