element-of
¶What this is
element-of is a positional frontmatter relation declaring that the subject entity is a named element of the object structure X — a specific member of X, identified by an explicit evaluation term under the evaluation map for X.
The evaluation map MUST be nameable. This distinguishes element-of from mere conceptual association: there must be a specific evaluation procedure that maps the element’s syntactic representation to its position within X.
¶Coarseness ordering
Write the most specific level the current math supports:
| Coarsest | Refine toward |
|---|---|
entity |
tightest named algebra with explicit evaluation term |
Examples of refinement:
entity→relational-universe(if the entity is an object in R)relational-universe→relational-history-fiber(if the entity is a proposition in H_t)relational-history-fiber→relational-history-fiber-fixed-layer(if it is a fixed point of both nuclei)
Blank is never correct. Blank asserts no membership relation — which is a stronger and usually false claim.
¶Relation to component-of and fiber-of
The three positional frontmatter fields are distinct:
component-of: X— X has a named retraction π: X → E (structural projection)fiber-of: F/base: t— E = F(t) for a named functor F (categorical fiber)element-of: X— E ∈ X as a named element under an evaluation map (algebraic membership)
A single entity may carry all three with different objects.
¶Open questions
- Formal specification of what counts as a “nameable” evaluation map in the flatfile context.
- Whether the coarseness ordering is always a chain or can branch (e.g., an entity that is simultaneously an element of two incomparable algebras).