NMG-代数中同态核的结构刻画

Characterizations of Homomorphic Nuclei on NMG-Algebras

逻辑代数上的Bosbach态与Rie?an态是经典概率论中Kolmogorov公理的两种不同方式的多值化推广,也是概率计量逻辑中语义计量化方法的代数公理化,是非经典数理逻辑领域中的重要研究分支.现已证明具有Glivenko性质的逻辑代数上的Bosbach态与Rie?an态等价,并且逻辑代数的Glivenko性质是研究态算子的构造和存在性的重要工具,因而是态理论中的研究热点之一.研究了NMG-代数基于核算子的Glivenko性质,证明NMG-代数具有核基Glivenko性质的充要条件是该核算子是从此NMG-代数到其像集代数的同态,并给出NMG-代数中同态核的结构刻画.这里,NMG-代数是刻画序和三角模([0,1 2],T_(NM)),([1 2,1],T_M)的逻辑系统NMG的语义逻辑代数.

Bosbach states and Riecan states are two different types of many-valued generalizations of classical probability measures on Boolean algebras by extending the prominent Kolmogorov axioms in different ways. Being regarded as algebraic and axiomatic counterparts of the semantic quantification in probabilistically quantitative logic, both states draw great interests of researchers in the community of non-classical mathematical logics. It has been proved in the literature that Bosbaeh states and Riecan states coincide on many-valued logical algebras having the Glivenko property, and that the Glivanko property plays a key role in the study of construction and existence of states on logical algebras. This paper studies the Glivanko property of NMG-algebras with respect to a nucleus, providing several necessary and sufficient conditions for the underlying nucleus to be a homomorphism into the NMG-algebra with its range as the supporting set. A particularly interesting eharaeterization shows that a nucleus on an NMG-algebra is such a homomorphism if and only if it is a double relative negation defined by an involutive element whose （canonical） negation is a fixpoint of the t-norm square operation.