In this paper we generalize some results about the chaos-related properties on the product of two semiflows, which appeared in the lite\-rature in the last few years, to the case of the most general possible acting monoids. In order to do that we introduce some new notions, namely the notions of a directional, psp and sip monoid, and the notion of a strongly transitive semiflow. In particular, we obtain a sufficient condition for the Devaney chaoticity of a product, which works for the (very large) class of the psp acting monoids.