Terminology about trees
$begingroup$
In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered. I am interested in the class of partial orders $P$ such that for every $p in P$, the set of $q leq p$ is just linearly ordered. Does this have a name?
set-theory terminology posets trees
asked 3 hours ago
Monroe EskewMonroe Eskew
7,63012157
$endgroup$
add a comment |
$begingroup$
In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered. I am interested in the class of partial orders $P$ such that for every $p in P$, the set of $q leq p$ is just linearly ordered. Does this have a name?
set-theory terminology posets trees
asked 3 hours ago
Monroe EskewMonroe Eskew
7,63012157
$endgroup$
1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
2 mins ago
add a comment |
$begingroup$
In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered. I am interested in the class of partial orders $P$ such that for every $p in P$, the set of $q leq p$ is just linearly ordered. Does this have a name?
set-theory terminology posets trees
asked 3 hours ago
Monroe EskewMonroe Eskew
7,63012157
$endgroup$
In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered. I am interested in the class of partial orders $P$ such that for every $p in P$, the set of $q leq p$ is just linearly ordered. Does this have a name?
set-theory terminology posets trees
set-theory terminology posets trees
asked 3 hours ago
Monroe EskewMonroe Eskew
7,63012157
asked 3 hours ago
Monroe EskewMonroe Eskew
7,63012157
edited 14 mins ago
Monroe Eskew
asked 3 hours ago
Monroe EskewMonroe Eskew
7,63012157
asked 3 hours ago
Monroe EskewMonroe Eskew
7,63012157
asked 3 hours ago
Monroe EskewMonroe Eskew
7,63012157
7,63012157
1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
2 mins ago
add a comment |
1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
2 mins ago
1
1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
2 mins ago
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
2 mins ago
add a comment |
1 Answer
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$begingroup$
They are also called trees.
In that terminology, trees of your first kind are known as the well-founded trees, since they are trees where the tree order is well-founded (and well-founded linear orders are the same as well-orders).
I think that the situation is that because set theorists are mainly interested in the well-founded case, the terminology evolved to drop the adjective from well-founded trees.
There are many competing definitions of tree in mathematics, not all equivalent. For graph-theorists, for example, a tree is a certain kind of cycle-free graph.
answered 2 hours ago
Joel David HamkinsJoel David Hamkins
164k25502869
$endgroup$
$begingroup$
This is also what I thought off the top of my head, but Jech, Kunen, and Kanamori all put well-foundedness into the definition of a tree. Do you know of a reference the defines trees as the more general class?
$endgroup$
– Monroe Eskew
15 mins ago
add a comment |
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1 Answer
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$begingroup$
They are also called trees.
In that terminology, trees of your first kind are known as the well-founded trees, since they are trees where the tree order is well-founded (and well-founded linear orders are the same as well-orders).
I think that the situation is that because set theorists are mainly interested in the well-founded case, the terminology evolved to drop the adjective from well-founded trees.
There are many competing definitions of tree in mathematics, not all equivalent. For graph-theorists, for example, a tree is a certain kind of cycle-free graph.
answered 2 hours ago
Joel David HamkinsJoel David Hamkins
164k25502869
$endgroup$
$begingroup$
This is also what I thought off the top of my head, but Jech, Kunen, and Kanamori all put well-foundedness into the definition of a tree. Do you know of a reference the defines trees as the more general class?
$endgroup$
– Monroe Eskew
15 mins ago
add a comment |
$begingroup$
They are also called trees.
In that terminology, trees of your first kind are known as the well-founded trees, since they are trees where the tree order is well-founded (and well-founded linear orders are the same as well-orders).
I think that the situation is that because set theorists are mainly interested in the well-founded case, the terminology evolved to drop the adjective from well-founded trees.
There are many competing definitions of tree in mathematics, not all equivalent. For graph-theorists, for example, a tree is a certain kind of cycle-free graph.
answered 2 hours ago
Joel David HamkinsJoel David Hamkins
164k25502869
$endgroup$
$begingroup$
This is also what I thought off the top of my head, but Jech, Kunen, and Kanamori all put well-foundedness into the definition of a tree. Do you know of a reference the defines trees as the more general class?
$endgroup$
– Monroe Eskew
15 mins ago
add a comment |
$begingroup$
They are also called trees.
In that terminology, trees of your first kind are known as the well-founded trees, since they are trees where the tree order is well-founded (and well-founded linear orders are the same as well-orders).
I think that the situation is that because set theorists are mainly interested in the well-founded case, the terminology evolved to drop the adjective from well-founded trees.
There are many competing definitions of tree in mathematics, not all equivalent. For graph-theorists, for example, a tree is a certain kind of cycle-free graph.
answered 2 hours ago
Joel David HamkinsJoel David Hamkins
164k25502869
$endgroup$
They are also called trees.
In that terminology, trees of your first kind are known as the well-founded trees, since they are trees where the tree order is well-founded (and well-founded linear orders are the same as well-orders).
I think that the situation is that because set theorists are mainly interested in the well-founded case, the terminology evolved to drop the adjective from well-founded trees.
There are many competing definitions of tree in mathematics, not all equivalent. For graph-theorists, for example, a tree is a certain kind of cycle-free graph.
answered 2 hours ago
Joel David HamkinsJoel David Hamkins
164k25502869
answered 2 hours ago
Joel David HamkinsJoel David Hamkins
164k25502869
answered 2 hours ago
Joel David HamkinsJoel David Hamkins
164k25502869
answered 2 hours ago
Joel David HamkinsJoel David Hamkins
164k25502869
164k25502869
$begingroup$
This is also what I thought off the top of my head, but Jech, Kunen, and Kanamori all put well-foundedness into the definition of a tree. Do you know of a reference the defines trees as the more general class?
$endgroup$
– Monroe Eskew
15 mins ago
add a comment |
$begingroup$
This is also what I thought off the top of my head, but Jech, Kunen, and Kanamori all put well-foundedness into the definition of a tree. Do you know of a reference the defines trees as the more general class?
$endgroup$
– Monroe Eskew
15 mins ago
$begingroup$
This is also what I thought off the top of my head, but Jech, Kunen, and Kanamori all put well-foundedness into the definition of a tree. Do you know of a reference the defines trees as the more general class?
$endgroup$
– Monroe Eskew
15 mins ago
$begingroup$
This is also what I thought off the top of my head, but Jech, Kunen, and Kanamori all put well-foundedness into the definition of a tree. Do you know of a reference the defines trees as the more general class?
$endgroup$
– Monroe Eskew
15 mins ago
add a comment |
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1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
2 mins ago