Coherency for monoids and purity for their acts

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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.