Timeline for Terminology question for poset maps
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Apr 15 at 18:36 | comment | added | Tri | I have heard Esakia morphisms, at least if the posets are the Priestley duals of Heyting algebras and the maps are also continuous. | |
| Jul 21, 2013 at 11:36 | vote | accept | Benjamin Steinberg | ||
| Jul 20, 2013 at 16:18 | comment | added | Joel David Hamkins | Posets are used extensively in forcing, including many detailed morphism types (and to this extent I could be considered a "poset person"), but this particular notion does not arise significantly in any forcing context to my knowledge. In particular, every poset is forcing equivalent to a poset with no such maps in either direction. And furthermore, we won't generally find such maps from a poset to its Boolean completion, although the projection map from a large forcing notion to a complete subalgebra will have the property. | |
| Jul 20, 2013 at 3:16 | answer | added | Mike Shulman | timeline score: 4 | |
| Jul 19, 2013 at 22:00 | comment | added | Joseph Van Name | Down sets are called ideals. Furthermore, ideals and filters are usually required to be directed (or downward directed). Also, the term "filter map" could possibly refer to morphisms in the category of filters. Perhaps the term "Alexandrov closed" (or "Alexandrov open" depending on whether you want you lower sets to be open or closed) would be better. | |
| Jul 19, 2013 at 21:43 | comment | added | The Masked Avenger | If I were developing the subject (and if I could remember whether down sets were called filters or ideals, let's say filters), then I would call such maps filter maps. I am not a poset person and currently don't talk in a knowing way to poset people, so consider this an opinion. | |
| Jul 19, 2013 at 21:30 | comment | added | Benjamin Steinberg | Yes it is the open maps for the Alexandrov topology but I want to know if poset people have a name. | |
| Jul 19, 2013 at 21:22 | comment | added | Joseph Van Name | @The Masked Avenger. Now that you mentioned it, it is actually a topological condition. If we give posets the topology where the lower sets are precisely the open sets, then the order preserving maps are precisely the continuous functions, and the open maps are precisely the maps that map lower sets to lower sets. Similarly, if we give each poset the topology where the lower sets are precisely the closed sets, then the closed maps are precisely the maps that map lower sets to lower sets. | |
| Jul 19, 2013 at 21:12 | comment | added | The Masked Avenger | This "feels" like a topological condition, like a subbase maps to a subbase, a sort of subopen map. Have you considered this perspective? | |
| Jul 19, 2013 at 20:43 | comment | added | Vidit Nanda | @BenjaminSteinberg it seems as though you want something at least related to residuated mappings, see en.wikipedia.org/wiki/Residuated_mapping which are $f:P \to Q$ for which the pre-images of down sets are down sets. I can't find a "dual" notion in the literature which does this in the forward direction. | |
| Jul 19, 2013 at 18:07 | comment | added | Benjamin Steinberg | Infs and sups need not exist in my context nor be preserved. | |
| Jul 19, 2013 at 17:46 | comment | added | Vidit Nanda | @BenjaminSteinberg Is it possible for you to be more specific about your posets? For instance, finite sups and infs exist for face posets of regular CW complexes but not in general. I suspect that if your posets have semilattice structure, then you might have a morphism of semilattices, in which case another possible avenue is "limit preserving maps". In general, I don't think there is a standard name. | |
| Jul 19, 2013 at 16:37 | comment | added | Benjamin Steinberg | @JoelDavidHamkins, by order preserving I mean $a\leq b$ implies $f(a)\leq f(b)$. | |
| Jul 19, 2013 at 16:19 | comment | added | Joel David Hamkins | I suppose it could also be ambiguous whether one means $p\leq q\iff f(p)\leq f(q)$, or $p\leq q\to f(p)\leq f(q)$ or $p\lt q\to f(p)\lt f(q)$. | |
| Jul 19, 2013 at 15:47 | history | edited | Benjamin Steinberg | CC BY-SA 3.0 |
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| Jul 19, 2013 at 15:47 | comment | added | Benjamin Steinberg | By maps of posets I meant order preserving. I'll edit. | |
| Jul 19, 2013 at 15:40 | comment | added | Joel David Hamkins | You don't also insist that the map is order-preserving? | |
| Jul 19, 2013 at 15:20 | history | edited | Benjamin Steinberg | CC BY-SA 3.0 |
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| Jul 19, 2013 at 15:12 | history | asked | Benjamin Steinberg | CC BY-SA 3.0 |