报告题目：Coherency for monoids and purity for their acts

报 告 人：杨丹丹教授 (西安电子科技大学)

报告时间：2023年6月28日（星期三）16: 00-17: 00

腾讯会议：625-842-301

主 持 人：乔守红

报告摘要：

In this talk, we study the relationship between coherency of a monoid and purity properties of its acts. An underlying motivation comes from the following question for an algebra: when does the guaranteed solution of a finite consistent set of equations in one variable lift to 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 and semigroups) but not fully understood for modules over rings or acts over monoids.

       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. 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.

      This talk is based on joint work with Victoria Gould [Advances in Mathematics, 2023].

报告人简介：杨丹丹，西安电子科技大学教授，博士生导师，陕西省杰青。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、陕西省杰出青年科学基金项目；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,陕西省科协青年人才托举计划。研究成果发表在Advances in Mathematics(2篇)等国际著名期刊上。