Sheaves and $\mathfrak{Ta}$-bicompactifications of mappings

The paper is devoted to an investigation of relations between bicompactifications of mappings and sheaves of algebras. Bicompactifications of mappings are a generalization of compactifications of topological spaces, and sheaves of algebras take place of algebras of continuous bounded functions on topological spaces.
The first section contains a historical review of main constructions and notions used in the paper as well as a short introduction to the theory of bicompactifications of mappings. In particular, we state here basic definitions and recall some statements about bicompactifications of mappings that were obtained earlier.
In the second section some new topological properties of the fan product and the inverse limit are proved.
The third section contains important constructions which are used for an upbuilding of bicompactifications of mappings. Several new properties of these constructions are proved.
The fourth section is devoted to a definition and an investigation of algebras of functions on mappings. In this section a natural topology on these algebras is defined; the class of globally completely regular mappings is singled out for which such algebras play a role similar to that of algebras of continuous bounded functions on completely regular spaces; a functor from the category of mappings to the category of perfect globally completely regular mappings is constructed which preserves algebras of continuous “bounded” functions on mappings; a correspondence between “mappings” of mappings and homomorphisms of their algebras is investigated.
In the fifth section sheaves of algebras connected with mappings are defined and investigated.
The sixth section contains a proof of the main result of the paper: there exists a one-to-one correspondence preserving the order between the set of all $\mathfrak{Ta}$-bicompactifications of a given mapping and the set of all sheaves of a special kind.
In the seventh section we define maximal closed ideals of sheaves of algebras; relations between these ideals and points of $\mathfrak{Ta}$ of a given mapping are investigated.