A note in inverse and dual semigroups
C. K. Lai, K. P. Shum · Czechoslovak Mathematical Journal · 1992
In studying the structure of inverse semigroups, L. Markl [2] has proved the following conditions on semigroups with 0 are equivalent:(i) S is an inverse semigroup and the union of a finite number of its 0-minimal left (right) ideals, (ii) S is the union of a finite number of its quasi-ideals, and all these quasi-ideals form a special complete system, (iii) S is an inverse semigroup and the union of a finite number of its 0-minimal quasi-ideals.(iv) S is an inverse semigroup with finitely many idempotents and every non-zero idempotent is primitive.However, it has been noticed by him that the following condition: (v) S is a 0-direct union of finitely many two-sided ideals which are completely 0-simple inverse subsemigroups of S. is weaker than any one of the conditions (i) to (iv).In this note, we observe that if each summand in (v) satisfies min-r (that is, Minimum condition on right ideals), then the above conditions are in fact all equaivalent.We shall prove that any one of these conditions is a necessary and sufficiency condition for a semigroup to be semisimple and dual.Thus a characterization theorem for semisimple dual semigroup is obtained.Throughout the paper, every semigroup has 0 and contains more than one element.The reader is referred to O. Steinfeld [7] for all terminology and definitions not given here.