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Is this ordered set of vectors a dcpo?
Clash Royale CLAN TAG#URR8PPP
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The definition of a dcpo, or directed complete partial order, can be found here.
A real vector is formed with a basis set $B = e_1, e_2 ldots $, and the real numbers $mathbbR$. A vector is a formal sum of the form $v = Sigma_i a_i e_i$, for $a_i in mathbbR$. Let us define the "terms" as a span $mathbbR leftarrow Term rightarrow mathbbR$. Where the left arrow gives the real coefficient and the right arrow gives an integer, identifying the basis vector. This allows us to have a set of "terms", binding the basis vector to the coefficient. Two vectors may have some of the same terms. We can even say that, for $v,w$, $Term_v subseteq Term_w$ which means that $w$ has all the terms of $v$ and perhaps some more (or no more)
Define the following ordering:
$v le w$ if $Term_v subseteq Term_w$
Does this define a dcpo? What kind of ordering is this? What are the compact elements of the dcpo?
vector-spaces order-theory
asked 45 mins ago
Ben Sprott
398211
add a comment |Â
up vote
0
down vote
favorite
The definition of a dcpo, or directed complete partial order, can be found here.
A real vector is formed with a basis set $B = e_1, e_2 ldots $, and the real numbers $mathbbR$. A vector is a formal sum of the form $v = Sigma_i a_i e_i$, for $a_i in mathbbR$. Let us define the "terms" as a span $mathbbR leftarrow Term rightarrow mathbbR$. Where the left arrow gives the real coefficient and the right arrow gives an integer, identifying the basis vector. This allows us to have a set of "terms", binding the basis vector to the coefficient. Two vectors may have some of the same terms. We can even say that, for $v,w$, $Term_v subseteq Term_w$ which means that $w$ has all the terms of $v$ and perhaps some more (or no more)
Define the following ordering:
$v le w$ if $Term_v subseteq Term_w$
Does this define a dcpo? What kind of ordering is this? What are the compact elements of the dcpo?
vector-spaces order-theory
asked 45 mins ago
Ben Sprott
398211
1
directed-complete partial order?
– mvw
40 mins ago
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The definition of a dcpo, or directed complete partial order, can be found here.
A real vector is formed with a basis set $B = e_1, e_2 ldots $, and the real numbers $mathbbR$. A vector is a formal sum of the form $v = Sigma_i a_i e_i$, for $a_i in mathbbR$. Let us define the "terms" as a span $mathbbR leftarrow Term rightarrow mathbbR$. Where the left arrow gives the real coefficient and the right arrow gives an integer, identifying the basis vector. This allows us to have a set of "terms", binding the basis vector to the coefficient. Two vectors may have some of the same terms. We can even say that, for $v,w$, $Term_v subseteq Term_w$ which means that $w$ has all the terms of $v$ and perhaps some more (or no more)
Define the following ordering:
$v le w$ if $Term_v subseteq Term_w$
Does this define a dcpo? What kind of ordering is this? What are the compact elements of the dcpo?
vector-spaces order-theory
asked 45 mins ago
Ben Sprott
398211
The definition of a dcpo, or directed complete partial order, can be found here.
A real vector is formed with a basis set $B = e_1, e_2 ldots $, and the real numbers $mathbbR$. A vector is a formal sum of the form $v = Sigma_i a_i e_i$, for $a_i in mathbbR$. Let us define the "terms" as a span $mathbbR leftarrow Term rightarrow mathbbR$. Where the left arrow gives the real coefficient and the right arrow gives an integer, identifying the basis vector. This allows us to have a set of "terms", binding the basis vector to the coefficient. Two vectors may have some of the same terms. We can even say that, for $v,w$, $Term_v subseteq Term_w$ which means that $w$ has all the terms of $v$ and perhaps some more (or no more)
Define the following ordering:
$v le w$ if $Term_v subseteq Term_w$
Does this define a dcpo? What kind of ordering is this? What are the compact elements of the dcpo?
vector-spaces order-theory
asked 45 mins ago
Ben Sprott
398211
edited 39 mins ago
asked 45 mins ago
Ben Sprott
398211
asked 45 mins ago
Ben Sprott
398211
asked 45 mins ago
Ben Sprott
398211
398211
1
directed-complete partial order?
– mvw
40 mins ago
add a comment |Â
1
directed-complete partial order?
– mvw
40 mins ago
1
1
directed-complete partial order?
– mvw
40 mins ago
directed-complete partial order?
– mvw
40 mins ago
add a comment |Â
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1
directed-complete partial order?
– mvw
40 mins ago