On commonly-used magic systems (4/?)

Dual-elemental magics are relatively uncommon: I'm not offhand aware of any in any of the Final Fantasy games, for instance, although Chrono Trigger has the Antipode series of Double Techniques. Part of this is because it may require special handling: what happens if an enemy is weak to one of the elements, but strong against another? The dissection described in the previous post is silent on the matter effectively assumes that this does not happen, and that identical coefficients do not stack.

The structure we have so far isn't quite enough to describe the case where they do stack. If stacking is to take place, you'll have to define a multiplication operation on R; this is easy if R⊆ℝ, but somewhat less so in, e.g., Persona 3, where R = {2, 1, 0.5, 0, -1, Reflect}. (Just making Reflect override everything [∀r∈R: Reflect * r = Reflect = r * Reflect] would work, but it would be a design decision. P3 evades the issue entirely: it has no multielemental spells.) Then an element h is a hybrid element iff there exists some set of elements V = {v₁, ... v_n} such that for all t∈T, D(h,t) = ΠD(v,t).

(I have really got to find a better way to type up math; that product just looks wrong, and I can't even fit the index in properly without a full-blown table.)

Actually, that's probably a better way to work than protoelements and derived protoelements; the non-stacking case reduces to defining * as being idempotent on R, 1 as being the identity on *, and leaving all other operations of * undefined. Then you don't need to differentiate between protoelements and derived protoelements and construct elements out of them; just let what was described as a protoelement be an element instead, and all the old dual-elementals are hybrid elements. N is, more comprehensibly, the unique element such that ∀t∈T, D(N,t) = 1. (Although there may be t∈T that are "weak against magic" — but in that case "melée" is just another element.)