An incline S is a commutative semiring where $r+1=1$ for any $r\in S$. 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.

中文翻译：

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倾斜上半模块的结构和表示

斜度S是交换半环，其中$\mathrm{[R}+\mathrm{1个}=\mathrm{1个}$ 对于任何 $\mathrm{[R}\in \mathrm{小号}$。我们注意到，S-半模的理想晶格自然是S-半模，当S是传递性时，其全等晶格也是如此。我们证明，完全S-半模的类别以及对偶函子，内部同态和张量积都是⋆-自治类别。我们定义与Birkhoff子直接乘积分解有关的局部和全局最大同余。我们证明S-半模块，代数S-半模块和拓扑S-半模块的类别是等效的。最后，我们得到任何S-半模块的捆表示。