  Regular dilation has recently been extended to graph product of $\mathbb{N}$,
where having a *-regular dilation is equivalent to having a minimal isometric
Nica-covariant dilation. In this paper, we first extend the result to right LCM
semigroups, and establish a similar equivalence among *-regular dilation,
minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This
result can be applied to various semigroups to establish conditions for
*-regular dilation.
