报告名称：Coherency for monoids and purity for their acts

报告专家：杨丹丹

专家所在单位：西安电子科技大学

报告时间：2023年5月20日下午2：30-4:30

报告地点：威廉希尔203

专家简介：杨丹丹，西安电子科技大学教授，博导。2014年获得英国约克大学的数学博士学位。主要研究方向为半群理论。目前主持国家自然科学基金面上项目和陕西省杰出青年科学基金项目各一项；获陕西省青年科技奖，入选陕西省高校青年杰出人才支持计划。研究成果发表在Adv. Math., Quart. J. Math., J. Algebra等期刊。

报告摘要：

In this talk, we study the relationship between coherency of a monoid and purity properties of its acts.An underlyingmotivation comes from the following question for an algebra: when does the guaranteed solution of a finite consistent set of equations in one variable liftto the guarantee of solutions of finite consistent sets equations in any(finite) number of variables? This is a long-standing and intriguing problem,with a positive answer for some algebraic structures (e.g. groups andsemigroups) but not fully understood for modules over rings or acts overmonoids.

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 andabsolutely pure $S$-acts coincide. We give specific examples of monoids $S$ that are not right coherentyet are such that the classes ofalmost 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.