Primitive Idempotents in Semigroups
by
Aleksandar Stamenkovic
University of Nis, Nis, Yugoslavia
Coauthors: Stojan Bogdanovic (University of Nis), Miroslav \'Ciric (University of Nis)

The concepts of primitivity and 0-primitivity play an outstanding role in theory of semigroups. For example, they are used to define completely simple and completely 0-simple semigroups. Semigroups without zero all of whose idempotents are primitive, called primitive semigroups , and semigroups with zero all of whose nonzero idempotents are 0-primitive, called 0-primitive semigroups , have been a subject of interest of many authors. Primitive regular semigroups are exactly completely simple ones, whereas 0-primitive regular semigroups were characterized as orthogonal sums of completely 0-simple semigroups, by Venkatesan (1966) and Steinfeld (1966). These semigroups have been also studied by Hall (1970), Lallement and Petrich (1966), Preston (1969), and more information about them can be found in the books by Clifford and Preston, Steinfeld, Bogdanovi\'c and \'Ciri\'c and Howie. Primitive \pi-regular semigroups were characterized by Bogdanovi\'c and Mili\'c (1984) as nil-extensions of completely simple semigroups, and 0-primitive \pi-regular semigroups were described by Bogdanovi\'c and \'Ciri\'c (1992) as certain extensions of 0-primitive regular semigroups.

The main our aim is to give certain general properties of primitive and 0-primitive idempotents and to describe primitive and 0-primitive semigroups in some classes more general than the class of \pi-regular semigroups, such as E-inversive, B-inversive, 0-inversive and 0-B-inversive semigroups. On the other hand, it is known that the primitivity and 0-primitivity of idempotents are closely related to the minimality and 0-minimality of twosided, onesided and bi-ideals of a semigroup, and we study this relationship, too. For example, we determine some necessary and sufficient conditions for a left ideal of a semigroup with zero generated by a nonzero idempotent to be a left 0-simple semigroup. The obtained results generalize the above mentioned results of Venkatesan, Steinfeld, Bogdanovi\'c and Mili\'c, Bogdanovi\'c and \'Ciri\'c, Lallement and Petrich and others, as well as the results of Mitsch and Petrich, announced in 1997, concerning 0-primitive 0-inversive and primitive E-inversive semigroups.

Date received: May 6, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-20.