Basic semigroups: theory and applications

A concept of basic matrix semigroups over fields (with some variations) is introduced and throughly investigated. Sections 1 and 2 contain main definitions, Section 3 treats some properties of basic semigroups, Section 4 is devoted to some application of basic semigroups: matrix representations (including faithful representations), finiteness theorems, the problem of Korjakov (when a matrix semigroup over field K is conjugate to a matrix semigroup over a proper subfield of K). The paper is a survey and contains no proofs (which may be found in papers from References)

Retract extensions of ordered sets

The concept of retract extension of an ordered set is introduced and studied. We give an example of a retract extension and an example that is not a retract extension. Bibliography: 12 titles. ©2006 Springer Science+Business Media, Inc

Bi-ideals in ordered semigroups and ordered groups

It is shown that an ordered semigroup is right and left simple if and only if it does not contain proper bi-ideals. An example showing that an ordered semigroup without proper bi-ideals need not be an ordered group is constructed. ©2002 Plenum Publishing Corporation