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 Jan 18 at 12:42
Monroe EskewMonroe Eskew
7,65012158
$endgroup$
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 Jan 18 at 12:42
Monroe EskewMonroe Eskew
7,65012158
$endgroup$
1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
Jan 18 at 15:45
1
$begingroup$
Prefix orders seem to be relevant, if for no other reason than they appear to fit the required definitional niche. See en.m.wikipedia.org/wiki/Prefix_order
$endgroup$
– Not Mike
Jan 18 at 17:18
$begingroup$
Thanks @NotMike! It's nice to see this coming from outside of pure set theory. Feel free to put this as an answer.
$endgroup$
– Monroe Eskew
Jan 18 at 17:26
$begingroup$
Adeleke and Neumann have a Memoir of the AMS 'Relations related to betweenness' which considers various structures along these lines. In particular, a partially ordered set satisfying the property above which is also assumed to be connected (every pair of elements has a common lower bound) is called a semilinearly ordered set. I don't know if they have a name for such objects if they're not connected.
$endgroup$
– shane.orourke
Jan 18 at 19:22
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 Jan 18 at 12:42
Monroe EskewMonroe Eskew
7,65012158
$endgroup$
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 Jan 18 at 12:42
Monroe EskewMonroe Eskew
7,65012158
asked Jan 18 at 12:42
Monroe EskewMonroe Eskew
7,65012158
edited Jan 18 at 15:32
Monroe Eskew
asked Jan 18 at 12:42
Monroe EskewMonroe Eskew
7,65012158
asked Jan 18 at 12:42
Monroe EskewMonroe Eskew
7,65012158
asked Jan 18 at 12:42
Monroe EskewMonroe Eskew
7,65012158
7,65012158
1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
Jan 18 at 15:45
1
$begingroup$
Prefix orders seem to be relevant, if for no other reason than they appear to fit the required definitional niche. See en.m.wikipedia.org/wiki/Prefix_order
$endgroup$
– Not Mike
Jan 18 at 17:18
$begingroup$
Thanks @NotMike! It's nice to see this coming from outside of pure set theory. Feel free to put this as an answer.
$endgroup$
– Monroe Eskew
Jan 18 at 17:26
$begingroup$
Adeleke and Neumann have a Memoir of the AMS 'Relations related to betweenness' which considers various structures along these lines. In particular, a partially ordered set satisfying the property above which is also assumed to be connected (every pair of elements has a common lower bound) is called a semilinearly ordered set. I don't know if they have a name for such objects if they're not connected.
$endgroup$
– shane.orourke
Jan 18 at 19:22
add a comment |
1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
Jan 18 at 15:45
1
$begingroup$
Prefix orders seem to be relevant, if for no other reason than they appear to fit the required definitional niche. See en.m.wikipedia.org/wiki/Prefix_order
$endgroup$
– Not Mike
Jan 18 at 17:18
$begingroup$
Thanks @NotMike! It's nice to see this coming from outside of pure set theory. Feel free to put this as an answer.
$endgroup$
– Monroe Eskew
Jan 18 at 17:26
$begingroup$
Adeleke and Neumann have a Memoir of the AMS 'Relations related to betweenness' which considers various structures along these lines. In particular, a partially ordered set satisfying the property above which is also assumed to be connected (every pair of elements has a common lower bound) is called a semilinearly ordered set. I don't know if they have a name for such objects if they're not connected.
$endgroup$
– shane.orourke
Jan 18 at 19:22
1
1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
Jan 18 at 15:45
$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
Jan 18 at 15:45
1
1
$begingroup$
Prefix orders seem to be relevant, if for no other reason than they appear to fit the required definitional niche. See en.m.wikipedia.org/wiki/Prefix_order
$endgroup$
– Not Mike
Jan 18 at 17:18
$begingroup$
Prefix orders seem to be relevant, if for no other reason than they appear to fit the required definitional niche. See en.m.wikipedia.org/wiki/Prefix_order
$endgroup$
– Not Mike
Jan 18 at 17:18
$begingroup$
Thanks @NotMike! It's nice to see this coming from outside of pure set theory. Feel free to put this as an answer.
$endgroup$
– Monroe Eskew
Jan 18 at 17:26
$begingroup$
Thanks @NotMike! It's nice to see this coming from outside of pure set theory. Feel free to put this as an answer.
$endgroup$
– Monroe Eskew
Jan 18 at 17:26
$begingroup$
Adeleke and Neumann have a Memoir of the AMS 'Relations related to betweenness' which considers various structures along these lines. In particular, a partially ordered set satisfying the property above which is also assumed to be connected (every pair of elements has a common lower bound) is called a semilinearly ordered set. I don't know if they have a name for such objects if they're not connected.
$endgroup$
– shane.orourke
Jan 18 at 19:22
$begingroup$
Adeleke and Neumann have a Memoir of the AMS 'Relations related to betweenness' which considers various structures along these lines. In particular, a partially ordered set satisfying the property above which is also assumed to be connected (every pair of elements has a common lower bound) is called a semilinearly ordered set. I don't know if they have a name for such objects if they're not connected.
$endgroup$
– shane.orourke
Jan 18 at 19:22
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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 Jan 18 at 13:26
Joel David HamkinsJoel David Hamkins
164k25502870
$endgroup$
1
$begingroup$
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
Jan 18 at 15:32
$begingroup$
Well, those are all set theory references, and so they use the set theory convention, which as we said is focused on the well-founded case. I've seen the linearly ordered version sometimes in talks, but I don't know a specific reference. And in truth, I think that the non-well-founded tree case is simply much less studied. For finite trees, however, saying linear is enough (for every finite order is well-founded), and one can definitely find that.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:12
1
$begingroup$
I guess the complication also is that "well-founded tree" now means something else in set theory, where the tree is growing downward and has no infinite branches.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:58
2
$begingroup$
For one reference: Keisler uses the linear order definition in his article "Models with tree structures". Full citation: H. Jerome Keisler. “Models with tree structures”. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) . Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348.
$endgroup$
– Kameryn Williams
Jan 18 at 22:27
$begingroup$
But set-theoreticians, who use the word "tree" to mean well-founded tree, must occasionally have to refer to the more general trees. What do they call them? If I use "tree" to mean a poset in which the predecessors of any element are well-ordered, then what should I call a poset in which the predecessors of any element are linearly ordered? I have some vague recollection of hearing them called "pseudotrees" or something like that.
$endgroup$
– bof
1 hour ago
add a comment |
$begingroup$
Upgraded from a comment:
After a little bit of searching, the notion of prefix order seems to be relevant; if for no other reason than that it appears to fit the required definitional niche.
(Also, it seemed worth pointing out the notion of prefix-order is precisely that of a "first-order tree".)
answered Jan 18 at 19:19
Not MikeNot Mike
1,3651528
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
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 Jan 18 at 13:26
Joel David HamkinsJoel David Hamkins
164k25502870
$endgroup$
1
$begingroup$
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
Jan 18 at 15:32
$begingroup$
Well, those are all set theory references, and so they use the set theory convention, which as we said is focused on the well-founded case. I've seen the linearly ordered version sometimes in talks, but I don't know a specific reference. And in truth, I think that the non-well-founded tree case is simply much less studied. For finite trees, however, saying linear is enough (for every finite order is well-founded), and one can definitely find that.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:12
1
$begingroup$
I guess the complication also is that "well-founded tree" now means something else in set theory, where the tree is growing downward and has no infinite branches.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:58
2
$begingroup$
For one reference: Keisler uses the linear order definition in his article "Models with tree structures". Full citation: H. Jerome Keisler. “Models with tree structures”. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) . Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348.
$endgroup$
– Kameryn Williams
Jan 18 at 22:27
$begingroup$
But set-theoreticians, who use the word "tree" to mean well-founded tree, must occasionally have to refer to the more general trees. What do they call them? If I use "tree" to mean a poset in which the predecessors of any element are well-ordered, then what should I call a poset in which the predecessors of any element are linearly ordered? I have some vague recollection of hearing them called "pseudotrees" or something like that.
$endgroup$
– bof
1 hour 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 Jan 18 at 13:26
Joel David HamkinsJoel David Hamkins
164k25502870
$endgroup$
1
$begingroup$
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
Jan 18 at 15:32
$begingroup$
Well, those are all set theory references, and so they use the set theory convention, which as we said is focused on the well-founded case. I've seen the linearly ordered version sometimes in talks, but I don't know a specific reference. And in truth, I think that the non-well-founded tree case is simply much less studied. For finite trees, however, saying linear is enough (for every finite order is well-founded), and one can definitely find that.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:12
1
$begingroup$
I guess the complication also is that "well-founded tree" now means something else in set theory, where the tree is growing downward and has no infinite branches.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:58
2
$begingroup$
For one reference: Keisler uses the linear order definition in his article "Models with tree structures". Full citation: H. Jerome Keisler. “Models with tree structures”. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) . Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348.
$endgroup$
– Kameryn Williams
Jan 18 at 22:27
$begingroup$
But set-theoreticians, who use the word "tree" to mean well-founded tree, must occasionally have to refer to the more general trees. What do they call them? If I use "tree" to mean a poset in which the predecessors of any element are well-ordered, then what should I call a poset in which the predecessors of any element are linearly ordered? I have some vague recollection of hearing them called "pseudotrees" or something like that.
$endgroup$
– bof
1 hour 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 Jan 18 at 13:26
Joel David HamkinsJoel David Hamkins
164k25502870
$endgroup$
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 Jan 18 at 13:26
Joel David HamkinsJoel David Hamkins
164k25502870
answered Jan 18 at 13:26
Joel David HamkinsJoel David Hamkins
164k25502870
answered Jan 18 at 13:26
Joel David HamkinsJoel David Hamkins
164k25502870
answered Jan 18 at 13:26
Joel David HamkinsJoel David Hamkins
164k25502870
164k25502870
1
$begingroup$
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
Jan 18 at 15:32
$begingroup$
Well, those are all set theory references, and so they use the set theory convention, which as we said is focused on the well-founded case. I've seen the linearly ordered version sometimes in talks, but I don't know a specific reference. And in truth, I think that the non-well-founded tree case is simply much less studied. For finite trees, however, saying linear is enough (for every finite order is well-founded), and one can definitely find that.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:12
1
$begingroup$
I guess the complication also is that "well-founded tree" now means something else in set theory, where the tree is growing downward and has no infinite branches.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:58
2
$begingroup$
For one reference: Keisler uses the linear order definition in his article "Models with tree structures". Full citation: H. Jerome Keisler. “Models with tree structures”. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) . Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348.
$endgroup$
– Kameryn Williams
Jan 18 at 22:27
$begingroup$
But set-theoreticians, who use the word "tree" to mean well-founded tree, must occasionally have to refer to the more general trees. What do they call them? If I use "tree" to mean a poset in which the predecessors of any element are well-ordered, then what should I call a poset in which the predecessors of any element are linearly ordered? I have some vague recollection of hearing them called "pseudotrees" or something like that.
$endgroup$
– bof
1 hour ago
add a comment |
1
$begingroup$
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
Jan 18 at 15:32
$begingroup$
Well, those are all set theory references, and so they use the set theory convention, which as we said is focused on the well-founded case. I've seen the linearly ordered version sometimes in talks, but I don't know a specific reference. And in truth, I think that the non-well-founded tree case is simply much less studied. For finite trees, however, saying linear is enough (for every finite order is well-founded), and one can definitely find that.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:12
1
$begingroup$
I guess the complication also is that "well-founded tree" now means something else in set theory, where the tree is growing downward and has no infinite branches.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:58
2
$begingroup$
For one reference: Keisler uses the linear order definition in his article "Models with tree structures". Full citation: H. Jerome Keisler. “Models with tree structures”. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) . Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348.
$endgroup$
– Kameryn Williams
Jan 18 at 22:27
$begingroup$
But set-theoreticians, who use the word "tree" to mean well-founded tree, must occasionally have to refer to the more general trees. What do they call them? If I use "tree" to mean a poset in which the predecessors of any element are well-ordered, then what should I call a poset in which the predecessors of any element are linearly ordered? I have some vague recollection of hearing them called "pseudotrees" or something like that.
$endgroup$
– bof
1 hour ago
1
1
$begingroup$
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
Jan 18 at 15:32
$begingroup$
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
Jan 18 at 15:32
$begingroup$
Well, those are all set theory references, and so they use the set theory convention, which as we said is focused on the well-founded case. I've seen the linearly ordered version sometimes in talks, but I don't know a specific reference. And in truth, I think that the non-well-founded tree case is simply much less studied. For finite trees, however, saying linear is enough (for every finite order is well-founded), and one can definitely find that.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:12
$begingroup$
Well, those are all set theory references, and so they use the set theory convention, which as we said is focused on the well-founded case. I've seen the linearly ordered version sometimes in talks, but I don't know a specific reference. And in truth, I think that the non-well-founded tree case is simply much less studied. For finite trees, however, saying linear is enough (for every finite order is well-founded), and one can definitely find that.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:12
1
1
$begingroup$
I guess the complication also is that "well-founded tree" now means something else in set theory, where the tree is growing downward and has no infinite branches.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:58
$begingroup$
I guess the complication also is that "well-founded tree" now means something else in set theory, where the tree is growing downward and has no infinite branches.
$endgroup$
– Joel David Hamkins
Jan 18 at 17:58
2
2
$begingroup$
For one reference: Keisler uses the linear order definition in his article "Models with tree structures". Full citation: H. Jerome Keisler. “Models with tree structures”. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) . Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348.
$endgroup$
– Kameryn Williams
Jan 18 at 22:27
$begingroup$
For one reference: Keisler uses the linear order definition in his article "Models with tree structures". Full citation: H. Jerome Keisler. “Models with tree structures”. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) . Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348.
$endgroup$
– Kameryn Williams
Jan 18 at 22:27
$begingroup$
But set-theoreticians, who use the word "tree" to mean well-founded tree, must occasionally have to refer to the more general trees. What do they call them? If I use "tree" to mean a poset in which the predecessors of any element are well-ordered, then what should I call a poset in which the predecessors of any element are linearly ordered? I have some vague recollection of hearing them called "pseudotrees" or something like that.
$endgroup$
– bof
1 hour ago
$begingroup$
But set-theoreticians, who use the word "tree" to mean well-founded tree, must occasionally have to refer to the more general trees. What do they call them? If I use "tree" to mean a poset in which the predecessors of any element are well-ordered, then what should I call a poset in which the predecessors of any element are linearly ordered? I have some vague recollection of hearing them called "pseudotrees" or something like that.
$endgroup$
– bof
1 hour ago
add a comment |
$begingroup$
Upgraded from a comment:
After a little bit of searching, the notion of prefix order seems to be relevant; if for no other reason than that it appears to fit the required definitional niche.
(Also, it seemed worth pointing out the notion of prefix-order is precisely that of a "first-order tree".)
answered Jan 18 at 19:19
Not MikeNot Mike
1,3651528
$endgroup$
add a comment |
$begingroup$
Upgraded from a comment:
After a little bit of searching, the notion of prefix order seems to be relevant; if for no other reason than that it appears to fit the required definitional niche.
(Also, it seemed worth pointing out the notion of prefix-order is precisely that of a "first-order tree".)
answered Jan 18 at 19:19
Not MikeNot Mike
1,3651528
$endgroup$
add a comment |
$begingroup$
Upgraded from a comment:
After a little bit of searching, the notion of prefix order seems to be relevant; if for no other reason than that it appears to fit the required definitional niche.
(Also, it seemed worth pointing out the notion of prefix-order is precisely that of a "first-order tree".)
answered Jan 18 at 19:19
Not MikeNot Mike
1,3651528
$endgroup$
Upgraded from a comment:
After a little bit of searching, the notion of prefix order seems to be relevant; if for no other reason than that it appears to fit the required definitional niche.
(Also, it seemed worth pointing out the notion of prefix-order is precisely that of a "first-order tree".)
answered Jan 18 at 19:19
Not MikeNot Mike
1,3651528
answered Jan 18 at 19:19
Not MikeNot Mike
1,3651528
answered Jan 18 at 19:19
Not MikeNot Mike
1,3651528
answered Jan 18 at 19:19
Not MikeNot Mike
1,3651528
1,3651528
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1
$begingroup$
rd.springer.com/article/10.1007/BF00571186
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– Asaf Karagila
Jan 18 at 15:45
1
$begingroup$
Prefix orders seem to be relevant, if for no other reason than they appear to fit the required definitional niche. See en.m.wikipedia.org/wiki/Prefix_order
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– Not Mike
Jan 18 at 17:18
$begingroup$
Thanks @NotMike! It's nice to see this coming from outside of pure set theory. Feel free to put this as an answer.
$endgroup$
– Monroe Eskew
Jan 18 at 17:26
$begingroup$
Adeleke and Neumann have a Memoir of the AMS 'Relations related to betweenness' which considers various structures along these lines. In particular, a partially ordered set satisfying the property above which is also assumed to be connected (every pair of elements has a common lower bound) is called a semilinearly ordered set. I don't know if they have a name for such objects if they're not connected.
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– shane.orourke
Jan 18 at 19:22