On the classification of the finite left-duo \(P\)-(\(\Delta\)-) semigroups (Q1175615)

A semigroup \(S\) is called a left-duo semigroup if every left ideal of \(S\) is a two-sided ideal. A semigroup \(S\) is called a \(P\)-semigroup (\(\Delta\)-semigroup) if its congruences are permutable (form a chain). This paper reports the classification of finite left-duo \(P\)-semigroups and the structure of these semigroups without proof. The starting point of this study is based on the fact that such a semigroup consists of at most two archimedean components. These components are completely described. Also, finite left-duo \(\Delta\)-semigroups are treated as a special case of \(P\)-semigroups.