Cubical homology and the Leech dimension of free partially commutative monoids

The paper is devoted to problems arising when applying homological algebra to computer science. It is proved that the Leech dimension of a free partially commutative monoid is equal to the least upper bound of the cardinalities of finite subsets of pairwise commuting generators of the monoid. For an arbitrary free partially commutative monoid M(E,I) in which every subset of pairwise commuting generators is finite and for any contravariant natural system F on M(E,I) we construct a semicubical set T(E, I) with a homological system (F) over bar on this set such that the Leech homology groups H(n)(M(E, I), F) are isomorphic to the cubical homology groups H(n)(T(E, I), (F) over bar). Complexes of Abelian groups are also constructed enabling one to obtain (under additional finiteness conditions) algorithms for computing the Leech homology groups and homology groups with coefficients in right M(E, I)-modules.