正则FI-代数的刻画及成为Boole代数的条件

Characterizations of regular FI-algebras and condition for them to be Boolean algebras

正则FI-代数是仅基于蕴涵算子在一般集合上建立的逻辑代数。基于正则FI-代数的公理组以及诸多性质之间的内部联系,给出了正则FI-代数的两个公理组条件更少的刻画定理,简化了正则FI-代数的定义形式。在正则FI-代数中引入蕴涵分配性,探讨了蕴涵分配正则FI-代数的若干性质,证明了蕴涵分配正则FI-代数与Boole代数是相互等价的代数系统,给出了Boole代数的一种新的刻画,使其在形式上更接近于二值逻辑代数。

The regular FI-algebras are built up on general sets merely by the implication operation.In this paper,based on axiom groups and internal relations of properties of regular FI-algebras,two characterizations of regular FI-algebras with fewer axioms are given,which simplify the definition of regular FI-algebras.Moreover,implicative distributivity is introduced into regular FI-algebras and some properties of implicative distributive FI-algebras are discussed.It is proved that implicative distributive regular FI-algebras and Boolean algebras are equivalent algebraic structures.A new characterization for Boolean algebras is given,showing them to be more similar to two-valued logic algebras in form.