元素个数不超过6的真伪BCK-代数
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摘要
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 poset 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 poset 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 posets 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.
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 poset 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 poset 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 posets 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.
引文
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[3]Meng J,Jun Y B.BCK-algebras[M].Korea:Kyung Moon Sa Co Seoul,1994.
[4]Zhang X H,Dudek W A.Fuzzy BIK+-logic and non-commutative fuzzy logics[J].Fuzzy Systems and Mathemat-ics,2009,23(4):8～20.
[5]张小红.模糊逻辑及其代数分析[M].北京:科学出版社,2008.
[6]Georgescu G,Iorgulescu A.Pseudo-BCK algebras:An extension of BCK algebras[C]//Proceeding of DMTCS’01:Combinatorics computability and logic.London:Springer-Verlag,2001:97～114.
[7]Iorgulescu A.Classes of pseudo-BCK algebras-part I[J].Journal of Multiple-Valued Logic and Soft Computing,2006,12(1～2):71～130.
[8]Iorgulescu A.Classes of pseudo-BCK algebras-part II[J].Journal of Multiple-Valued Logic and Soft Comput-ing,2006,12(5～6):575～629.
[9]Iorgulescu A.Pseudo-Iséki algebras.Connection with pseudo-BL algebras[J].Journal of Multiple-Valued Logicand Soft Computing,2005,11(3～4):263～308.
[10]Zhang X H,Li W H.On pseudo-BL algebras and BCC-algebra[J].Soft Computing,2006,10(10):941～952.
[11]Liu Y L,Liu S Y,Xu Y.Pseudo-BCK algebras and PD-posets[J].Soft Computing,2007,11(1):91～101.
[12]王丽丽,张小红.剩余格及有界psBCK-代数成为布尔代数的充要条件[J].模糊系统与数学,2010,24(5):8～13.
[13]Vetterlein T.Pseudo-BCK algebras as partial algebras[J].Information Sciences,2010,180(24):5101～5114.
[14]Zhang X H.Pseudo-BCK part and anti-grouped part of pseudo-BCI algebras[C]//IEEE International Conferenceon Progress in Informatics and Computing,2010:127～131.
[15]Radomir H.On the variety generated by bounded pseudo-BCK-algebras[J].International Journal of TheoreticalPhysics,2010,49(12):3131～3138.
[16]Belohlavek R,Vychodil V.Residuated lattices of size≤12[J].Order,2010,27:147～161.
[2]Komori Y.The class of BCC-algebras is not a variety[J].Math Japonicae,1984,29:391～394.
[3]Meng J,Jun Y B.BCK-algebras[M].Korea:Kyung Moon Sa Co Seoul,1994.
[4]Zhang X H,Dudek W A.Fuzzy BIK+-logic and non-commutative fuzzy logics[J].Fuzzy Systems and Mathemat-ics,2009,23(4):8～20.
[5]张小红.模糊逻辑及其代数分析[M].北京:科学出版社,2008.
[6]Georgescu G,Iorgulescu A.Pseudo-BCK algebras:An extension of BCK algebras[C]//Proceeding of DMTCS’01:Combinatorics computability and logic.London:Springer-Verlag,2001:97～114.
[7]Iorgulescu A.Classes of pseudo-BCK algebras-part I[J].Journal of Multiple-Valued Logic and Soft Computing,2006,12(1～2):71～130.
[8]Iorgulescu A.Classes of pseudo-BCK algebras-part II[J].Journal of Multiple-Valued Logic and Soft Comput-ing,2006,12(5～6):575～629.
[9]Iorgulescu A.Pseudo-Iséki algebras.Connection with pseudo-BL algebras[J].Journal of Multiple-Valued Logicand Soft Computing,2005,11(3～4):263～308.
[10]Zhang X H,Li W H.On pseudo-BL algebras and BCC-algebra[J].Soft Computing,2006,10(10):941～952.
[11]Liu Y L,Liu S Y,Xu Y.Pseudo-BCK algebras and PD-posets[J].Soft Computing,2007,11(1):91～101.
[12]王丽丽,张小红.剩余格及有界psBCK-代数成为布尔代数的充要条件[J].模糊系统与数学,2010,24(5):8～13.
[13]Vetterlein T.Pseudo-BCK algebras as partial algebras[J].Information Sciences,2010,180(24):5101～5114.
[14]Zhang X H.Pseudo-BCK part and anti-grouped part of pseudo-BCI algebras[C]//IEEE International Conferenceon Progress in Informatics and Computing,2010:127～131.
[15]Radomir H.On the variety generated by bounded pseudo-BCK-algebras[J].International Journal of TheoreticalPhysics,2010,49(12):3131～3138.
[16]Belohlavek R,Vychodil V.Residuated lattices of size≤12[J].Order,2010,27:147～161.