Mixing
Partially ordered sets cardinality
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What is the cardinality of the set of all partially ordered sets of natural numbers which have one least element and infinity number of maximal elements?
I only noticed that upperbound for this set is $P(mathbb{N}timesmathbb{N}) $ which cardinality is $mathfrak{C} $.
elementary-set-theory order-theory cardinals
edited Jan 18 at 3:47
Andrés E. Caicedo
65.5k8159250
asked Jan 17 at 21:55
avan1235avan1235
3297
$endgroup$
add a comment |
$begingroup$
What is the cardinality of the set of all partially ordered sets of natural numbers which have one least element and infinity number of maximal elements?
I only noticed that upperbound for this set is $P(mathbb{N}timesmathbb{N}) $ which cardinality is $mathfrak{C} $.
elementary-set-theory order-theory cardinals
edited Jan 18 at 3:47
Andrés E. Caicedo
65.5k8159250
asked Jan 17 at 21:55
avan1235avan1235
3297
$endgroup$
add a comment |
$begingroup$
What is the cardinality of the set of all partially ordered sets of natural numbers which have one least element and infinity number of maximal elements?
I only noticed that upperbound for this set is $P(mathbb{N}timesmathbb{N}) $ which cardinality is $mathfrak{C} $.
elementary-set-theory order-theory cardinals
edited Jan 18 at 3:47
Andrés E. Caicedo
65.5k8159250
asked Jan 17 at 21:55
avan1235avan1235
3297
$endgroup$
What is the cardinality of the set of all partially ordered sets of natural numbers which have one least element and infinity number of maximal elements?
I only noticed that upperbound for this set is $P(mathbb{N}timesmathbb{N}) $ which cardinality is $mathfrak{C} $.
elementary-set-theory order-theory cardinals
elementary-set-theory order-theory cardinals
edited Jan 18 at 3:47
Andrés E. Caicedo
65.5k8159250
asked Jan 17 at 21:55
avan1235avan1235
3297
edited Jan 18 at 3:47
Andrés E. Caicedo
65.5k8159250
asked Jan 17 at 21:55
avan1235avan1235
3297
edited Jan 18 at 3:47
Andrés E. Caicedo
65.5k8159250
edited Jan 18 at 3:47
Andrés E. Caicedo
65.5k8159250
edited Jan 18 at 3:47
Andrés E. Caicedo
65.5k8159250
65.5k8159250
asked Jan 17 at 21:55
avan1235avan1235
3297
asked Jan 17 at 21:55
avan1235avan1235
3297
asked Jan 17 at 21:55
avan1235avan1235
3297
3297
add a comment |
add a comment |
1 Answer
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Given $Ssubset Bbb N$, we turn it into a poset by defining $apreceq biff amid b$. Now let $S$ contain $1$, all prime-squares, and an arbitrary set of primes. Then $S$ is of the kind we are interested in. As we can pick an arbitrary subset of the primes, there are $2^{aleph_0}$ such posets.
answered Jan 17 at 22:07
Hagen von EitzenHagen von Eitzen
281k23272505
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1 Answer
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1 Answer
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$begingroup$
Given $Ssubset Bbb N$, we turn it into a poset by defining $apreceq biff amid b$. Now let $S$ contain $1$, all prime-squares, and an arbitrary set of primes. Then $S$ is of the kind we are interested in. As we can pick an arbitrary subset of the primes, there are $2^{aleph_0}$ such posets.
answered Jan 17 at 22:07
Hagen von EitzenHagen von Eitzen
281k23272505
$endgroup$
add a comment |
$begingroup$
Given $Ssubset Bbb N$, we turn it into a poset by defining $apreceq biff amid b$. Now let $S$ contain $1$, all prime-squares, and an arbitrary set of primes. Then $S$ is of the kind we are interested in. As we can pick an arbitrary subset of the primes, there are $2^{aleph_0}$ such posets.
answered Jan 17 at 22:07
Hagen von EitzenHagen von Eitzen
281k23272505
$endgroup$
add a comment |
$begingroup$
Given $Ssubset Bbb N$, we turn it into a poset by defining $apreceq biff amid b$. Now let $S$ contain $1$, all prime-squares, and an arbitrary set of primes. Then $S$ is of the kind we are interested in. As we can pick an arbitrary subset of the primes, there are $2^{aleph_0}$ such posets.
answered Jan 17 at 22:07
Hagen von EitzenHagen von Eitzen
281k23272505
$endgroup$
Given $Ssubset Bbb N$, we turn it into a poset by defining $apreceq biff amid b$. Now let $S$ contain $1$, all prime-squares, and an arbitrary set of primes. Then $S$ is of the kind we are interested in. As we can pick an arbitrary subset of the primes, there are $2^{aleph_0}$ such posets.
answered Jan 17 at 22:07
Hagen von EitzenHagen von Eitzen
281k23272505
answered Jan 17 at 22:07
Hagen von EitzenHagen von Eitzen
281k23272505
answered Jan 17 at 22:07
Hagen von EitzenHagen von Eitzen
281k23272505
answered Jan 17 at 22:07
Hagen von EitzenHagen von Eitzen
281k23272505
281k23272505
add a comment |
add a comment |
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