Structure and representation of semimodules over inclines

An incline S is a commutative semiring where r+1=1 for any r∈S. We note that the ideal lattice of an S-semimodule is naturally an S-semimodule and so is its congruence lattice when S is transitive. We prove that the categories of complete S-semimodules, together with dual functor, internal hom and tensor product, is a ⋆-autonomous category. We define the locally and globally maximal congruences which are related to Birkhoff subdirect product decomposition. We show that the categories of S-semimodules, algebraic S-semimodules and topological S-semimodules are equivalent. Finally, we get a sheaf representation of any S-semimodule.