报告题目：Pseudo L-algebras and non-commutative logic

报告人：辛小龙教授，西北大学数学学院

报告时间：2024年5月7日（周二）11:00

报告地点：20-404（第一会议室）

报告摘要： In [W. Rump, L-algebras, self-similarity, and l-groups, Journal of Algebra 320(2008) 2328-2348], the concept of L-algebras was introduced by Rump, which form the algebraic solutions of the quantum Yang-Baxter equation.

In this talk, we firstly introduce generalized structures of L-algebras, called pseudo L-algebras, which are the multiplication reduction of pseudo hoops and are structures combining two L-algebras with one compatible order. We prove that every pseudo hoop gives rise to a pseudo L-algebra and every pseudo effect algebra gives rise to a pseudo L-algebra. The self-similarity is the most important property of an L-algebra L, which guarantees to induce a multiplication on L. We introduce a notion of self-similar pseudo L-algebras and prove that a self-similar pseudo L-algebra becomes an L-algebra if and only if the multiplication is commutative.

Secondly, we use pseudo L-algebra to unify several important classes of fuzzy logic and quantum logic. We get some interesting results: (1) Every ortho-pseudo Heyting algebra is a pseudo L-algebra, which shows that pseudo L-algebras are a generalized algebraic model of non-commutative intuitionistic logic. (2) Each pseudo-MV algebra is a pseudo L-algebra, which means that pseudo L-algebras are a generalized algebraic model of non-commutative Lukasiewicz logic. (3) Every lattice-ordered pseudo effect algebra gives rise to a pseudo L-algebra and every pseudo orthomodular lattice is a pseudo L-algebra. The results shows that pseudo L-algebras are a generalized algebraic model of non-commutative quantum logic.

Finally, we use self-similar pseudo L-algebras as common model of non-classical logic. For this topic, some interesting results are obtained: (1) Every self-similar pseudo L-algebra is a pseudo BCK-algebra. (2) Every self-similar pseudo L-algebra  with a negation is a pseudo residuated lattice. (3) Every self-similar pseudo L-algebra is a pseudo-BE algebra. These results indicate that self-similar pseudo L-algebras form the common algebraic model of combinatorial logic, substructure logic and basic logic.

报告人简介：辛小龙，西北大学数学学院，博士生导师。目前主要从事模糊逻辑、逻辑代数、序代数及不确定性数学理论等方向的研究工作。在国内外学术刊物发表论文130余篇(SCI收录80余篇，EI收录50余篇)，共招收培养代数学和信息论与密码学方向的硕士研究生60余名，博士研究生23名。近年共主持国家自然科学基金面上项目、教育部留学回国人员基金项目、国家天元数学会议基金项目、陕西省自然科学基金项目等省部级项目10余项，主持并获得陕西省高等学校科技进步一等奖2项、陕西省科学技术奖二等奖1项、陕西省优秀教学成果奖二等奖1项、陕西省精品课程《数学建模》课程建设项目1项。2019年在科学出版社出版专著“逻辑代数上的非概率测度”1部，2014年，在高等教育出版社出版教材“信息论与编码理论”；2007年出版，2012年再版系列教材“高等数学”、“线性代数”、“概率论与数理统计”。