论文标题
单素和纯度的连贯性
Coherency for monoids and purity for their acts
论文作者
论文摘要
本文探讨了Monoid $ s $的正确相干性，$ s $ acts的方程解决方案与$ s $ acts的注射属性属性之间的三向关系。如果每个有限的（正确）$ s $ ACT本身的每个有限生成的亚法本身都有有限的演示文稿，则Monoid $ s $是正确的连贯性。 $ s $ act $ a $的纯度属性可以用$ a $ a $的一致方程的$ a $ a $ a表示。例如，如果$ a $ a $（几乎是纯）的$ s $ act $ a $（几乎是纯），那么每$ a $（一个变量）上的所有有限的方程组都有$ a $的解决方案。同等地，如果$ a $在有限生成的亚乳汁包含在有限呈现的（单基因有限呈现的）$ s $ acts中，则绝对是纯净的（几乎是纯）。 我们的第一个主要结果表明，对于正确的连贯的单型$ s $，几乎纯净，绝对纯净的$ s $ acts的类别重合。我们的第二个主要结果是，当且仅当MFP-Pure和绝对纯$ S $ - acts的类别相吻合时，Monoid $ s $是正确的连贯性：如果有限地呈现在有限的包含在单一有限的一单基因中$ s $ s $ s $ s $ s $ s-acts中，则$ s $ act是mfp-pure。我们给出了不正确连贯的单型$ s $的具体示例，使得几乎纯净且绝对纯净的$ s $ acts的类别相吻合。最后，我们在所有几乎所有纯$ s $ acts的单体$ s $上给出条件，这是绝对纯粹的，就有限的$ s $ acts，其有限生成的亚表演和某些规范扩展而言。
This article examines the three-way relationship between right coherency of a monoid $S$, solutions of equations over $S$-acts, and injectivity properties of $S$-acts. A monoid $S$ is right coherent if every finitely generated subact of every finitely presented (right) $S$-act itself has a finite presentation. Purity properties of an $S$-act $A$ may either be expressed in terms of solutions in $A$ of certain consistent sets of equations over $A$, or in terms of injectivity properties. For example, an $S$-act $A$ is absolutely pure (almost pure) if every finite consistent set of equations over $A$ (in one variable) has a solution in $A$. Equivalently, $A$ is absolutely pure (almost pure) if it is injective with respect to inclusions of finitely generated subacts into finitely presented (monogenic finitely presented) $S$-acts. Our first main result shows that for a right coherent monoid $S$ the classes of almost pure and absolutely pure $S$-acts coincide. Our second main result is that a monoid $S$ is right coherent if and only if the classes of mfp-pure and absolutely pure $S$-acts coincide: an $S$-act is mfp-pure if it is injective with respect to inclusions of finitely presented subacts into monogenic finitely presented $S$-acts. We give specific examples of monoids $S$ that are not right coherent yet are such that the classes of almost pure and absolutely pure $S$-acts coincide. Finally we give a condition on a monoid $S$ for all almost pure $S$-acts to be absolutely pure in terms of finitely presented $S$-acts, their finitely generated subacts, and certain canonical extensions.
