元素个数不超过6的真伪BCK-代数

Proper Pseudo-BCK Algebras of Elememts Number n≤6

研究元素个数不超过6的真伪BCK-代数的计数问题。首先,证明了在元素个数不超过3的偏序集上不存在真伪BCK-代数。其次,引入NP-型偏序集(不存在真伪BCK-代数的含最大元的偏序集)、偏序集的层、次余原子等概念,证明了在一个层数n≤3的NP-型偏序集上添加孤立余原子(或孤立次余原子或上邻元的个数n≥3的极小次余原子)后得到的偏序集也是NP-型偏序集,由此得到26种NP-型偏序集(元素个数n≤6)。最后,借助Matlab软件编程计算得出所有非同构的元素个数不超过6的真伪BCK-代数,其中元素个数为4的真伪BCK-代数2个,元素个数为5的真伪BCK-代数34个,元素个数为6的真伪BCK-代数631个。

This paper represents the enumeration problem on proper pseudo-BCK algebras of elements number n≤6. Firstly, it is shown that there is no proper pseudo-BCK algebra in a poset of elements number n≤3. Secondly, the notions of NP-type poset （the poser with the greatest element 1 in which proper pseudo-BCK algebra does not exist）, layer and sub-coatom etc are introduced. And then it is also proved that a NP-type poser of layer n≤ 3, after adding a isolated coatom, a isolated sub-coatom or a minimum sub-coatom with at least three upper adjacent elements, is still a NP-type poset. Thereby, 26 NP-type posers of elements number n ≤6 are obtained. Finally, by virtue of Matlab software, all non-isomorphic proper pseudo-BCK algebras of elements number n≤6 are calculated, and among them there are 2 algebras of elements number n=4, 34 algebras of elements number n=5 and 631 algebras of elements number n= 6.