Timeline for Theorem constructing a mathematical structure from a set of internal isomorphisms
Current License: CC BY-SA 4.0
11 events
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| Jul 14, 2023 at 8:58 | comment | added | Keith Kearnes | @AlexKruckman: I meant that a 'partial isomorphism' is often defined to be a certain kind of partial function $f\colon M\to N$ between different structures. Also, the domain and codomain are often taken to be subsets rather than substructures. | |
| Jul 14, 2023 at 0:33 | comment | added | Alex Kruckman | @KeithKearnes That's good to know, thanks for the reference. I didn't mean to suggest the terminology hadn't been used, just that I hadn't heard it. I'm curious about the other meaning of "partial isomorphism". Are you referring to the back-and-forth notion? I've seen people define structure to be "partially isomorphic" if there is a back-and-forth system between them, but define a "partial isomorphism" to be a single isomorphism between substructures - so e.g. the maps in the back-and-forth system are partial isomorphisms. (This is not ideal terminology!) | |
| Jul 13, 2023 at 21:02 | comment | added | Keith Kearnes | @AlexKruckman: The phrase 'internal isomorphism' for 'isomorphism between substructures' has a long history. It appears on the first page of A.F.Pixley, The ternary discriminator function in universal algebra. Math. Ann. 191 (1971), 167-180. This is probably not the first use of it. On the other hand, 'partial isomorphism' typically means something different than 'isomorphism between substructures'. | |
| Jul 11, 2023 at 20:25 | comment | added | Alex Kruckman | I've never heard an isomorphism between substructures called an internal isomorphism. These are usually called partial isomorphisms, or partial automorphisms, in the case that the domain and codomain are substructures of a single structure. Searching for this keyword indicates that there's a reasonably large literature on "inverse semigroups" and "inverse monoids" which are algebraic structures abstracting partial automorphism monoids. The theorem you're looking for is probably in that literature. | |
| Jul 11, 2023 at 19:17 | comment | added | Pablo | By internal isomorphism, I mean an isomorphism between substructures. | |
| Jul 11, 2023 at 19:13 | history | edited | Pablo | CC BY-SA 4.0 |
Add explanation about intarnal isomorphism
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| Jul 10, 2023 at 22:29 | comment | added | tomasz | Like Alex, I'm not sure what you mean by "internal isomorphism", but perhaps you are talking about the construction which, given a set $X$ and a closed $G\leq \operatorname{Sym}(X)$, yields a structure with universe $X$, whose automorphism group is exactly $G$? In that case, it suffices to add (for each natural $n$) a predicate for each orbit of $G$ in $X^n$. | |
| Jul 10, 2023 at 17:00 | comment | added | Alex Kruckman | Also, what is an internal isomorphism in this context? | |
| Jul 10, 2023 at 16:54 | comment | added | Alex Kruckman | Can you give a more precise reference? On what page of this book is the theorem mentioned? | |
| S Jul 10, 2023 at 1:10 | review | First questions | |||
| Jul 10, 2023 at 2:35 | |||||
| S Jul 10, 2023 at 1:10 | history | asked | Pablo | CC BY-SA 4.0 |