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Mathematics 2019, 7(3), 252; https://doi.org/10.3390/math7030252

On (α,β)-US Sets in BCK/BCI-Algebras

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India
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Received: 26 January 2019 / Revised: 4 March 2019 / Accepted: 6 March 2019 / Published: 11 March 2019
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications)
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Abstract

Molodtsov originated soft set theory, which followed a general mathematical framework for handling uncertainties, in which we encounter the data by affixing the parameterized factor during the information analysis. The aim of this paper is to establish a bridge to connect a soft set and the union operations on sets, then applying it to B C K / B C I -algebras. Firstly, we introduce the notion of the ( α , β ) -Union-Soft ( ( α , β ) -US) set, with some supporting examples. Then, we discuss the soft B C K / B C I -algebras, which are called ( α , β ) -US algebras, ( α , β ) -US ideals, ( α , β ) -US closed ideals, and ( α , β ) -US commutative ideals. In particular, some related properties and relationships of the above algebraic structures are investigated. We also provide the condition of an ( α , β ) -US ideal to be an ( α , β ) -US closed ideal. Some conditions for a Union-Soft (US) ideal to be a US commutative ideal are given by means of ( α , β ) -unions. Moreover, several characterization theorems of (closed) US ideals and US commutative ideals are given in terms of ( α , β ) -unions. Finally, the extension property for an ( α , β ) -US commutative ideal is established. View Full-Text
Keywords: BCK/BCI-algebra; (α,β)-US set; (α,β)-US subalgebra; (α,β)-US (closed) ideal; (α,β)-US commutative ideal BCK/BCI-algebra; (α,β)-US set; (α,β)-US subalgebra; (α,β)-US (closed) ideal; (α,β)-US commutative ideal
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Jana, C.; Pal, M. On (α,β)-US Sets in BCK/BCI-Algebras. Mathematics 2019, 7, 252.

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