Abstract
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.
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