序代数与半环理论中的几个基础问题研究

Having essentially advanced the open question of Birkhoff and Pierce (1956) whether the field of complex numbers can be lattice-ordered, having proved Anderson's conjecture (1989) between integral domains and partially ordered algebras, having answered Belluce question (1986) on prime ideals and annihilators of MV-algebras, having established quantum B-algebras to unify non-commutative logical algebras, having generalized "the fundamental theorem of arithmetic" into non-commutative case, etc., we have obtained a wealth of related research experience. ..In this project, we will comprehensively apply the essential connections (i. e., Di Nola-Gerla-Lu-Shang correlation, refined Jaffard-Ohm-Nakayama-Kaplansky-Krull correspondence, and Chang-Mundici-Dvurevcenskij equivalence) between algebraic logics and semirings, between partially ordered algebras and integral domains, between partially ordered algebraic systems, combined with the basis of our preliminary work, to investigate several fundamental questions in partially-ordered algebras and semirings: first study whether the complex algebraic number field, then the division-rings and semirings can be directed partially ordered, and then try to develop a semimodule and directed partially ordered semimodule theory over semirings.

我们前期工作本质推进了Birkhoff-Pierce复数域格序化公开问题（1956年）的研究、证明了整环与序代数间的的Anderson猜想（1989年）、解决了Belluce关于逻辑代数理想零化子问题（1986年）、建立了统一非交换逻辑代数的量子-B代数、推广“算数基本定理”到非交换情形等、积累了较为丰富的研究经验。.本项目将在前期工作基础上，综合应用序代数、整环、半环和逻辑代数间内在联系的Di Nola-Gerla-Lu-Shang关联、精细化Jaffard-Ohm-Nakayama-Kaplansky-Krull对应、和Chang-Mundici-Dvurevcenskij等价等桥梁，研讨序代数与半环理论中的几个基础问题：首先探究复代数数域上定向序的存在性，再尝试研究除环和半环上的定向序存在性问题，进而力图发展半环上的半模及定向半模理论。

我们前期工作本质推进了Birkhoff-Pierce复数域格序化公开问题（1956年）的研究、证明了整环与序代数间的的Anderson猜想（1989年）、解决了Belluce关于逻辑代数理想零化子问题（1986年）、建立了统一非交换逻辑代数的量子-B代数、推广“算数基本定理”到非交换情形等、积累了较为丰富的研究经验。本项目在前期工作基础上，综合应用序代数、整环、半环和逻辑代数间内在联系的Di Nola-Gerla-Lu-Shang关联、精细化Jaffard-Ohm-Nakayama-Kaplansky-Krull对应、和Chang-Mundici-Dvurevcenskij等价等桥梁，研究了序代数与半环理论中的几个基础问题，得到了系列创新结果。探究了数域、（矩阵）环上的定向序和格序的存在性以及L*-序，建立了半环上的半模及定向半模等理论，并探讨了序代数图像处理等方面的应用做了探究。