Activity for Prime Moverâ€
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Comment | Post #286572 |
Added as suggested (more) |
— | over 2 years ago |
| Edit | Post #286572 |
Post edited: added definition of entropic structure |
— | over 2 years ago |
| Edit | Post #286572 | Initial revision | — | over 2 years ago |
| Question | — |
Endomorphisms on an entropic structure whose pointwise product is the identity automorphism - entropic idempotent structure? Context: self-study from Warner's "Modern Algebra (1965): Exercise 16.27. >Let $\alpha$ and $\beta$ be endomorphisms of an entropic structure $(S, \odot)$ such that $\alpha \odot \beta$ is the identity automorphism, and let $\otimes$ be the operation on $S$ defined by: $$x \otimes y = \alpha (x) ... (more) |
— | over 2 years ago |
| Edit | Post #286571 |
Post edited: |
— | over 2 years ago |
| Edit | Post #286571 | Initial revision | — | over 2 years ago |
| Answer | — |
A: In "if and only if" proofs, why's 1 direction easier to prove than the other? Here's a good one: Every set is well-orderable iff the Axiom of Choice holds. It is quite challenging to prove that the Axiom of Choice implies that every set is well-orderable. Smullyan and Fitting build up to it in a focused but leisurely manner over the course of 3 chapters in their "Set Theory... (more) |
— | over 2 years ago |