Coherency, free inverse monoids and free left ample monoids
Coherency, free inverse monoids and free left ample monoids
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一致性、自由逆幺半群和自由左充足幺半群
DOI：
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发表时间：
2015
期刊：
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通讯作者：
V. Gould
中科院分区：
文献类型：
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作者：
Miklós Hartmann;V. Gould
么半群$S$是右相干的，如果每个有限表现的右$S$-作用的每个有限生成的子作用都是有限表现的。一个环R的相应概念是：每个R-模的每个R-生成子模都是R-模。对于幺半群（和环），右凝聚性是一个有限性质，它决定了右S-作用（右R-模）类的模型伴侣的存在性，因此存在闭右S-作用（右R-模）类是可公理化的。
Choo、Lam和Luft证明了自由环是右（和左）凝聚的;他们与Ruskuc一起证明了群和自由幺半群具有相同的性质。我们证明，自由逆幺半群没有。
任何自由逆幺半群包含自由左充足幺半群作为子幺半群，实际上自由幺半群在同一个生成元集合上。本文的主要目的是证明自由左充足么半群是右凝聚的。此外，通过使用相同的技巧，我们证明了自由逆和自由左充足幺半群都满足$（{\bfR}）$，$（{\bfr）}$，$（{\bfL}）$和$（{\bfl）}$，这些条件来自于右$S$-作用类和左$S$-作用类的可公理化性。
A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. The corresponding notion for a ring $R$ states that every finitely generated submodule of every finitely presented right $R$-module is finitely presented. For monoids (and rings) right coherency is a finitary property which determines the existence of a model companion of the class of right $S$-acts (right $R$-modules) and hence that the class of existentially closed right $S$-acts (right $R$-modules) is axiomatisable.
Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruskuc, have shown that groups, and free monoids, have the same properties. We demonstrate that free inverse monoids do not.
Any free inverse monoid contains as a submonoid the free left ample monoid, and indeed the free monoid, on the same set of generators. The main objective of the paper is to show that the free left ample monoid is right coherent. Furthermore, by making use of the same techniques we show that both free inverse and free left ample monoids satisfy $({\bf R})$, $({\bf r)}$, $({\bf L})$ and $({\bf l)}$, conditions arising from the axiomatisability of classes of right $S$-acts and of left $S$-acts.