Timeline for On the notion of partial semigroup
Current License: CC BY-SA 3.0
5 events
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| Mar 6, 2013 at 9:09 | comment | added | Salvo Tringali | Yes, but now that I read myself, I must say that it's expressed in a wrong way. What I mean is that either (xy)z and x(yz) are both defined (and then they're equal to each other), one, and only one, of them is defined, or none of them is defined. In the first two cases, we have a unique unambiguous way to interpret the expression xyz. I'll delete my previous comment to avoid confusion. | |
| Mar 5, 2013 at 21:37 | comment | added | Hans | @Salvo: mhm, does this also hold if we take condition 2) in your list and (x y) z is defined, but y z is not defined? - ED: ok, you mean that we know that we must interpret xyz as (xy)z because the other possibility makes no sense. Ok, I see. Good argument. It will be hard to work with such expressions practically, but it's interesting. Maybe they should indeed also be considered. | |
| Mar 5, 2013 at 19:33 | comment | added | Hans | @Salvo: Yes, it depends on what you like. There is probably no final choice, it depends. That is why we have also groups, groupoids etc. - A stronger reason for the above partial semigroup definition by @Andreas is that you do not need brackets any more in word expressions. I think that's the point for that definition. - On the other hand it is also somewhat limiting. For example, most (arbitrary) subsets of groups are not partial semigroups in this sense. Think about it in Z. | |
| Mar 5, 2013 at 14:54 | comment | added | Salvo Tringali | Yours is precisely my notion of dissociativity, which I've not included in the list in the OP since I myself don't find it particularly "natural". And although we agree that one "right" definition, of course, does not exist, I'd be quite surprised if a "categorial" perspective on the question cannot provide a conceptual motivation, going beyond the scope of a mere "principle of local utility", for favoring one choice over another (which is really what I'm looking for). In any case, thank you much for your answer and the reference. | |
| Mar 5, 2013 at 13:43 | history | answered | Andreas Blass | CC BY-SA 3.0 |