overlinehyx

Overlinehyx is a notational construct described in a fictional or hypothetical mathematical framework referred to as hyx-calculus. It denotes a dual or conjugation-like operation applied to expressions built from three basic symbols, h, y, and x. The term combines the conventional notion of an overline with the compact triple hyx, signaling that the operation targets hyx-expressions as a whole and interacts with their duals in a structured way.

Formal definition: In the imagined hyx-algebra, let the generators h, y, and x be combined under addition and multiplication. Overlinehyx is defined as an involutive anti-automorphism that acts linearly on expressions and satisfies, for the generators, overlinehyx(h) = h*, overlinehyx(y) = y*, overlinehyx(x) = x*. For any two hyx-terms E1 and E2, overlinehyx(E1 E2) = overlinehyx(E2) overlinehyx(E1). By linearity, overlinehyx(aE1 + bE2) = a overlinehyx(E1) + b overlinehyx(E2) for scalars a and b. The operation is involutive in the sense that overlinehyx(overlinehyx(E)) = E for all hyx-terms E.

Interpretation and use: Overlinehyx serves to define dual or conjugate objects within hyx-calculus, enabling formal expressions of symmetry and correspondence between primal and dual entities. It is analogous to familiar notions such as complex conjugation or transposition, but situated in the fictional hyx framework and applied at the level of symbolic expressions rather than scalar values alone.

Examples: If E = h y x, then overlinehyx(E) = h* y* x*. If E = h + yx, then overlinehyx(E) = h* + y* x*. Note that this construction is hypothetical and not part of standard mathematical practice.

See also: overline, conjugation, dual space, anti-automorphism, closure (notions in established mathematics).