TY - JOUR

T1 - Semigroups whose idempotents form a subsemigroup

AU - Birget, Jean Camille

AU - Margolis, Stuart

AU - Rhodes, John

PY - 1990/4

Y1 - 1990/4

N2 - 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[formula omitted] * G if and only if Sn belongs to u[formula omitted]. Here u[formula omitted] 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.

AB - 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[formula omitted] * G if and only if Sn belongs to u[formula omitted]. Here u[formula omitted] 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.

UR - http://www.scopus.com/inward/record.url?scp=84971790237&partnerID=8YFLogxK

U2 - 10.1017/S0004972700017986

DO - 10.1017/S0004972700017986

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AN - SCOPUS:84971790237

SN - 0004-9727

VL - 41

SP - 161

EP - 184

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

IS - 2

ER -