Understanding minimum/maximum and minimal'maximal elements in a partial order.
$begingroup$
I don't understand, are "minimal/minimum/maximal/maxium" elements properties of a partial order or properties of base sets of partial orders? Given any partial order $(X,leq)$ from what I can gather, the following equivalences seem to hold for any $Ssubseteq X$ and any $tin S$ :
$$ttext{ is a }leqtext{minimal element of }Siff ttext{ is a minimal element of }(S,leq )$$
$$ttext{ is a }leqtext{minimum element of }Siff ttext{ is a minimum element of }(S,leq )$$
$$ttext{ is a }leqtext{maximal element of }Siff ttext{ is a maximal element of }(S,leq )$$
$$ttext{ is a }leqtext{maximum element of }Siff ttext{ is a maximum element of }(S,leq )$$
If this is the case, then why define two different notions of say "maximal elements on subsets" and "maximal elements on orders themselves"? It just seems to complicate the notations even more. Why not just pick one notation and be done with it?
notation order-theory
edited Jan 21 at 13:35
Andrés E. Caicedo
65.5k8158249
asked Jan 21 at 6:41
user3865391user3865391
6161215
$endgroup$
add a comment |
$begingroup$
I don't understand, are "minimal/minimum/maximal/maxium" elements properties of a partial order or properties of base sets of partial orders? Given any partial order $(X,leq)$ from what I can gather, the following equivalences seem to hold for any $Ssubseteq X$ and any $tin S$ :
$$ttext{ is a }leqtext{minimal element of }Siff ttext{ is a minimal element of }(S,leq )$$
$$ttext{ is a }leqtext{minimum element of }Siff ttext{ is a minimum element of }(S,leq )$$
$$ttext{ is a }leqtext{maximal element of }Siff ttext{ is a maximal element of }(S,leq )$$
$$ttext{ is a }leqtext{maximum element of }Siff ttext{ is a maximum element of }(S,leq )$$
If this is the case, then why define two different notions of say "maximal elements on subsets" and "maximal elements on orders themselves"? It just seems to complicate the notations even more. Why not just pick one notation and be done with it?
notation order-theory
edited Jan 21 at 13:35
Andrés E. Caicedo
65.5k8158249
asked Jan 21 at 6:41
user3865391user3865391
6161215
$endgroup$
3
$begingroup$
These are exactly the same things, just phrased in English a little differently. Either way requires reference to both $S$ and $leq$, and you'd write them in first order logic exactly the same. As to why, that's just asking why languages can phrase the same things different ways; some ways seem more natural, some ways seem more concise, some ways parallel language used for similar ideas the author wants to highlight, etc.
$endgroup$
– Malice Vidrine
Jan 21 at 6:57
$begingroup$
@bof Soz let me re-add it. I was worried no one was going to help me, and didn't know if I should keep it up.
$endgroup$
– user3865391
Feb 9 at 2:30
add a comment |
$begingroup$
I don't understand, are "minimal/minimum/maximal/maxium" elements properties of a partial order or properties of base sets of partial orders? Given any partial order $(X,leq)$ from what I can gather, the following equivalences seem to hold for any $Ssubseteq X$ and any $tin S$ :
$$ttext{ is a }leqtext{minimal element of }Siff ttext{ is a minimal element of }(S,leq )$$
$$ttext{ is a }leqtext{minimum element of }Siff ttext{ is a minimum element of }(S,leq )$$
$$ttext{ is a }leqtext{maximal element of }Siff ttext{ is a maximal element of }(S,leq )$$
$$ttext{ is a }leqtext{maximum element of }Siff ttext{ is a maximum element of }(S,leq )$$
If this is the case, then why define two different notions of say "maximal elements on subsets" and "maximal elements on orders themselves"? It just seems to complicate the notations even more. Why not just pick one notation and be done with it?
notation order-theory
edited Jan 21 at 13:35
Andrés E. Caicedo
65.5k8158249
asked Jan 21 at 6:41
user3865391user3865391
6161215
$endgroup$
I don't understand, are "minimal/minimum/maximal/maxium" elements properties of a partial order or properties of base sets of partial orders? Given any partial order $(X,leq)$ from what I can gather, the following equivalences seem to hold for any $Ssubseteq X$ and any $tin S$ :
$$ttext{ is a }leqtext{minimal element of }Siff ttext{ is a minimal element of }(S,leq )$$
$$ttext{ is a }leqtext{minimum element of }Siff ttext{ is a minimum element of }(S,leq )$$
$$ttext{ is a }leqtext{maximal element of }Siff ttext{ is a maximal element of }(S,leq )$$
$$ttext{ is a }leqtext{maximum element of }Siff ttext{ is a maximum element of }(S,leq )$$
If this is the case, then why define two different notions of say "maximal elements on subsets" and "maximal elements on orders themselves"? It just seems to complicate the notations even more. Why not just pick one notation and be done with it?
notation order-theory
notation order-theory
edited Jan 21 at 13:35
Andrés E. Caicedo
65.5k8158249
asked Jan 21 at 6:41
user3865391user3865391
6161215
edited Jan 21 at 13:35
Andrés E. Caicedo
65.5k8158249
asked Jan 21 at 6:41
user3865391user3865391
6161215
edited Jan 21 at 13:35
Andrés E. Caicedo
65.5k8158249
edited Jan 21 at 13:35
Andrés E. Caicedo
65.5k8158249
edited Jan 21 at 13:35
Andrés E. Caicedo
65.5k8158249
65.5k8158249
asked Jan 21 at 6:41
user3865391user3865391
6161215
asked Jan 21 at 6:41
user3865391user3865391
6161215
asked Jan 21 at 6:41
user3865391user3865391
6161215
6161215
3
$begingroup$
These are exactly the same things, just phrased in English a little differently. Either way requires reference to both $S$ and $leq$, and you'd write them in first order logic exactly the same. As to why, that's just asking why languages can phrase the same things different ways; some ways seem more natural, some ways seem more concise, some ways parallel language used for similar ideas the author wants to highlight, etc.
$endgroup$
– Malice Vidrine
Jan 21 at 6:57
$begingroup$
@bof Soz let me re-add it. I was worried no one was going to help me, and didn't know if I should keep it up.
$endgroup$
– user3865391
Feb 9 at 2:30
add a comment |
3
$begingroup$
These are exactly the same things, just phrased in English a little differently. Either way requires reference to both $S$ and $leq$, and you'd write them in first order logic exactly the same. As to why, that's just asking why languages can phrase the same things different ways; some ways seem more natural, some ways seem more concise, some ways parallel language used for similar ideas the author wants to highlight, etc.
$endgroup$
– Malice Vidrine
Jan 21 at 6:57
$begingroup$
@bof Soz let me re-add it. I was worried no one was going to help me, and didn't know if I should keep it up.
$endgroup$
– user3865391
Feb 9 at 2:30
3
3
$begingroup$
These are exactly the same things, just phrased in English a little differently. Either way requires reference to both $S$ and $leq$, and you'd write them in first order logic exactly the same. As to why, that's just asking why languages can phrase the same things different ways; some ways seem more natural, some ways seem more concise, some ways parallel language used for similar ideas the author wants to highlight, etc.
$endgroup$
– Malice Vidrine
Jan 21 at 6:57
$begingroup$
These are exactly the same things, just phrased in English a little differently. Either way requires reference to both $S$ and $leq$, and you'd write them in first order logic exactly the same. As to why, that's just asking why languages can phrase the same things different ways; some ways seem more natural, some ways seem more concise, some ways parallel language used for similar ideas the author wants to highlight, etc.
$endgroup$
– Malice Vidrine
Jan 21 at 6:57
$begingroup$
@bof Soz let me re-add it. I was worried no one was going to help me, and didn't know if I should keep it up.
$endgroup$
– user3865391
Feb 9 at 2:30
$begingroup$
@bof Soz let me re-add it. I was worried no one was going to help me, and didn't know if I should keep it up.
$endgroup$
– user3865391
Feb 9 at 2:30
add a comment |
0
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$begingroup$
These are exactly the same things, just phrased in English a little differently. Either way requires reference to both $S$ and $leq$, and you'd write them in first order logic exactly the same. As to why, that's just asking why languages can phrase the same things different ways; some ways seem more natural, some ways seem more concise, some ways parallel language used for similar ideas the author wants to highlight, etc.
$endgroup$
– Malice Vidrine
Jan 21 at 6:57
$begingroup$
@bof Soz let me re-add it. I was worried no one was going to help me, and didn't know if I should keep it up.
$endgroup$
– user3865391
Feb 9 at 2:30