On commonly-used magic systems (3/?)

(continued from last)

So an elemental magic system, as commonly implemented, has at its base a mapping D from spells s ∈ S and spell-targets t ∈ T to an effect-modifier α∈R (where R is usually a small subset of the real line, but must contain 1). We call β = D(s,t) the coefficient of effectiveness for that spell and target.

Note that any of S, T, or R may be infinite, or even merely absurdly large. In the case of T specifically, we consider a player character with two different sets of equipment to be two distinct targets; likewise considered distinct are entities with statuses such as Reflect, Ice Break, or Zombie applied, or the various states of a boss which switches around its elemental weaknesses.

A collection of spells E⊆S all of which have the same β for any target [∀s₁, s₂∈E: ∀t∈T: D(s₁,t) = D(s₂,t)] we call coelemental; if no superset of E shares this quality, E defines a protoelement. For convenience, we extend the mapping D such that we may write D(E,t) for a protoelement E to refer to that protoelement's shared coefficient for that target. Note that any two distinct protoelements are disjoint.

We say a target t is affected by a protoelement E exactly when D(E,t) ≠ 1. Given two protoelements E and E', if any target affected by E is affected identically by E' [∀t∈T: D(E,t) ≠ 1 → D(E,t) = D(E',t)], we say that E' is a derived protoelement of E.

Finally, an element V is an ordered pair: a single protoelement v which is not a derived protoelement (called the base protoelement), and the collection of all protoelements which are derived protoelements of v. Once again, we extend the mapping D, such that we may write D(V,t) for D(π₁(V),t) (that is, identifying the β of an element with that of its base protoelement). We say that protoelements E belong to V if they are any of the protoelements that compose it, base or derived.

The unique protoelement N that belongs to every element, if it exists, is actually the set of non-elemental spells. (It would include things like Apocalypse, Flare, Giga Flare, Gravity Well, Megido, Meteo, NUKE, Ultima... you may possibly see something of a pattern here.)

Derived protoelements other than N may not exist, in which case we can simply say that every protoelement defines an element (including one special non-element from N), and then essentially forget about the protoelemental structure. (This is the usual case.) However, if another (necessarily derived) protoelement exists which belongs to more than one element, that protoelement is a set of multi-elemental spells (e.g., D&D4e's common Fire and Radiant combination).

A derived protoelement that belongs to only one element would be curious indeed, and doesn't seem to have any concrete examples.

(To be continued, with examples and extensions.)