Timeline for Cryptomorphisms
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29, 2010 at 5:12 | comment | added | Gerhard Paseman | Since the subject was about cryptomorphisms, I thought alternate characterizations (e.g. ternary operations used to represent Abelian groups, single functions that encode mutliplication and inverse) were of interest. With a symbol that again represents the group multiplication, I too know of no pure equational characterization without another function symbol. If you only intend to say such a characterization does not have the same multiplication because it includes monoids, I agree, but think such reading of this answer misses the point. Gerhard Paseman, 2010.11.28 | |
| Nov 28, 2010 at 23:00 | comment | added | Andreas Blass | I was writing about fact, not preference. I believe the last sentence of my previous comment suffices to show that there is no axiomatization of the notion of group that simultaneously (1) uses only multiplication and identity element (not inversion) and (2) consists entirely of universally quantified equations. If you know of such an axiomatization, I'd like to see it, so that I can track down where my argument goes wrong. | |
| Nov 26, 2010 at 22:29 | comment | added | Gerhard Paseman | Andreas, you may choose the axiomatization you prefer. I do not recall an equational characterization which has only multiplication and identity for groups, but I do know that there are those with a constant symbol for the identity, as well as one without, which will characterize groups, and I believe that there is an equational characterization of groups using just the similarity type <2,0>. So if you are disagreeing with my preference, that is your perogative; be careful that you are not disagreeing with fact. Gerhard "Austin Identities Come To Mind" Paseman, 2010.11.26 | |
| Nov 26, 2010 at 17:26 | comment | added | Andreas Blass |
If the group example in the second sentence was intended to connect with "If you don't mind working in equational logic" in the first sentence, then I disagree. It seems to me that, in order to axiomatize groups in a language with only multiplication and identity (no inverse), one needs sentences that contain both universal and existential quantifiers, like $\forall x\exists y\,(xy=e)$. The point is that, in this restricted language, a subalgebra (in the sense of universal algebra) of a group need not be a group (but merely a monoid).
|
|
| Feb 19, 2010 at 6:29 | history | made wiki | Post Made Community Wiki by Anton Geraschenko | ||
| Feb 18, 2010 at 19:34 | history | answered | Gerhard Paseman | CC BY-SA 2.5 |