Vol. 68, No. 2, 1977

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On a representation theory for ideal systems

Paul Ezust

Vol. 68 (1977), No. 2, 347–367

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.

Mathematical Subject Classification 2000
Primary: 18A05
Received: 30 June 1975
Revised: 30 November 1976
Published: 1 February 1977
Paul Ezust