Vol. 28, No. 1, 1969

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ISSN: 0030-8730
On Croisot’s theory of decompositions

Kenneth Kapp

Vol. 28 (1969), No. 1, 105–115
Abstract

Croisot gave a definition of (m,n)-regularity which he then showed defined four logically distinct classes of semi-groups. However, semigroups with nilpotent elements did not fall within his classification. Our generalization of (m,n)0-regularity remedies this exclusion; countably many distinct classes of semi-groups are defined.

In particular we investigate the structure of semigroups which are (2,2)0-regular. We show that a semigroup S is in this class precisely when for each x S either x2 = 0 or x2 Hx. Further, each regular 𝒟-class together with 0 of such a semigroup is itself a completely 0-simple semigroup. The (2,2)0-regularity condition is specialized to that of absorbency: for each a,b S either ab = 0 or ab (Ra Lb). We show that a regular absorbent semigroup is just a mutually annihilating collection of completely 0-simple semigroups.

Mathematical Subject Classification
Primary: 20.93
Milestones
Received: 31 July 1967
Published: 1 January 1969
Authors
Kenneth Kapp