New Type of Soft (Prime) Ideals in Commutative BCK-Algebras

doi:10.3778/j.issn.1673-9418.2012083

Journal of Frontiers of Computer Science and Technology ›› 2022, Vol. 16 ›› Issue (7): 1673-1680.DOI: 10.3778/j.issn.1673-9418.2012083

• Theory and Algorithm • Previous Articles

New Type of Soft (Prime) Ideals in Commutative BCK-Algebras

HUANG Yu¹^,⁺(), LIAO Zuhua²

1. Department of Basic Course Teaching, Wuxi Taihu University, Wuxi, Jiangsu 214064, China
2. School of Science, Jiangnan University, Wuxi, Jiangsu 214122, China

Received:2020-12-21 Revised:2021-03-22 Online:2022-07-01 Published:2021-04-01
Supported by:
the National Natural Science Foundation of China(61673193);the National Natural Science Foundation of China(61170121);the Natural Science Foundation of Jiangsu Province(BK20151117);the Natural Science Foundation of Jiangsu Higher Education Institutions(19KJD100006)

交换BCK-代数的新型软（素）理想

黄昱¹^,⁺(), 廖祖华²

1.无锡太湖学院基础课教学部,江苏无锡 214064
2.江南大学理学院,江苏无锡 214122

作者简介:黄昱（1981—）,女,江苏盐城人,硕士,副教授,CCF会员,主要研究方向为模糊集、人工智能等。
HUANG Yu, born in 1981, M.S., associate professor, member of CCF. Her research interests include fuzzy sets, a.pngicial intelligence, etc.
廖祖华（1957—）,男,江西南昌人,博士,教授,硕士生导师,主要研究方向为模糊集、人工智能等。
LIAO Zuhua, born in 1957, Ph.D., professor, M.S. supervisor. His research interests include fuzzy sets, a.pngicial intelligence, etc.
基金资助:
国家自然科学基金(61673193);国家自然科学基金(61170121);江苏省自然科学基金(BK20151117);江苏省高等学校自然科学研究项目(19KJD100006)

Abstract

Abstract:

The soft set theory is an important mathematical tool for dealing with uncertainty. By endowing a par-ameter set as a commutative BCK-algebra (that is commutative weak-BCI-algebra), the notions of a new type of soft prime ideals, annihilators of soft sets and new type of involutory soft ideals in commutative BCK-algebras are introduced. Two new compositional operations are defined and used to characterize the new type of soft ideals in commutative BCK-algebras. By using partial ordering on commutative BCK-algebras, some properties of the new type of soft ideals are studied. Properties of annihilators of soft sets and new type of involutory soft ideals are obtained. The existence of a new type of soft prime ideals in commutative BCK-algebras and its difference from the standard soft prime ideals are illustrated with examples. It is shown that a soft set is a new type of soft prime ideals in commutative BCK-algebras and its level set is a prime ideal is not a necessary and sufficient condition, which is different from the results of the usual fuzzy algebra. Some equivalent characterizations of the new type of soft prime ideals in commutative BCK-algebras are given. Furthermore, the properties of its homomorphism image and inverse image are discussed.

Key words: BCK-algebras, soft sets, prime ideals, annihilators, compositional operation

摘要：

软集是处理不确定性问题的一个重要的数学工具。将软集的参数集赋予交换BCK-代数（即为交换亚BCI-代数）,提出了交换BCK-代数的新型软素理想、软集的零化子、新型对合软理想等新概念。定义了两个新的合成运算,并利用合成运算对交换BCK-代数的新型软理想进行刻画。研究了交换BCK-代数的新型软理想在偏序关系下的若干性质以及软集的零化子和新型对合软理想的性质。举例说明了交换BCK-代数的新型软素理想的存在性,以及它与通常的软素理想的不同。举例说明了一个软集是交换BCK-代数的新型软素理想与它的水平集是素理想不是充要条件,这与通常的模糊代数的结果不同。给出了交换BCK-代数的新型软素理想的一些等价刻画, 并讨论了它的同态像与原像的性质。

关键词: BCK-代数, 软集, 素理想, 零化子, 合成运算

CLC Number:

O159

HUANG Yu, LIAO Zuhua. New Type of Soft (Prime) Ideals in Commutative BCK-Algebras[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(7): 1673-1680.

黄昱, 廖祖华. 交换BCK-代数的新型软（素）理想[J]. 计算机科学与探索, 2022, 16(7): 1673-1680.

Figures/Tables 1

References 20

[1]	IMAI Y, ISEKI K. On axiom systems of propositional calculi XIV[J]. Proceedings of the Japan Academy, 1996, 42: 19-22.
[2]	ISEKI K. An algebra related with a propositional calculus[J]. Proceedings of the Japan Academy, 1996, 42: 26-29.
[3]	XI O G. Fuzzy BCK-algebras[J]. Mathematica Japonica, 1991, 36: 935-942.
[4]	彭家寅. BCI-代数的扰动模糊q-理想[J]. 计算机科学与探索, 2019, 13(5): 892-900.
	PENG J Y. Disturbing fuzzy q-ideals of BCI-algebras[J]. Journal of Frontiers of Computer Science and Technology, 2019, 13(5): 892-900.
[5]	JUN Y B. Soft BCK/BCI algebras[J]. Computers and Mathe-matics with Applications, 2008, 56(5): 1408-1413.
[6]	JUN Y B, PARK C H. Applications of soft sets in ideal theory of BCK/BCI-algebras[J]. Information Sciences, 2008, 178: 2466-2475.
[7]	KHADEMAN S, ZAHEDI M M, JUN Y B, et al. Fuzzy soft positive implicative hyper BCK-ideals in hyper BCK-algebras[J]. Journal of Intelligent and Fuzzy Systems, 2019, 36(3): 2605-2613. DOI URL
[8]	ISEKI K. On some ideals in BCK-algebras[J]. Mathematics Seminar Notes, 1975, 3: 65-70.
[9]	BORZOOEI R A, ZAHIRI O. Prime ideals in BCI and BCK-algebras[J]. Annals of the University of Craiova, Mathematics and Computer Science Series, 2012, 39(2): 266-276.
[10]	JUN Y B, XIN X L. Fuzzy prime ideals and invertible fuzzy ideals in BCK-algebrs[J]. Fuzzy Sets and Systems, 2001, 117: 471-476. DOI URL
[11]	温永川. 关于软集的研究[D]. 大连: 辽宁师范大学, 2008.
	WEN Y C. The study on soft set[D]. Dalian: Liaoning Normal University, 2008.
[12]	AMERI R, HEDAYATI H, GHASEMIAN E. Prime idealistic soft BCK-BCI-algebras[R]. Babolsar: University of Mazan-daran, 2010: 1-15.
[13]	黄昱, 廖祖华. 亚BCI-代数的新型软理想[J]. 模糊系统与数学, 2018, 32(5): 55-62.
	HUAGN Y, LIAO Z H. A new type of soft ideals of weak-BCI-algebras[J]. Fuzzy Systems and Mathematics, 2018, 32(5): 55-62.
[14]	黄益生. BCI-代数[M]. 北京: 科学出版社, 2006.
	HUANG Y S. BCI-algebra[M]. Beijing: Science Press, 2006.
[15]	NEGGERS J, AHN S S, KIM H S. On Q-algebras[J]. Inter-national Journal of Mathematics and Mathematical Scie-nces, 2001, 27(12): 749-757.
[16]	陈露, 蒲义书. 亚BCI-代数及其理想[J]. 纯粹数学与应用数学, 2005, 21(3): 250-254.
	CHEN L, PU Y S. Sub-BCI-algebra and its ideal[J]. Pure and Applied Mathematics, 2005, 21(3): 250-254.
[17]	MOLODTSOW D. Soft set theory—first results[J]. Com-puters and Mathematics with Applications, 1999, 37(4/5): 19-31.
[18]	黄昱, 廖祖华, 李论. 新型软亚BCI-代数的进一步研究[J]. 计算机工程与应用, 2018, 54(23): 31-35.
	HUAGN Y, LIAO Z H, LI L. Further study on new type of soft weak-BCI-algebras[J]. Computer Engineering and App-lications, 2018, 54(23): 31-35.
[19]	ALI M I, FENG F, LIU X Y, et al. On some new operations in soft set theory[J]. Computers and Mathematics with App-lications, 2009, 57(9): 1547-1553.
[20]	ZHENG G P, LIAO Z H, WANG N N, et al. Soft lattice implication subalgebras[J]. Applied Mathematics and Infor-mation Sciences, 2013, 7(3): 1181-1186.

New Type of Soft (Prime) Ideals in Commutative BCK-Algebras

交换BCK-代数的新型软（素）理想

RichHTML

PDF

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 1

References 20

Related Articles 0

Recommended Articles

Metrics