Abstract: Firstly, by introducing the concept of filters in m-semilattices, some properties of filters in m-semilattices were discussed, then the filter topology on m-semilattices was constructed, and a series of properties of filter spaces were obtained. Secondly, we proved that each filter space was connected, and  the filter spaces on two-sided m-semilattices satisfied the first countability axiom. The necessary and sufficient conditions for them to be T₀ spaces and satisfied the second countability axiom were given respectively. Finally, by introducing  the concept of prime filters in m-semilattices, we discussed  the dual prime spectral spaces on m-semilattices, proved that the dual prime spectral spaces on two-sided m-semilattices were T₀ space, and gave its equivalent characterization as T₁ space.

Key words: m-semilattice, filter, prime filter, topology, dual prime spectrum