We devise a framework which leads to the formulation of a unified theory of normality (regularity), semi-normality (semi-regularity), s-normality (s-regularity), feeblynormality (feebly-regularity), pre-normality (pre-regularity), and others. Certain aspects of theory are given by unified proof.

In the present paper, we define regularity and normality like strong S1 regularity, S1 regularity, weak S1 regularity, strong S1 normality, S1 normality as well as weak S1 normality on L-topological spaces. Also we investigate some of their properties and the relations between them. 2010 Mathematics Subject Classification: 54A40, 54D10, 54D15, 54D65, 03E72.