Timeline for Closing Subsets Under Operations
Current License: CC BY-SA 3.0
6 events
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| Apr 29, 2018 at 11:31 | comment | added | Andreas Blass | @GerhardPaseman By "the clone of f", do you mean (as I would) the smallest clone of operations on $A$ that has f as a member? If so, then your comment is correct but doesn't apply to the examples in the question or the one in Noah Schweber's comment. For example, in a group, the operation of taking inverses is not generally in the clone of multiplication. | |
| Apr 28, 2018 at 23:42 | comment | added | Gerhard Paseman | I am not seeing how this is different from generating subalgebras. One can consider this from the viewpoint of lattices of subalgebras. If g is part of the clone of f, then g adds nothing to the concept. Have you studied closure operators in general? Gerhard "I Recommend 'Algebras, Lattices, Varieties'" Paseman, 2018.04.28. | |
| Apr 28, 2018 at 21:45 | comment | added | Eran | Adding to @NoahSchweber 's comment, Pigozzi found the 17 different closure operators coming from $H, S$, and $P$ in his paper ``On some operations on classes of algebras.'' | |
| Apr 28, 2018 at 21:28 | comment | added | Noah Schweber | Another interesting example, from universal algebra: the HSP monoid. Let $H, S, P$ be the operators on classes of structures closed under isomorphism, given by closing under homomorphic images, substructures, and (arbitrarily indexed) products, respectively. Then $HSP=SHSP=PHSP=HHSP$. On the other hand, order matters: e.g. $PSH$ applied to the class containing only the two-element Boolean algebras doesn't contain any atomless Boolean algebras, which are in HSP applied to that class. | |
| Apr 28, 2018 at 21:27 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
Removed deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)
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| Apr 28, 2018 at 21:12 | history | asked | user30211 | CC BY-SA 3.0 |