A centered right $S$ -polygon (synonyms: $S$ -operand, $S$ -system) $A$ is called right compressive if $AS\ne0$ and $\alpha a=\alpha b\to\alpha=0\vee(a,b)\in(\operatorname{Ker}A)_S$ and leftt compressive if $AS\ne0$ and $\alpha a=\beta a\to\alpha=\beta\vee Aa=0$ . Here $(\operatorname{Ker}A)_S$ is the congruence on the semigroup $S$ called the kernel of the $S$ -polygon $A$ which is defined as follows: $(a,b)\in(\operatorname{Ker}A)_S\leftrightarrow(\forall\,\alpha\in A)(\alpha a=\alpha b)$. The intersection of the kernels of all right (left) compressive$S$-polygons is called the right (left) compressive radical of$S$. In this paper we study compressively semisimple and compressively radical semigroups.Bibliography: 11 titles.