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In this paper, we investigate some special regularities and decay propertiesof solutions to the initial value problem(IVP) of the Benjamin equation. Themain result shows that: for initial datum $u_{0}\in H^{s}(\mathbb{R})$ with$s>3/4,$ if the restriction of $u_{0}$ belongs to $H^{l}((x_{0}, \infty))$ forsome $l\in \mathbb{Z}^{+}$ and $x_{0}\in \mathbb{R},$ then the restriction ofthe corresponding solution $u(\cdot, t)$ belongs to $H^{l}((\alpha, \infty))$for any $\alpha\in \mathbb{R}$ and any $t\in(0, T)$. Consequently, this type ofregularity travels with infinite speed to its left as time evolves.

On the propagation of regularity and decay of solutions to the Benjamin equation