有界BCK-代数中有限生成的对偶理想

Finitely Generated Dual Ideals in BCK—algebras

本文是作者[8]的继续,在[8]中我们引进了对偶理想的概念,本文主要讨论了,在有界BCK-代数中如何由一个集合生成对偶理想,并给出了有限生成对偶理想的一些特征。我们顺便指出[3]中主要结果是错误的,我们的定理3包含了对[3]中错误的纠正。

We begin by pointing out the error in B Ahmad's 《Dual ideals in BCK-algebras Ⅱ (Math. Seminor Notes; vol 10(1982), 243-250)》In this note, we mainly discuss the finitely generated dual ideals in bounded BCK=algebras. The main results are as follows.Definition 2. ([8]) A nonempty subset D in bounded BCK-algebra X is called a dual ideal if(1) 1∈D(2) N(Ny*Nx)∈D and x∈D imply y∈D, Vx,y∈X.Theorem 1. If A is a nonempty subset of bounded BCK=algebra X, then the set [A) composed of all x∈X such that there are a_1, a_2, a_3…, a_n∈A saisfying ((Nx*Na_1) *…) * Na_n=0 is the least dual ideal containing A.This [A) is called the dual ideal generated by A.We denote [{a})=[a).Definition 3. Let D be any dual ideal of X. If there is a in X such that D= {x∈X: a≤x}, then D is called a principal dual ideal.The finitely generated dual ideals have the following characterizations.Theorem 2. If X is a bounded implicative BCK-algebra, then any dual ideal D of X is finitely generated iff D is a principal dual ide.Defiuition 4. If for every sequace D_1(?)D_2(?)…(?)D_n(?)… of sets consisting of dual ideals of bounded BCK-algebra X, there is a natural number k such that D_k=D_(k+1)=… then we call X satisfying the condition DACC.Theorem 3. If X is a bounded BCK-algebra, then the following conditions are equivalent:(26) Every dual ideal is finitely generated; (27) X satisfies DACC(28) Every nonempty set of dual ideals exists maximal element.This theorem corrects B. Ahmad's error.Theorem 4. If X is a positive implicative and involutory BCK-algebra, D a dual ideal of X, and d a fixed element of X, then A={x∈X: N(Nx * Nd)∈D} is the least dual ideal containg D and d.