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Fuzzy Lattice Ordered G-modules

Ursala Paul, Paul Isaac

Source Title: International Journal of Fuzzy System Applications (IJFSA) 8(3)

DOI: 10.4018/IJFSA.2019070104

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Abstract

The study of mathematics emphasizes precision, accuracy, and perfection, but in many of the real-life situations, people face ambiguity, vagueness, imprecision, etc. Fuzzy set theory and rough set theory are two innovative tools in mathematics which are used for decision-making in vague and uncertain information systems. Fuzzy algebra has a significant role in the current era of mathematical research and it deals with the algebraic concepts and models of fuzzy sets. The study of various ordered algebraic structures like lattice ordered groups, Riesz spaces, etc., are of great importance in algebra. The theory of lattice ordered G-modules is very useful in the study of lattice ordered groups and similar algebraic structures. In this article, the theories of fuzzy sets and lattice ordered G-modules are synchronized in a suitable manner to evolve a novel concept in mathematics i.e., fuzzy lattice ordered G-modules which would pave the way for new researchers in fuzzy mathematics to explore much more in this field.

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2. Preliminaries

In this section, some basic definitions that will be needed in the sequel are given. For crisp algebraic concepts one may refer the books (Curties & Reiner, 1962) and (Fraleigh, 1986) and for more basic definitions related with lattice ordered G-modules one may refer (Tremblay & Manohar, 1975).

2.1. Definition

(Tremblay & Manohar, 1975) A lattice is a partially ordered set in which every pair of elements has a greatest lower bound (called their ) and a least upper bound (called their ). Here is the partial order on the set L.