For a monoid $M$, we denote by $\mathbb G(M)$ the group of units, $\mathbb E(M)$ the submonoid generated by the idempotents, and $\mathbb G_L(M)$ and $\mathbb G_R(M)$ the submonoids consisting of all left or right units. Writing $\mathcal M$ for the (monoidal) category of monoids, $\mathbb G$, $\mathbb E$, $\mathbb G_L$ and $\mathbb G_R$ are all (monoidal) functors $\mathcal M\to\mathcal M$. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We show that this monoid has size $15$, and describe its algebraic structure. We also show how to associate certain lattice invariants to a monoid, and classify the lattices that arise in this fashion. A number of examples are discussed throughout, some of which are essential for the proofs of the main theoretical results.

中文翻译：

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幂等和单边单位：格不变量和幺半群范畴的函子半群

对于幺半群 $M$，我们用 $\mathbb G(M)$ 表示单位群，$\mathbb E(M)$ 表示由幂等项生成的次幺半群，以及 $\mathbb G_L(M)$ 和 $\ mathbb G_R(M)$ 由所有左单元或右单元组成的子幺半群。为幺半群的（幺半群）范畴写 $\mathcal M$，$\mathbb G$、$\mathbb E$、$\mathbb G_L$ 和 $\mathbb G_R$ 都是（幺半群）函子 $\mathcal M\to \mathcal M$。还有其他自然函子与由幂等和单边或双边单元组合生成的子幺半群相关联。上述函子生成了一个以组合为运算的幺半群。我们证明了这个幺半群的大小为 $15$，并描述了它的代数结构。我们还展示了如何将某些格不变量与幺半群相关联，并对以这种方式出现的格进行分类。整个过程中讨论了许多示例，