Timeline for Equational theories determined by "identities without variables"
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 10, 2015 at 12:08 | comment | added | მამუკა ჯიბლაძე | Should I go ahead and just insert the word "distributive"? | |
| Sep 10, 2015 at 12:06 | comment | added | მამუკა ჯიბლაძე | Seems to be fine. Except some might not be embeddable into a product of copies of the initial algebra but rather be a subquotient. But all free ones should be so embeddable - in fact, the collection of all $n$-ary functions from your clone must be the free algebra on $n$ generators ($=$ projections). | |
| Sep 9, 2015 at 18:34 | comment | added | Will Sawin | @მამუკაჯიბლაძე Yes, I was not sure I got all the identities. I guess you can check that you get all of them by checking that every algebra embeds into a proudct of copies of $\{0,1\}$. Wikipedia just told me that every bounded distributive lattice is a lattice of sets, so that's fine. Clearly every $\mathbb F_2$-vector space lives in a product of copise of $\mathbb F_2$. With join or with meet orders there's a similar embedding into sets (this time respecting only the join or only the meet). The last two are easy. | |
| Sep 9, 2015 at 16:22 | comment | added | მამუკა ჯიბლაძე | I am on the verge of accepting it :D It took me some time to convince myself and then some more to convince myself that no further details need to be added. There remains just a minor question regarding examples - there might be further identities hidden, no? At least with the first example, as was pointed to me today at the seminar, one seemingly actually gets the subvariety of distributive bounded lattices... | |
| Sep 7, 2015 at 13:47 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| Sep 7, 2015 at 13:45 | comment | added | Will Sawin | @მამუკაჯიბლაძე My first appearance of the word "algebra" in the third line was supposed to be "variety", but I mistyped it. Does that clear everything up? | |
| Sep 7, 2015 at 5:52 | comment | added | მამუკა ჯიბლაძე | Sorry I feel this answer holds the key but don't quite understand why :D You established a correspondence between what and what? On one side you seemingly have all clones on any sets containing all constant maps from these sets to themselves. On the other side you seemingly have certain algebras in certain varieties. Do you mean that these algebras are in one-to-one correspondence with varieties in question? Specifically I am puzzled by your third line. I would somehow presume that the algebra is itself free on an empty set in the corresponding to be pinned down variety, no? | |
| Sep 6, 2015 at 23:24 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| Sep 6, 2015 at 23:07 | history | answered | Will Sawin | CC BY-SA 3.0 |