Timeline for Shape of axioms in algebraic structures
Current License: CC BY-SA 4.0
32 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 6, 2020 at 6:25 | history | edited | YCor |
edited tags
|
|
| Apr 8, 2020 at 7:43 | history | edited | YCor |
edited tags
|
|
| Apr 8, 2020 at 1:19 | comment | added | YCor | @NoahSchweber (for some reason it refuses to move to chat) I don't really see your point (if any, since you initially removed the tag as not general enough, then say it's too general... the only common denominator I can feel is that you don't like it). Anyway, let me mention that an example of a question where I'd find it useful is precisely your question where it's a natural framework. | |
| Apr 8, 2020 at 0:41 | comment | added | Noah Schweber | @YCor Honestly it really doesn't make any sense to me as a tag, but we should probably stop discussing this in the comments. | |
| Apr 7, 2020 at 23:31 | comment | added | YCor | @NoahSchweber are all questions tagged "set-theory" because there's a set ? Are all questions about rings tagged "gr.group-theory" because of the additive group? No. This tag purports to be useful when it's a natural framework in its generality. For instance, a model-theory question about some specific context (groups, fields) doesn't fit. This question definitely fits. A few questions in model theory would fit, e.g. "is there a structure such...". I guess that certainly more than 12 questions fit (I did a quite rough search then) but not a huge number, probably less than 'universal-algebra'. | |
| Apr 7, 2020 at 23:02 | comment | added | Noah Schweber | @YCor But if it's that broad, what doesn't fall under the "relational-structures" heading? At a glance, everything seems like a relational structure (at least if you allow infinitary relations). | |
| Apr 7, 2020 at 22:25 | comment | added | YCor | @NoahSchweber the problem with 'model-theory' is that it's first-order-logic-oriented. I created the tag recently in order to have a non-logic-motivated name for this broad but useful notion. I think one of the reasons which made model-theory papers uneasily accessible to me for a long time is that the framework (namely relational structure in the broadest interpretation) is quite implicit and not considered as interesting for itself. (Group theory is quite the opposite, we spend months to dissect the definition, a whole subdomain is to construct exotic examples, etc.) | |
| Apr 7, 2020 at 21:52 | comment | added | Noah Schweber | @YCor That seems way too broad to be useful (especially given that "model theory" already exists as a tag); also, it seems to only be used about 12 times, each of them added by you. Is this really a useful tag? That said, if you revert I won't re-revert. | |
| Apr 7, 2020 at 21:15 | comment | added | YCor | @NoahSchweber you might have commented before editing... one can view every $n$-ary law as a $(n+1)$-ary relation (with explicitly telling it's a law with respect to, say, the $0$-th variable). Thus I view universal algebra is a particular case of the realm of relational structures, and relational structures as the general framework for model theory. (Removing the tag sounds to me like saying that pointed topological spaces don't fit in the framework/tag "topological-spaces".) | |
| Apr 7, 2020 at 20:59 | comment | added | Noah Schweber | @YCor I've removed the "relational-structures" tag - I don't see any role of relational structures here, and in fact the question seems to be implicitly focusing on non-relational structures. | |
| Apr 7, 2020 at 20:59 | history | edited | Noah Schweber |
edited tags
|
|
| Apr 7, 2020 at 20:42 | history | edited | YCor | CC BY-SA 4.0 |
changed tags (removed deprecated tag, added two), minor reformulations
|
| Jul 25, 2013 at 16:40 | answer | added | The Masked Avenger | timeline score: 4 | |
| Jul 25, 2013 at 15:57 | history | edited | user9072 |
topl-level tags
|
|
| Jul 25, 2013 at 15:41 | answer | added | Michael Barr | timeline score: 10 | |
| Jul 25, 2013 at 4:59 | history | reopened |
Joel David Hamkins Joseph O'Rourke Eric Wofsey Theo Johnson-Freyd Emerton |
||
| Jul 25, 2013 at 0:53 | review | Reopen votes | |||
| Jul 25, 2013 at 5:04 | |||||
| S Jul 24, 2013 at 22:42 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
type checking and correcting
|
| S Jul 24, 2013 at 22:42 | history | suggested | The Masked Avenger | CC BY-SA 3.0 |
type checking and correcting
|
| Jul 24, 2013 at 22:26 | review | Suggested edits | |||
| Jul 24, 2013 at 22:42 | |||||
| Jul 24, 2013 at 20:20 | comment | added | Todd Trimble | I am a little surprised this was shut down so quickly. Meta: meta.mathoverflow.net/a/538/2926 | |
| Jul 24, 2013 at 20:13 | comment | added | Todd Trimble | The specific example you gave is an example of what category theorists call an essentially algebraic theory, where partially defined operations (with domains specified as loci of equations of previously given operations) may be admitted. The theory of categories is an example of an essentially algebraic theory. Categories of models of (finitary) essentially algebraic theories are characterized as locally finitely presentable categories. See the book on Locally Presentable and Accessible Categories by Adamek and Rosicky. | |
| Jul 24, 2013 at 19:37 | history | closed |
Andy Putman Joel David Hamkins Andrés E. Caicedo Andrey Rekalo David White |
Not suitable for this site | |
| Jul 24, 2013 at 17:05 | comment | added | Denis | @JoelDavidHamkins : yes, I meant the class of all structures $M$ defined by these axioms is not a variety. But then, does it make sense to say that adding new equations defines varieties of structures M (i.e. once we accepted the original axioms, we still say that equations define varieties). | |
| Jul 24, 2013 at 17:02 | comment | added | The Masked Avenger | There is a descriptive term which I have forgotten which I will call x. It might be "representation" but I don't think so. Birkhoff's HSP theorem is an x theorem, as is the one which says the class of models axiomatized by Horn sentences (eq. 1 implies eq. 2) is a quasivariety, a class closed under cetain algebraic constructions. There are many x theorems in Model Theory. Solve for x and I think you will have a general answer to your question. | |
| Jul 24, 2013 at 17:00 | comment | added | Joel David Hamkins | A variety is a class of structures, not a single structure (and I see that the Wikipedia page could be made more clear about this---someone should edit it), so it doesn't make sense to say that "the whole structure is a variety" or "is not a variety". If you are defining a class of structures that cannot be defined by an equational theory, then you do not have a variety. But you do have a first-order definable class of structures $\text{Mod}(T)$, the class of models of a first-order theory $T$, and this is what much of model theory is about. | |
| Jul 24, 2013 at 16:50 | comment | added | Denis | Thanks, I edited the question to be more specific. I am in fact interested specifically in varieties, but I need such an axiom for the general structure. | |
| Jul 24, 2013 at 16:50 | review | Close votes | |||
| Jul 24, 2013 at 19:37 | |||||
| Jul 24, 2013 at 16:50 | history | edited | Denis | CC BY-SA 3.0 |
added 160 characters in body
|
| Jul 24, 2013 at 16:47 | comment | added | Joel David Hamkins | In the model theory of first-order logic, of course any kind of axioms are allowed and studied. Yet, often one can say much more when the axiomatizations have a special form. In universal algebra, for example, one often fruitfully restricts to equational axiomatizations, leading to the concept of a variety: see en.wikipedia.org/wiki/Universal_algebra#Varieties | |
| Jul 24, 2013 at 16:33 | comment | added | Włodzimierz Holsztyński | *** Be my guest. *** | |
| Jul 24, 2013 at 16:21 | history | asked | Denis | CC BY-SA 3.0 |