Conjugateyi

Conjugateyi is a hypothetical concept in abstract algebra describing a family of conjugation-like operations on an algebraic structure. The term combines “conjugation” with the suffix “yi,” reflecting a class of automorphism-induced maps defined relative to a chosen automorphism or subgroup of automorphisms.

Definition: Let S be a structure (such as a field or vector space) and let Y be a specified automorphism or a subgroup of Aut(S). For a fixed y in Y, the conjugateyi of an element x in S is defined as y(x). The map conj_yi: x ↦ y(x) is called the conjugateyi with respect to y.

Properties: If y has order two (y^2 = id_S), then conj_yi is an involution. If y preserves the operations of S (as an automorphism does), conj_yi is a structure-preserving map. In a Galois extension L/K with Galois group Gal(L/K), taking x to y(x) (for y in Gal(L/K)) produces the conjugates of x over K, which is a special case of the conjugateyi idea.

Examples and usage: In the complex field C, ordinary complex conjugation is the conjugateyi with respect to the complex conjugation automorphism. In more general contexts, conjugateyi can be used pedagogically to discuss how different automorphisms produce conjugate elements and how their properties relate to the symmetry of the structure.

Notes: Conjugateyi is not a standard term in mainstream mathematical literature; it appears in some didactic discussions or as a naming convenience for a family of conjugation-like maps. See also conjugation, automorphism, Galois theory.