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Semigroups whose idempotents form a subsemigroup

Published online by Cambridge University Press:  17 April 2009

Jean-Camille Birget ,
Stuart Margolis  and
John Rhodes
Jean-Camille Birget
Affiliation:
Computer Science DepartmentUniversity of NebraskaLincoln, NE 68588USA
Stuart Margolis
Affiliation:
Computer Science DepartmentUniversity of NebraskaLincoln, NE 68588USA
John Rhodes
Affiliation:
Mathematics DepartmentUniversity of CaliforniaBerkeley, CA 94720USA
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Abstract

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We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:

(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.

(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.

(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.

For these three classes of semigroups, type-II is equal to type-II construct.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 2 , April 1990 , pp. 161 - 184
Copyright
Copyright © Australian Mathematical Society 1990

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