We prove that the Leech dimension of any free partially commutative monoid is equal to the supremum of numbers of its mutually commuting generators. As a consequence, we confirm a conjecture that if a free partially commutative monoid does not contain more than $n$ mutually commuting generators, then it is of homological dimension $\leqslant n$. We apply this result to the homological dimension of asynchronous transition systems. We positively answer the question whether the homological dimension of an asynchronous transition system is not greater than the maximal number of its mutually independent events.
"On the Leech dimension of a free partially commutative monoid." Tbilisi Math. J. 1 71 - 87, 2008. https://doi.org/10.32513/tbilisi/1528768824