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In the last decade, some researches on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the interest of the nite power property to achieve partial solutions of this problem. We prove that the star problem is decidable in some trace monoid if and only if in the same monoid, it is decidable whether a recognizable language has the nite power property. Intermediary results allow us to give a shorter proof for the decidability of thedoi:10.1007/s00224-001-0006-x fatcat:5grot4tre5g2hhs26mcr4cirje