伪补MS-代数的素理想及同余性质
|
摘要
素理想是研究Ockham代数类结构的一个重要工具。伪补MS-代数是同时具有伪补代数和MS-代数特征的一类代数。首先在伪补MS-代数上引入两类素理想,以伪补MS-代数本身的运算属性为基础获得了伪补MS-代数素理想的运算特征。其次,利用素理想构造出了伪补MS-代数上的一类同余关系等式,借助素理想集刻画伪补MS-代数的每一个同余关系,获得了伪补MS-代数上的同余关系判别定理。最后,得到次直不可约的伪补MS-代数的结构特征,其元素个数小于或等于6。所得结论为其他Ockham代数类核理想性质的研究提供了方法,丰富了Ockham代数的发展,为进一步研究Ockham代数类的代数结构提供理论支持。
Prime Ideals is an important tool for studying the class structure of Ockham algebras. Pseudocomplemented MS-algebra is a class of Algebra with the characteristics of both pseudocomplement and MS-algebra. Firstly,two prime ideals are introduced into the pseudocomplemented MS-algebra,and the operation characteristics of the prime ideal on pseudocomplement MS-algebra are obtained on the basis of the operation attributes of pseudo complement MS-algebra itself. Secondly,a kind of congruence equation on the pseudo complement MS-algebra is constructed by using the prime ideal,and the congruence relation of the pseudocomplemented MS-algebra is portrayed by the prime ideal set,and the congruence relation theorem on the pseudocomplemented MS-algebra is obtained. Finally,the structural characteristics of the subdirectly irreducible pseudocomplemented MS-algebras are obtained,and its elements are less than or equal to 6. The conclusion provids a method for the study of the properties of the other Ockham algebras,and enriched the theory of ordered algebraic structures.
Prime Ideals is an important tool for studying the class structure of Ockham algebras. Pseudocomplemented MS-algebra is a class of Algebra with the characteristics of both pseudocomplement and MS-algebra. Firstly,two prime ideals are introduced into the pseudocomplemented MS-algebra,and the operation characteristics of the prime ideal on pseudocomplement MS-algebra are obtained on the basis of the operation attributes of pseudo complement MS-algebra itself. Secondly,a kind of congruence equation on the pseudo complement MS-algebra is constructed by using the prime ideal,and the congruence relation of the pseudocomplemented MS-algebra is portrayed by the prime ideal set,and the congruence relation theorem on the pseudocomplemented MS-algebra is obtained. Finally,the structural characteristics of the subdirectly irreducible pseudocomplemented MS-algebras are obtained,and its elements are less than or equal to 6. The conclusion provids a method for the study of the properties of the other Ockham algebras,and enriched the theory of ordered algebraic structures.
引文
[1]BLYTH T S,VARLET J C.Ockham algebras[M].Oxford:Oxford University Press,1994.
[2]BLYTH T S,VARLET J C.On a common abstraction of de Morgan algebras and Stone algebras[J].Proc.Roy.Soc.Edinburgh,1983,94A:301-308.
[3]FANG J.Distributive lattices with unary operations[M].北京:科学出版社,2011.
[4]方捷.格论导引[M].北京:高等教育出版社,2014.
[5]GRATZER G.Lattice theory[M].New York:W.H.Freeman and Company,1971.
[6]赵秀兰,初元红.平衡双重伪补Ockham代数的滤子同余特征[J].模糊系统与数学,2017,31(3):44-48.
[7]赵秀兰,陈丽娟.双重Stone代数的核理想[J].四川理工学院学报:自然科学版,2017,30(1):88-91.
[8]罗从文.MS-代数的核理想[J].应用数学,2001,14(1):39-41.
[9]赵秀兰,初元红,史西专.双重伪补Ockham代数的理想与滤子同余关系的注记[J].汕头大学学报:自然科学版,2016,31(88):35-40.
[10]王雷波,方捷.双重伪补代数的假值理想的一点注记[J].纯粹数学与应用数学,2012,28(1):119-122.
[11]童娟,廖祖华,赵衍才,等.格的反软理想[J],浙江大学学报:理学版,2017,44(1):33-38.
[12]牛超群,吴洪博.BRo代数中的*理想及其诱导的商代数[J].江西师范大学学报:自然科学版,2013,37(3):221-224.
[13]HAVESHKI M.Onαfilter of BL-algebras[J].Journal of intelligent and Fuzzy Systems,2015,28(1):373-382.
[14]BORZOOZI R.Fuzzy n-fold fantastic filters in BL-algebras[J].Afrika Matematika,2014,25(4):1001-1012.
[15]赵萃魁,吴润衡.分配格上的素理想与同余关系[J].内蒙古大学学报,1989,20(1):26-29.
[16]黎爱平.双重MS-代数的素理想与次直不可约类[J].南昌大学学报:理科版,2002,26(1):43-45.
[17]花秀娟,辛小龙,刘庆.(∈,∈∨q)-模糊素理想与(λ,μ)-模糊子格(理想)[J].模糊系统与数学,2010,24(4):33-38.
[18]黎爱平.MS-代数的素理想与同余关系[J].南昌大学学报:理科版,1999,23(4):357-361.
[19]叶婷,廖祖华,朱晓英.半群的软完全素理想[J].模糊系统与数学,2014,28(3):1-5.
[2]BLYTH T S,VARLET J C.On a common abstraction of de Morgan algebras and Stone algebras[J].Proc.Roy.Soc.Edinburgh,1983,94A:301-308.
[3]FANG J.Distributive lattices with unary operations[M].北京:科学出版社,2011.
[4]方捷.格论导引[M].北京:高等教育出版社,2014.
[5]GRATZER G.Lattice theory[M].New York:W.H.Freeman and Company,1971.
[6]赵秀兰,初元红.平衡双重伪补Ockham代数的滤子同余特征[J].模糊系统与数学,2017,31(3):44-48.
[7]赵秀兰,陈丽娟.双重Stone代数的核理想[J].四川理工学院学报:自然科学版,2017,30(1):88-91.
[8]罗从文.MS-代数的核理想[J].应用数学,2001,14(1):39-41.
[9]赵秀兰,初元红,史西专.双重伪补Ockham代数的理想与滤子同余关系的注记[J].汕头大学学报:自然科学版,2016,31(88):35-40.
[10]王雷波,方捷.双重伪补代数的假值理想的一点注记[J].纯粹数学与应用数学,2012,28(1):119-122.
[11]童娟,廖祖华,赵衍才,等.格的反软理想[J],浙江大学学报:理学版,2017,44(1):33-38.
[12]牛超群,吴洪博.BRo代数中的*理想及其诱导的商代数[J].江西师范大学学报:自然科学版,2013,37(3):221-224.
[13]HAVESHKI M.Onαfilter of BL-algebras[J].Journal of intelligent and Fuzzy Systems,2015,28(1):373-382.
[14]BORZOOZI R.Fuzzy n-fold fantastic filters in BL-algebras[J].Afrika Matematika,2014,25(4):1001-1012.
[15]赵萃魁,吴润衡.分配格上的素理想与同余关系[J].内蒙古大学学报,1989,20(1):26-29.
[16]黎爱平.双重MS-代数的素理想与次直不可约类[J].南昌大学学报:理科版,2002,26(1):43-45.
[17]花秀娟,辛小龙,刘庆.(∈,∈∨q)-模糊素理想与(λ,μ)-模糊子格(理想)[J].模糊系统与数学,2010,24(4):33-38.
[18]黎爱平.MS-代数的素理想与同余关系[J].南昌大学学报:理科版,1999,23(4):357-361.
[19]叶婷,廖祖华,朱晓英.半群的软完全素理想[J].模糊系统与数学,2014,28(3):1-5.