On regularity and intra regularity of the components of a Morita context of monoids

The notions of regular, intra regular and duo bi-acts of a Morita context of monoids have been introduced. Some characterizations of regular and intra regular bi-acts have been obtained in terms of b-subacts. Also connections among regularity of acts and k-regularity of power semimodules, intra regularity of acts and intra k-regularity of power semimodules have been observed using power functor. Then the study of the interplay among various components of a Morita context of monoids in terms of regularity, intra regularity and duoness have been accomplished. It has been deduced that regularity, intra regularity and being duo are Morita invariant properties. Finally regular (intra regular, duo) Morita context has been defined and it has been proved that for a duo Morita context of monoids regular and intra regular Morita context coincide.