On Neutrosophic filter and fantastic filter of BL-algebras

A. Ibrahim^*, S. Karunya Helen Gunaseeli

 Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India

Emails: dribrahimaadhil@gmail.com; karjes821@gmail.com

Abstract

The main aim of this paper is to present a few characteristics of the neutrosophic filter of BL-algebras. Also, we introduce the notion of the neutrosophic fantastic filter of BL-algebras with an illustration and discuss a few of its properties. Moreover, we prove every neutrosophic fantastic filter is a neutrosophic filter in BL-algebras. Finally, we acquire an extension property and equivalent condition of the neurosophic fantastic filter of BL-algebras.

Keywords: BL-algebra; filter; neutrosophic filter; fantastic filter, neutrosophic fantastic filter.

1.    Introduction

In 1998, Smarandache⁹ introduced the ideology of neutrosophy to characterise the neutralites. The notion of fuzzy sets introduced by Zadeh¹³and intuitionistic fuzzy sets by K.T. Atanassov^2,3 laid the foundation for neutrosophic sets. Neutrosophy further became a stepping stone for many mathematical theories like neutrosophic set theory, neutrosophic statistics, neutrosophic probability, and neutrosophic logic. They are widely used in medical diagnosis, image segmentation, decision-making, layout planning and design, security, robotics, and many other fields. Hajek⁴ introduced the notion of BL-algebras (basic logic). The unit interval [0, 1] furnished with the structure produced by a continuous t-norm is the primary illustration of a BL-algebra. The three most well-known classes of BL-algebras are MV-algebras, Godel algebras, and product algebras. The concept of filters was first initiated by Hajek⁴ in BL algebras. Some types of filters, including fantastic filters, were defined by him in BL-algebras. But Turunen¹⁰ was the first to study the theory of filters in BL-algebras well.

M. Haveshki, A. Borumand Saied and E. Eslami⁵ have investigated some types of filters in BL algebras.   S. Yahya Mohamed and P. Umamaheshwari¹² have investigated vague filters in BL-algebras. X. Zhang, X. Mao, Y. Wu, and X. Zhai¹⁴ have defined neutrosophic filters in pseudo-BCI algebras. The authors⁶ introduced the notion of a neutrosophic filter for BL-algebras and investigated some of its features with illustrations.

In this paper, we concentrate on neutrosophic filters and fantastic filters of BL-algebras. In Section 2, we explain some of the primary definitions and findings from the literature. In Section 3, we illustrate some features of neutrosophic filters. Also, we prove that the union of two neutrosophic filters need not be a neutrosophic filter. In Section 4, we introduce the notion of neutrosophic fantastic filters of BL-algebras along with some of their related features. We further prove that every fantastic neutrosophic filter is a neutrosophic filter.