Multiplicative order compact operators between vector lattices and Riesz algebras

In this paper, we present and examine the concept of multiplicative order compact operators from vector lattices to Riesz algebras. Specifically, a linear operator T from a vector lattice X to an Riesz algebra E is deemed mmm-compact, if every net x(alpha) in an m-bounded subset of X possesses a subnet x alpha beta such that Tx(alpha beta) (mo)(->) y for some y E E. Moreover, we introduce and investigate mmm-M- and mmm-L-weakly compact operators.