L-fuzzy set
Discover the definition of l-fuzzy set and the operations of l-fuzzy set. This post explains lattice-ordered membership values and provides an l-fuzzy set with example for advanced algebraic analysis.
L-Fuzzy Set (Lattice-Fuzzy Set)
Introduction
An L-Fuzzy Set is a generalization of fuzzy sets where the membership degrees are taken from a lattice L rather than the standard unit interval [0, 1]. This is useful when membership cannot be expressed by a single number—for example, when membership is a vector of attributes, a set of linguistic labels, or a partially ordered hierarchy.
Definition of L-Fuzzy Set
Let X be a universe of discourse and (L, ≤) be a complete lattice. An L-Fuzzy Set A on X is defined by a membership function:
The set is represented as the collection of pairs:
Where μA(x) is an element in the lattice L. A complete lattice must have a least element (⊥) and a greatest element (⊤).
L-Fuzzy Element
An L-Fuzzy Element is simply a value v ∈ L. Because L is a lattice, for any two elements, there exists a unique Least Upper Bound (Join) and a Greatest Lower Bound (Meet).
Mathematical Operations
Let a, b ∈ L be two membership degrees in the lattice. The operations on L-Fuzzy sets are governed by the lattice structure:
Calculated as the Supremum (Least Upper Bound) of the two elements in L.
Calculated as the Infimum (Greatest Lower Bound) of the two elements in L.
Based on the partial ordering defined by the lattice L.
Requires L to be a Complemented Lattice where a’ is the unique complement of a.
Consider a lattice L = { ⊥, a, b, ⊤ } where a and b are incomparable (neither is greater than the other).
Membership 1: μA(x) = a
Membership 2: μB(x) = b
Intersection (a ∧ b):
Since a and b are incomparable, their greatest lower bound is the bottom element: ⊥
Union (a ∨ b):
The least element that is greater than both a and b is the top element: ⊤
Result: Unlike standard fuzzy sets, the union of two partial memberships can result in “Total Membership” even if neither individual set is high.
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator