5 ms·

TIL of ≹, which "articulates a relationship where neither of the two compared entities is greater or lesser than the other, yet they aren't necessarily equal ei

by rob74 3y ago

TIL of ≹, which "articulates a relationship where neither of the two compared entities is greater or lesser than the other, yet they aren't necessarily equal either. This nuanced distinction is essential in areas where there are different ways to compare entities that aren't strictly numerical." (https://www.mathematics-monster.com/symbols/Neither-Greater-Than-nor-Less-Than.html https://www.mathematics-monster.com/symbols/Neither-Greater-...)

namanyayg 3y ago
Likewise, TIL. In your link however it states "Example 1: Numerical Context Let's consider two real numbers, a and b. If a is neither greater than nor less than b, but they aren't explicitly equal, the relationship is ≹" How can that be possible?
- magicalhippo 3y ago
  Perhaps this: if they represent angles, then 1 and 361 represents the same absolute orientation, but they're not the same as 361 indicates you went one full revolution to get there. Contrived, but only thing I could think of.
- thaumasiotes 3y ago
  It isn't. The real numbers are a totally ordered field. Any two real numbers are comparable to each other.
- shaunregenbaum 3y ago
  I would imagine trying to compare a purely imaginary number (3i) to a real number (3) would suffice.
  - thaumasiotes 3y ago
    An imaginary number wouldn't obey the stated constraint of being real.
    - coldtea 3y ago
      No, but if the parent's question goes beyond "how can this happen with reals" to "how can this happen with numbers in general", this answers his question.
      - thaumasiotes 3y ago
        The very next example on the page is "imagine two complex numbers with the same magnitude and different angles". For that to answer the parent's question, you'd have to assume he stopped reading immediately after seeing the part he quoted. The question is why the page says "imagine two real numbers that aren't comparable".
- rustybolt 3y ago
  I think they mean the case where a and b are variables for which you don't know the values.
  - didntcheck 3y ago
    Yeah, that's how I understood it. E.g one might write (a+b)^2 != a^2 + b^2 To mean that in general the equality doesn't hold. Despite exceptions like a=b=0 Strictly you should write something like ¬[∀ a,b (a+b)^2 != a^2 + b^2] But shorthand and abuse of notation are hardly rare
    - didntcheck 3y ago
      Noticed a copy and paste error too late - the != in the second expression should of course be =
  - coldtea 3y ago
    No, that can be the case also for mathematical entities for which you can know the values, not just for "unknown variables".
- salmonellaeater 3y ago
  That doesn't seem possible with the reals. An example from programming that comes to mind is NaN ≹ NaN (in some languages).
  - bruce343434 3y ago
    The floating point specification mandates that nan does not compare to nan in any way, so it should be all languages. If you want to know nam, use isnan()
  - jampekka 3y ago
    Isn't this what happens with infs (in maths and in many programming languages)? Edit: Not in many programming languages. In IEEE-754 inf == inf. In SymPy too oo == oo, although it's a bit controversial. Feels sketchy.
- yummypaint 3y ago
  This doesn't really pass the small test for me either, but to play devils advocate: Imagine you have 2 irrational numbers, and for some a priori reason you know they cannot be equal. You write a computer program to calculate them to arbitrary precision, but no matter how many digits you generate they are identical to that approximation. You know that there must be some point at which they diverge, with one being larger than the other, but you cannot determine when or by how much.
  - MikeDelta 3y ago
    Maybe you will find the proof that the infinite series 0.9999... exactly equals 1 interesting: https://en.wikipedia.org/wiki/0.999 https://en.wikipedia.org/wiki/0.999...
    - jcul 3y ago
      Wow, can't believe I've never realised this. How counterintuitive. The 1/3 * 3 argument, I found the most intuitive.
      - quchen 3y ago
        It's a flawed psychological argument though, because it hinges on accepting that 0.333...=1/3, for which the proof is the same as for 0.999...=1. People have less of a problem with 1/3 so they gloss over this - for some reason, nobody ever says "but there is always a ...3 missing to 1/3" or something.
        Scarblac 3y ago
        The problem is that there are two different ways to write the same number in infinite decimals notation. (0.999... and 1.000...). Thats what's counter intuitive to people, it's not an issue with 1/3. That has just one way to write it as decimals, 0.333...
      - thaumasiotes 3y ago
        I like the argument that observes "if you subtract 0.99(9) from 1, you get a number in which every decimal place is zero". The geometric series proof is less fun but more straightforward. As a fun side note, the geometric series proof will also tell you that the sum of every nonnegative power of 2 works out to -1, and this is in fact how we represent -1 in computers.
- omeid2 3y ago
  Think how any number on the z axis of complex plain isn't equal to the a number of same magnitude, on x and y axis. Now if you really think about, a number of a given magnitude on x axis also isn't exactly "equal" to a name of same magnitude on y axis or vice versa. Other wise, -5 and 5 should be equal, because they're the same magnitude from 0.
  - bruce343434 3y ago
    But |5|=|-5| so I don't exactly see your point. Edit: oh, I see what you mean. 1 is not larger or smaller than i, but it also doesn't equal i.
  - ska 3y ago
    This is just imposing an chosen (and in this case, pretty natural) order on a set without total order. But as you note, you have to give something up. In this case the order identifies each number on a circle in the complex plane with a single value, their magnitude. I've never really seen this notation used, but it could have some use in partially-ordered sets.
    - omeid2 3y ago
      It is formalisms all the way down.
- kevindamm 3y ago
  for the Reals, it is only hypothetical, the domain has a total order. \inf and $\inf + 1$ comes to mind but I don't think it really counts
  - thaumasiotes 3y ago
    > \inf and $\inf + 1$ comes to mind but I don't think it really counts That just depends on the numeric structure you're working with. In the extended reals, +inf is equal to +inf + 1. In a structure with more infinite values than that, it would generally be less. But they wouldn't be incomparable; nothing says "comparable values" quite like the pair "x" and "x + 1".
    - kaba0 3y ago
      I guess it depends on the exact definitions, but reals usually doesn’t include the infinities. At my uni we introduced infinities precisely as an extension of the reals with two values defined by `lim`.
- kaba0 3y ago
  It is false - real numbers fulfill the trichotomy property, which is precisely the lack of such a relationship: every two real number is either less than, equal or greater than. But the numerical context can still be correct: (edit: ~~imaginary~~) complex numbers for example don’t have such a property.
  - fsckboy 3y ago
    i'm learning about this right here as I read, but do you mean complex numbers rather than imaginary?
    - kaba0 3y ago
      Yep, that’s what I meant, sorry!
    - TeMPOraL 3y ago
      Same question. Or more generally, vectors. They don't have a total order, because as we define "less than"/"greater than" in terms of magnitude (length), this means for any vector V (other than 0), there's an infinitely many vectors that are not equal to V, but whose length is equal to length of V. Is this is what ≹ is talking about?
  - shemii 3y ago
    As I've written in another comment here, a great example of a number-y field which is not totally ordered is Games ⊂ Surreal Numbers ⊂ ℝ. There you have certain "numbers" which can can confused with (read incomparable) whole intervals of numbers. Games are really cool :)
    - SetTheorist 3y ago
      You've got the subset relationships backwards: Reals are a subset of the field of Surreal Numbers which is a subset of the group of Games. (Probably better phrased as embeddings, rather than subsets, but the point remains...) Note that Games do _not_ form a field: there is no general multiplication operation between arbitrary games.
- runiq 3y ago
  Hmm, what about +0 and -0?
  - RedNifre 3y ago
    +0 = -0
- tpoacher 3y ago
  Probably does not apply for real numbers, but could totally apply to, e.g., fuzzy numbers, whose 'membership function' bleeds beyond the 'crisp' number into nearby numbers. You could imagine two fuzzy numbers with the same 'crisp' number having different membership profiles, and thus not being "equal", while at the same time being definitely not less and not greater at the same time. Having said that, this all depends on appropriate definitions for all those concepts. You could argue that having the same 'crisp' representation would make them 'equal' but not 'equivalent', if that was the definition you chose. So a lot of this comes down to how you define equality / comparisons in whichever domain you're dealing with.
dfee 3y ago
I was so hoping I could win the day with 0 ≹ -0. But alas, 0 == -0 and 0 === -0.
- Figs 3y ago
  NaN ≹ NaN
shemii 3y ago
Reminds me of the concept of games in combinatorial game theory, they are a superset of surreal numbers (which are themselves a superset of the real numbers) in which the definition of the surreal numbers is loosened in a way which looses the property of they being totally ordered. This creates games (read weird numbers) which can be "confused with" or "fuzzy" with other numbers, the simplest example is * (star) which is confused with 0, i.e. not bigger or smaller than it, it's a fuzzy cloud around zero (notated 0║*). More complex games called switches can be confused with bigger intervals of numbers and are considered "hot". By creating numbers from switches you can create even more interesting hot games.
- cosmojg 3y ago
  > By creating numbers from switches you can create even more interesting hot games. Well, don't leave us hanging! What are some of your favorite hot games on top of switches?
  - shemii 3y ago
    I'm just starting to learn about all this stuff, but iirc thr game of Go is famously "hot". Also I will emphasize thats when talking about "games", usually what is meant is a game position. What spe cific game you are playing isn't too important as it can be shown that some positions in different games are equivalent.
- Akronymus 3y ago
  Heres a relevant video on the topic: https://www.youtube.com/watch?v=ZYj4NkeGPdM https://www.youtube.com/watch?v=ZYj4NkeGPdM I really love that video.
  - shemii 3y ago
    Exactly this video made me read more into this topic, I'm currently reading winning ways and lessons in play simultaneously. It's quite fun! I've just gotten started and am looking forward for what's left.
  - 3y ago
red_trumpet 3y ago
The example z_1 ≹ z_2 for complex numbers z_1, z_2 is weird. Imo it would be clearer to state |z_1| = |z_2|, that is both complex numbers have the same absolute value. > To conclude, the ≹ symbol plays a crucial role in providing a middle ground between the traditional relational operators. As a PhD student in math, I have never seen it before. I do not believe that it plays any crucial role.
- s-phi-nl 3y ago
  It sounds like the symbol "≹" just means "incomparable", which is a well-known concept in math. https://en.wikipedia.org/wiki/Comparability https://en.wikipedia.org/wiki/Comparability This symbol for it may be useful, but it's the concept that matters.
  - deleted 3y ago
    [deleted]
- VHRanger 3y ago
  Might play a crucial role in some arcane subfield of set theory or something of the sort
rollulus 3y ago
Apples ≹ pears.
brynbryn 3y ago
That article is misinterpreting the meaning of the symbol. It isn't useful in mathematics because it is a contradiction in terms: if "neither of the two compared entities is greater or lesser than the other" then they are equal. The author of the original article uses it correctly - think about it more in regards to importance for their example. The business is no more or less important than the developer, but they are NOT equal. It doesn't have to mean importance though, just the method by which you are comparing things. Monday ≹ Wednesday Come to think of it, it should be called the 'No better than' operator.
- denotational 3y ago
  > It isn't useful in mathematics because it is a contradiction in terms: if "neither of the two compared entities is greater or lesser than the other" then they are equal. That’s only true for a total order; there are many interesting orders that do not have this property. It holds for the usual ordering on N, Z, Q and R, but it doesn’t hold for more general partially ordered sets. In general one has to prove that an order is total, and this is frequently non-trivial: Cantor-Schröder-Bernstein can be seen as a proof that the cardinal numbers have a total order.
  - adastra22 3y ago
    Example: alphabetic ordering in most languages with diacritics. For example, "ea" < "éz", but also "éa" < "ez". That's because e and é are treated the same as far as the ordering function is concerned, but they are obviously also NOT the same glyph.
    - deleted 3y ago
      [deleted]
- Chirono 3y ago
  That’s only true for linearly ordered structures, but isn’t true for partially ordered ones. For example, set inclusion. Two different sets can be neither greater than not smaller than each other. Sets ordered by inclusion form a partially ordered lattice.
matharmin 3y ago
I find this concept is important in understanding causal ordering for distributed systems, for example in the context of CRDTs. For events generated on a single device, you always have a complete ordering. But if you generate events on two separate devices while offline, you can't say one came before the other, and end up with a ≹ relationship between the two. Or put differently, the events are considered concurrent. So you can end up with a sequence "d > b > a" and "d > c > a", but "c ≹ b". Defining how tie-breaking for those cases are deterministically performed is a big part of the problem that CRDTs solve.
nine_k 3y ago
I suppose that this glyph should result from a combination of emojis for apples and oranges.
- throwaway_08932 3y ago
  [flagged]
quietbritishjim 3y ago
One example would be if you define one set A to be "less than" another B if A is a subset of B. Then ∅ < {0} and {0} < {0, 1} but {0} ≹ {1}. Such a thing is called a partial ordering and a set of values with a partial ordering is called a partially ordered set or poset (pronounced Poe-set) for short. https://en.wikipedia.org/wiki/Partially_ordered_set https://en.wikipedia.org/wiki/Partially_ordered_set
lewisjoe 3y ago
The jargon from category theory for this phenomenon is - partial ordering. It really is an interesting thing. In fact, as human beings who by nature think in terms of abstract, non-concrete units (as opposed to mathematically precise units like a computer program), we tend to compare two related things. They might belong to the same category of things, but they might not be eligible for direct comparison at all. Once you internalize partial ordering, our brain gets a little more comfortable handling similar, yet incomparable analogies.
tpoacher 3y ago
Reminded me of this: https://math.stackexchange.com/q/586229/361068 https://math.stackexchange.com/q/586229/361068
lamontcg 3y ago
That suggests to me that you've got a multi objective optimization problem with conflicting objectives and a pareto optimal solution that balances the tradeoffs between the two objectives. If you swing it too far one way you've got a '>' and need to swing it back the other way, but go too far the other way and you've got a '<'. And they're definitely not equal since they pull in different directions.

by rob74 3y ago

namanyayg 3y ago

Likewise, TIL. In your link however it states "Example 1: Numerical Context Let's consider two real numbers, a and b. If a is neither greater than nor less than b, but they aren't explicitly equal, the relationship is ≹" How can that be possible?

magicalhippo 3y ago
Perhaps this: if they represent angles, then 1 and 361 represents the same absolute orientation, but they're not the same as 361 indicates you went one full revolution to get there. Contrived, but only thing I could think of.
thaumasiotes 3y ago
It isn't. The real numbers are a totally ordered field. Any two real numbers are comparable to each other.
shaunregenbaum 3y ago
I would imagine trying to compare a purely imaginary number (3i) to a real number (3) would suffice.
- thaumasiotes 3y ago
  An imaginary number wouldn't obey the stated constraint of being real.
  - coldtea 3y ago
    No, but if the parent's question goes beyond "how can this happen with reals" to "how can this happen with numbers in general", this answers his question.
    - thaumasiotes 3y ago
      The very next example on the page is "imagine two complex numbers with the same magnitude and different angles". For that to answer the parent's question, you'd have to assume he stopped reading immediately after seeing the part he quoted. The question is why the page says "imagine two real numbers that aren't comparable".
rustybolt 3y ago
I think they mean the case where a and b are variables for which you don't know the values.
- didntcheck 3y ago
  Yeah, that's how I understood it. E.g one might write (a+b)^2 != a^2 + b^2 To mean that in general the equality doesn't hold. Despite exceptions like a=b=0 Strictly you should write something like ¬[∀ a,b (a+b)^2 != a^2 + b^2] But shorthand and abuse of notation are hardly rare
  - didntcheck 3y ago
    Noticed a copy and paste error too late - the != in the second expression should of course be =
- coldtea 3y ago
  No, that can be the case also for mathematical entities for which you can know the values, not just for "unknown variables".
salmonellaeater 3y ago
That doesn't seem possible with the reals. An example from programming that comes to mind is NaN ≹ NaN (in some languages).
- bruce343434 3y ago
  The floating point specification mandates that nan does not compare to nan in any way, so it should be all languages. If you want to know nam, use isnan()
- jampekka 3y ago
  Isn't this what happens with infs (in maths and in many programming languages)? Edit: Not in many programming languages. In IEEE-754 inf == inf. In SymPy too oo == oo, although it's a bit controversial. Feels sketchy.
yummypaint 3y ago
This doesn't really pass the small test for me either, but to play devils advocate: Imagine you have 2 irrational numbers, and for some a priori reason you know they cannot be equal. You write a computer program to calculate them to arbitrary precision, but no matter how many digits you generate they are identical to that approximation. You know that there must be some point at which they diverge, with one being larger than the other, but you cannot determine when or by how much.
- MikeDelta 3y ago
  Maybe you will find the proof that the infinite series 0.9999... exactly equals 1 interesting: https://en.wikipedia.org/wiki/0.999 https://en.wikipedia.org/wiki/0.999...
  - jcul 3y ago
    Wow, can't believe I've never realised this. How counterintuitive. The 1/3 * 3 argument, I found the most intuitive.
    - quchen 3y ago
      It's a flawed psychological argument though, because it hinges on accepting that 0.333...=1/3, for which the proof is the same as for 0.999...=1. People have less of a problem with 1/3 so they gloss over this - for some reason, nobody ever says "but there is always a ...3 missing to 1/3" or something.
      - Scarblac 3y ago
        The problem is that there are two different ways to write the same number in infinite decimals notation. (0.999... and 1.000...). Thats what's counter intuitive to people, it's not an issue with 1/3. That has just one way to write it as decimals, 0.333...
    - thaumasiotes 3y ago
      I like the argument that observes "if you subtract 0.99(9) from 1, you get a number in which every decimal place is zero". The geometric series proof is less fun but more straightforward. As a fun side note, the geometric series proof will also tell you that the sum of every nonnegative power of 2 works out to -1, and this is in fact how we represent -1 in computers.
omeid2 3y ago
Think how any number on the z axis of complex plain isn't equal to the a number of same magnitude, on x and y axis. Now if you really think about, a number of a given magnitude on x axis also isn't exactly "equal" to a name of same magnitude on y axis or vice versa. Other wise, -5 and 5 should be equal, because they're the same magnitude from 0.
- bruce343434 3y ago
  But |5|=|-5| so I don't exactly see your point. Edit: oh, I see what you mean. 1 is not larger or smaller than i, but it also doesn't equal i.
- ska 3y ago
  This is just imposing an chosen (and in this case, pretty natural) order on a set without total order. But as you note, you have to give something up. In this case the order identifies each number on a circle in the complex plane with a single value, their magnitude. I've never really seen this notation used, but it could have some use in partially-ordered sets.
  - omeid2 3y ago
    It is formalisms all the way down.
kevindamm 3y ago
for the Reals, it is only hypothetical, the domain has a total order. \inf and $\inf + 1$ comes to mind but I don't think it really counts
- thaumasiotes 3y ago
  > \inf and $\inf + 1$ comes to mind but I don't think it really counts That just depends on the numeric structure you're working with. In the extended reals, +inf is equal to +inf + 1. In a structure with more infinite values than that, it would generally be less. But they wouldn't be incomparable; nothing says "comparable values" quite like the pair "x" and "x + 1".
  - kaba0 3y ago
    I guess it depends on the exact definitions, but reals usually doesn’t include the infinities. At my uni we introduced infinities precisely as an extension of the reals with two values defined by `lim`.
kaba0 3y ago
It is false - real numbers fulfill the trichotomy property, which is precisely the lack of such a relationship: every two real number is either less than, equal or greater than. But the numerical context can still be correct: (edit: ~~imaginary~~) complex numbers for example don’t have such a property.
- fsckboy 3y ago
  i'm learning about this right here as I read, but do you mean complex numbers rather than imaginary?
  - kaba0 3y ago
    Yep, that’s what I meant, sorry!
  - TeMPOraL 3y ago
    Same question. Or more generally, vectors. They don't have a total order, because as we define "less than"/"greater than" in terms of magnitude (length), this means for any vector V (other than 0), there's an infinitely many vectors that are not equal to V, but whose length is equal to length of V. Is this is what ≹ is talking about?
- shemii 3y ago
  As I've written in another comment here, a great example of a number-y field which is not totally ordered is Games ⊂ Surreal Numbers ⊂ ℝ. There you have certain "numbers" which can can confused with (read incomparable) whole intervals of numbers. Games are really cool :)
  - SetTheorist 3y ago
    You've got the subset relationships backwards: Reals are a subset of the field of Surreal Numbers which is a subset of the group of Games. (Probably better phrased as embeddings, rather than subsets, but the point remains...) Note that Games do _not_ form a field: there is no general multiplication operation between arbitrary games.
runiq 3y ago
Hmm, what about +0 and -0?
- RedNifre 3y ago
  +0 = -0
tpoacher 3y ago
Probably does not apply for real numbers, but could totally apply to, e.g., fuzzy numbers, whose 'membership function' bleeds beyond the 'crisp' number into nearby numbers. You could imagine two fuzzy numbers with the same 'crisp' number having different membership profiles, and thus not being "equal", while at the same time being definitely not less and not greater at the same time. Having said that, this all depends on appropriate definitions for all those concepts. You could argue that having the same 'crisp' representation would make them 'equal' but not 'equivalent', if that was the definition you chose. So a lot of this comes down to how you define equality / comparisons in whichever domain you're dealing with.

dfee 3y ago

I was so hoping I could win the day with 0 ≹ -0. But alas, 0 == -0 and 0 === -0.

Figs 3y ago
NaN ≹ NaN

shemii 3y ago

Reminds me of the concept of games in combinatorial game theory, they are a superset of surreal numbers (which are themselves a superset of the real numbers) in which the definition of the surreal numbers is loosened in a way which looses the property of they being totally ordered. This creates games (read weird numbers) which can be "confused with" or "fuzzy" with other numbers, the simplest example is * (star) which is confused with 0, i.e. not bigger or smaller than it, it's a fuzzy cloud around zero (notated 0║*). More complex games called switches can be confused with bigger intervals of numbers and are considered "hot". By creating numbers from switches you can create even more interesting hot games.

cosmojg 3y ago
> By creating numbers from switches you can create even more interesting hot games. Well, don't leave us hanging! What are some of your favorite hot games on top of switches?
- shemii 3y ago
  I'm just starting to learn about all this stuff, but iirc thr game of Go is famously "hot". Also I will emphasize thats when talking about "games", usually what is meant is a game position. What spe cific game you are playing isn't too important as it can be shown that some positions in different games are equivalent.
Akronymus 3y ago
Heres a relevant video on the topic: https://www.youtube.com/watch?v=ZYj4NkeGPdM https://www.youtube.com/watch?v=ZYj4NkeGPdM I really love that video.
- shemii 3y ago
  Exactly this video made me read more into this topic, I'm currently reading winning ways and lessons in play simultaneously. It's quite fun! I've just gotten started and am looking forward for what's left.
- 3y ago

red_trumpet 3y ago

The example z_1 ≹ z_2 for complex numbers z_1, z_2 is weird. Imo it would be clearer to state |z_1| = |z_2|, that is both complex numbers have the same absolute value. > To conclude, the ≹ symbol plays a crucial role in providing a middle ground between the traditional relational operators. As a PhD student in math, I have never seen it before. I do not believe that it plays any crucial role.

s-phi-nl 3y ago
It sounds like the symbol "≹" just means "incomparable", which is a well-known concept in math. https://en.wikipedia.org/wiki/Comparability https://en.wikipedia.org/wiki/Comparability This symbol for it may be useful, but it's the concept that matters.
- deleted 3y ago
  [deleted]
VHRanger 3y ago
Might play a crucial role in some arcane subfield of set theory or something of the sort

rollulus 3y ago

Apples ≹ pears.

brynbryn 3y ago

That article is misinterpreting the meaning of the symbol. It isn't useful in mathematics because it is a contradiction in terms: if "neither of the two compared entities is greater or lesser than the other" then they are equal. The author of the original article uses it correctly - think about it more in regards to importance for their example. The business is no more or less important than the developer, but they are NOT equal. It doesn't have to mean importance though, just the method by which you are comparing things. Monday ≹ Wednesday Come to think of it, it should be called the 'No better than' operator.

denotational 3y ago
> It isn't useful in mathematics because it is a contradiction in terms: if "neither of the two compared entities is greater or lesser than the other" then they are equal. That’s only true for a total order; there are many interesting orders that do not have this property. It holds for the usual ordering on N, Z, Q and R, but it doesn’t hold for more general partially ordered sets. In general one has to prove that an order is total, and this is frequently non-trivial: Cantor-Schröder-Bernstein can be seen as a proof that the cardinal numbers have a total order.
- adastra22 3y ago
  Example: alphabetic ordering in most languages with diacritics. For example, "ea" < "éz", but also "éa" < "ez". That's because e and é are treated the same as far as the ordering function is concerned, but they are obviously also NOT the same glyph.
  - deleted 3y ago
    [deleted]
Chirono 3y ago
That’s only true for linearly ordered structures, but isn’t true for partially ordered ones. For example, set inclusion. Two different sets can be neither greater than not smaller than each other. Sets ordered by inclusion form a partially ordered lattice.

matharmin 3y ago

I find this concept is important in understanding causal ordering for distributed systems, for example in the context of CRDTs. For events generated on a single device, you always have a complete ordering. But if you generate events on two separate devices while offline, you can't say one came before the other, and end up with a ≹ relationship between the two. Or put differently, the events are considered concurrent. So you can end up with a sequence "d > b > a" and "d > c > a", but "c ≹ b". Defining how tie-breaking for those cases are deterministically performed is a big part of the problem that CRDTs solve.

nine_k 3y ago

I suppose that this glyph should result from a combination of emojis for apples and oranges.

throwaway_08932 3y ago
[flagged]

quietbritishjim 3y ago

One example would be if you define one set A to be "less than" another B if A is a subset of B. Then ∅ < {0} and {0} < {0, 1} but {0} ≹ {1}. Such a thing is called a partial ordering and a set of values with a partial ordering is called a partially ordered set or poset (pronounced Poe-set) for short. https://en.wikipedia.org/wiki/Partially_ordered_set https://en.wikipedia.org/wiki/Partially_ordered_set

lewisjoe 3y ago

The jargon from category theory for this phenomenon is - partial ordering. It really is an interesting thing. In fact, as human beings who by nature think in terms of abstract, non-concrete units (as opposed to mathematically precise units like a computer program), we tend to compare two related things. They might belong to the same category of things, but they might not be eligible for direct comparison at all. Once you internalize partial ordering, our brain gets a little more comfortable handling similar, yet incomparable analogies.

tpoacher 3y ago

Reminded me of this: https://math.stackexchange.com/q/586229/361068 https://math.stackexchange.com/q/586229/361068

lamontcg 3y ago

That suggests to me that you've got a multi objective optimization problem with conflicting objectives and a pareto optimal solution that balances the tradeoffs between the two objectives. If you swing it too far one way you've got a '>' and need to swing it back the other way, but go too far the other way and you've got a '<'. And they're definitely not equal since they pull in different directions.