Generalizations of L- and M-weakly compact operators

L- and M-weakly compact operators were introduced by Meyer-Nieberg in the beginning of seventies in attempts of a diversification of the concept of weakly compact operators via imposing Banach lattice structure on the range or on the domain of operators. We investigate regularity and algebraic properties of various generalizations of L- and M-weakly compact operators from a unified point of view via using regularly P-operators.