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).