Discussion Overview
The discussion revolves around the concept of multiplication in projective space, specifically focusing on the validity and implications of defining multiplication through entrywise multiplication of elements in the context of projective geometry. Participants explore whether this operation can be considered well-defined and if it forms a group structure.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant proposes defining multiplication in projective space through entrywise multiplication of elements, questioning its well-defined nature and group structure.
- Another participant argues against the utility of entrywise multiplication, suggesting that it is not commonly used and referring to isomorphisms in higher dimensions instead.
- A different participant confirms that while the multiplication is well-defined, it does not create a group due to the non-invertibility of elements with an entry of zero.
- One participant expresses interest in the coordinatewise multiplication for its simplicity and potential geometric meaning, despite acknowledging its perceived lack of utility.
- Another participant mentions a geometric interpretation related to angles and tangents, suggesting a connection to the projective plane.
- A later post shares a personal success in defining a noncommutative and nonassociative addition that is distributive over the proposed multiplication, although the implications of this addition are still to be explored.
Areas of Agreement / Disagreement
Participants express differing views on the utility and implications of entrywise multiplication in projective space. There is no consensus on whether this operation should be considered a valid group operation, as some participants affirm its well-defined nature while others challenge its usefulness.
Contextual Notes
Participants mention limitations regarding the invertibility of elements and the noncommutative nature of the proposed addition, indicating that further exploration is needed to understand the implications of these definitions.
