In widely divergent branches of
mathematics, objects emerge which bear sufficient formal resemblance to the ideals of
rings for them to be called “ideals”. In a series of papers, Karl E. Aubert developed
an axiomatic theory of ideal systems which subsumes most of the existing “ideal”
theories. The goal of this paper is a representation theory for ideal systems in
commutative monoids which will allow the formation of a cohomology theory for
these systems. One of the results is a theorem which gives at once a monadic
(co)homology for each ideal system. The base category in the monad includes PTOP,
the category of pointed topological spaces and basepoint-preserving continuous maps,
as a full subcategory and, for each ideal system, the category of algebras
associated with the monad consists of the module systems over the ideal
system. It is the module systems which are the principal objects of this
study.
