Zhuchok A. Decompositions of free products of semigroups

The decompositions of free products of semigroups are described in the terms of bands of semigroups. The method of classification semiretractions of semigroups according to the qualities of mutations is offered. With the help of this method the existence of semiretractions of free monoids is proved. The qualities of mutations are studied in terms of theoretical constructions of semigroups. The translational hulls of free products of arbitrary semigroups are studied. The characteristic of semigroups of endomorphisms free products semigroups of idempotents is obtained in terms of semigroups matrix. Algebraic qualities the translations semigroups over the Rees matrix semigroups are examined and the results of Petrich about translational hulls of regular semigroups are modified.