Hereditary radicals and 0-bands of semigroups
With any abstract class \({\mathcal R}\) of semigroups is associated the map \(S\to \rho (S)\), the union of the ideals of S which belong to \({\mathcal R}\). This map is a ``radical'' (and \({\mathcal R}\) is a ``radical class'') if (i) \({\mathcal R}\) is closed under Rees quotients, (ii) \(S\in {\mathcal R}\) if and only if \(\rho (S)=S\) and (iii) \(\rho (S/\rho (S))=\{0\}\) for any S. For example, the class \({\mathcal N}\) of nilsemigroups determines the ``Clifford'' radical and the class \({\mathcal L}\) of locally nilpotent semigroups determines the ``Shevrin'' radical. The author studies the interaction between these, and other, radicals, such as the ``McCoy'' radical, with 0-band decompositions of semigroups. A radical \(\rho\) is ``restorable by the components of every 0-band'' if whenever S is a 0-band \(\{S_{\alpha}:\) \(\alpha\in \Omega \}\), (where the components \(S_{\alpha}\) are 0-disjoint), then \(\rho\) (S) is the largest ideal contained in \(\cup \{\rho (S_{\alpha}):\) \(\alpha\in \Omega \}\). Amongst other results, it is shown that the two radicals above have this property.
scientific article; zbMATH DE number 4079760
scientific article