On the structure of lower radical semigroups
Richard Wiegandt · Czechoslovak Mathematical Journal · 1972
The basic idea of this paper is due to the fact discovered in the recent categorytheoretically investigations cf [12], that the Wedderburn-Artinian decomposition theory of rings and STEINFELD'S [7] analogous results for semigroups are categorically duals of each other.The exquisite results of REES [5], SCHWARZ [6] and STEINFELD [7], [8] establish satisfactory characterizations of semigroups being unions of comple te 0-simple subsemigroups.Up to this time such "well-behaved" semigroups were considered and treated as semisimple semigroups, and indeed they are semi simple with respect to the nil radical property [6], but these semigroups do not satisfy those properties what a semisimple class ought to satisfy.Since the radical and semisimple properties are categorically dual notions (cf.[10]), so by the duality mentioned before it is reasonable to try to characterize all the semigroups of the smallest radical class which contains these well-behaved semigroups.Thus the well-behaved semigroups will be considered as radical semigroups according to an appropriate radical prop erty.Using the methods of the recent ring-theoretical developments, namely those of SULINSKI-ANDERSON-DIVINSKY [9], ARMENDARIZ-LEAVITT [1] and WATTERS [11], we shall construct the lower radical class determined by a class of simple idempotent semigroups with zero.It will turn out that this lower radical class is always hereditary, further the lower radical semigroups will be classified in Theorem 2. Thus for instance, if the lower radical class is determined by the class of completely 0-simple semigroups, then to any radical semigroup S there belong an ordinal number depending on S and a strictly ascending chain 0 = /Q cz ... с /^ = S of twosided ideals of S such that every Rees factor semigroup /я+хДя (0 ^ A < С) is the 0-disjoint union (i.e. the 0-direct union) of completely 0-simple subsemigroups (and the complete 0-simple semigroups are isomorphic to regular Rees matrix semigroups).