Vol. 51, No. 2, 1974

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Generalized ω −ℒ-unipotent bisimple semigroups

Ronson Joseph Warne

Vol. 51 (1974), No. 2, 631–648

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 , Green’s equivalence relation, is a left congruence on E(S), we term S a generalized ω--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--unipotent bisimple semigroups SE(S) is an ω-chain of right zero semigroups).

Mathematical Subject Classification 2000
Primary: 20M10
Received: 1 February 1973
Published: 1 April 1974
Ronson Joseph Warne