English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This article examines the three-way relationship between right coherency of amonoid $S$, solutions of equations over $S$-acts, and injectivity properties of$S$-acts. A monoid $S$ is right coherent if every finitely generated subact ofevery finitely presented (right) $S$-act itself has a finite presentation.Purity properties of an $S$-act $A$ may either be expressed in terms ofsolutions in $A$ of certain consistent sets of equations over $A$, or in termsof injectivity properties. For example, an $S$-act $A$ is absolutely pure(almost pure) if every finite consistent set of equations over $A$ (in onevariable) has a solution in $A$. Equivalently, $A$ is absolutely pure (almostpure) if it is injective with respect to inclusions of finitely generatedsubacts into finitely presented (monogenic finitely presented) $S$-acts.  Our first main result shows that for a right coherent monoid $S$ the classesof almost pure and absolutely pure $S$-acts coincide. Our second main result isthat a monoid $S$ is right coherent if and only if the classes of mfp-pure andabsolutely pure $S$-acts coincide: an $S$-act is mfp-pure if it is injectivewith respect to inclusions of finitely presented subacts into monogenicfinitely presented $S$-acts. We give specific examples of monoids $S$ that arenot right coherent yet are such that the classes of almost pure and absolutelypure $S$-acts coincide. Finally we give a condition on a monoid $S$ for allalmost pure $S$-acts to be absolutely pure in terms of finitely presented$S$-acts, their finitely generated subacts, and certain canonical extensions.

Coherency for monoids and purity for their acts