opfromO

opfromO is a theoretical operator used in formal discussions of duality and graph rewriting. It is not a single standardized construct; instead, it denotes a family of relative dualization operations defined with respect to a fixed base object O in the ambient structure. In practice, opfromO aims to produce a dual-like representation that reverses or reorients a subset of relationships tied to O.

Definition and construction: Let G = (V,E) be a directed graph and O ⊆ V a distinguished subset of vertices. The operator opfromO(G) yields a graph G' = (V,E') defined by reversing the direction of every edge incident to O, while leaving edges not touching O in their original orientation. Equivalently, E' contains (v,u) whenever E contains (u,v) and either u ∈ O or v ∈ O. This convention makes opfromO the edge reversal relative to O. In categorical language, opfromO is a form of dualization relative to the substructure determined by O, and different authors may encode the same idea with slightly different rules.

Properties: If the reversal is applied to the same O twice, opfromO(opfromO(G)) recovers G. If O is empty, opfromO is the identity. If O is V, opfromO corresponds to a global reversal of all edges in G. The exact properties can vary with the chosen convention for handling multi-edges, self-loops, or weighted edges.

Example: Consider a graph G with vertices {A, O, B} and edges A→O and O→B. Applying opfromO(G) reverses edges incident to O, yielding O→A and B→O as the edges of G'.

Usage and status: The term opfromO appears mainly in pedagogical or exploratory contexts to illustrate how dualization can be made relative to a base object O. It is not a widely standardized concept in mainstream literature, and definitions may vary by author.

See also: opposite category, duality, graph reversal.