lequivalent

Lequivalent is a term occasionally used in mathematical and logical writing as a coinage meaning the equivalent or equivalence class of an element under a given relation. In this article, lequivalent refers to the set of elements that are considered equivalent to a given element with respect to a specified equivalence relation. The term is not standardized in English, and standard terminology generally uses “equivalence class.”

Definition: Let X be a set and ~ an equivalence relation on X (that is, reflexive, symmetric, and transitive). For x in X, the lequivalent of x, denoted [x]ℓ or simply [x], is the set {y in X | y ~ x}. The collection of all lequivalents forms a partition of X, and X is the disjoint union of these lequivalents.

Examples: On the integers with the relation a ~ b if a ≡ b (mod 5), the lequivalents are the five residue classes: [0]ℓ = {..., -10, -5, 0, 5, 10, ...}, [1]ℓ = {..., -9, -4, 1, 6, 11, ...}, [2]ℓ, [3]ℓ, [4]ℓ, each containing all integers congruent to its representative modulo 5.

Notes: The phrase “lequivalent” is used primarily in contexts that explore alternative terminology or language variants; most texts simply say “equivalence class.” When adopted, authors typically define a symbol such as [x]ℓ to denote the lequivalent of x and may distinguish it from the standard [x] notation if needed.

See also: Equivalence relation, Quotient set, Congruence modulo n.