As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in $${\mathscr {I}}_{X}$$ I X , a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse $$\wedge$$ ∧ -semigroups is developed. This class of inverse $$\wedge$$ ∧ -semigroups, otherwise known as inverse algebras, includes partial automorphism monoids of entities such as graphs, vector spaces and modules. A workable theory of decompositions is reached; however complete distributivity is required for results approaching those of the classical case.

中文翻译：

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完全原子逆相遇半群中逆半群的表示

作为 $${\mathscr {I}}_{X}$$ IX 中逆半群表示的经典（Schein）理论特征的适当推广，将逆半群通过同态表示成完全原子逆的理论$$\wedge$$ ∧ -semigroups 已开发。这类逆 $$\wedge$$ ∧ -semigroups，也称为逆代数，包括实体的部分自同构幺半群，例如图、向量空间和模块。达到了一个可行的分解理论；然而，接近经典情况的结果需要完全分布。