Archimedean domains of skew generalized power series

A skew generalized power series ring R[[S, omega, <=]] consists of all functions from a strictly ordered monoid (S, <=) to a ring R whose support is artinian and narrow, with pointwise addition, and with multiplication given by convolution twisted by an action omega to of the monoid S on the ring R. Special cases of this ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal'cev-Neumann series rings, the "unskewed" versions of all of these, and generalized power series rings. In this paper, we characterize the skew generalized power series rings R[[S, omega, <=]] that are left (right) Archimedean domains in the case where the order <= is total, or <= is semisubtotal and the monoid S is commutative torsion-free cancellative, or <= is trivial and S is totally orderable. We also answer four open questions posed by Moussavi, Padashnik and Paykan regarding the rings in the title.