Terminology about trees












9












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    rd.springer.com/article/10.1007/BF00571186
    $endgroup$
    – Asaf Karagila
    12 hours ago






  • 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
    11 hours ago












  • $begingroup$
    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
    11 hours ago










  • $begingroup$
    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
    9 hours ago
















9












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    rd.springer.com/article/10.1007/BF00571186
    $endgroup$
    – Asaf Karagila
    12 hours ago






  • 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
    11 hours ago












  • $begingroup$
    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
    11 hours ago










  • $begingroup$
    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
    9 hours ago














9












9








9


1



$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 13 hours ago







Monroe Eskew

















asked 15 hours ago









Monroe EskewMonroe Eskew

7,64512157




7,64512157








  • 1




    $begingroup$
    rd.springer.com/article/10.1007/BF00571186
    $endgroup$
    – Asaf Karagila
    12 hours ago






  • 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
    11 hours ago












  • $begingroup$
    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
    11 hours ago










  • $begingroup$
    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
    9 hours ago














  • 1




    $begingroup$
    rd.springer.com/article/10.1007/BF00571186
    $endgroup$
    – Asaf Karagila
    12 hours ago






  • 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
    11 hours ago












  • $begingroup$
    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
    11 hours ago










  • $begingroup$
    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
    9 hours ago








1




1




$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
12 hours ago




$begingroup$
rd.springer.com/article/10.1007/BF00571186
$endgroup$
– Asaf Karagila
12 hours ago




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
11 hours ago






$begingroup$
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
11 hours ago














$begingroup$
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
11 hours ago




$begingroup$
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
11 hours ago












$begingroup$
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
9 hours ago




$begingroup$
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
9 hours ago










2 Answers
2






active

oldest

votes


















9












$begingroup$

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.






share|cite|improve this answer









$endgroup$









  • 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
    13 hours ago










  • $begingroup$
    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
    11 hours ago






  • 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
    10 hours ago










  • $begingroup$
    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
    6 hours ago





















2












$begingroup$

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".)






share|cite|improve this answer









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    2 Answers
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    2 Answers
    2






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    active

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    active

    oldest

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    9












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$









    • 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
      13 hours ago










    • $begingroup$
      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
      11 hours ago






    • 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
      10 hours ago










    • $begingroup$
      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
      6 hours ago


















    9












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$









    • 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
      13 hours ago










    • $begingroup$
      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
      11 hours ago






    • 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
      10 hours ago










    • $begingroup$
      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
      6 hours ago
















    9












    9








    9





    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$



    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.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 15 hours ago









    Joel David HamkinsJoel David Hamkins

    164k25502869




    164k25502869








    • 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
      13 hours ago










    • $begingroup$
      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
      11 hours ago






    • 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
      10 hours ago










    • $begingroup$
      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
      6 hours ago
















    • 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
      13 hours ago










    • $begingroup$
      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
      11 hours ago






    • 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
      10 hours ago










    • $begingroup$
      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
      6 hours 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
    13 hours 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
    13 hours ago












    $begingroup$
    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
    11 hours ago




    $begingroup$
    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
    11 hours ago




    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
    10 hours ago




    $begingroup$
    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
    10 hours ago












    $begingroup$
    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
    6 hours ago






    $begingroup$
    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
    6 hours ago













    2












    $begingroup$

    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".)






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      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".)






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        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".)






        share|cite|improve this answer









        $endgroup$



        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".)







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 9 hours ago









        Not MikeNot Mike

        1,3551528




        1,3551528






























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            Vallis Paradisi

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            Coordinata: .mw-parser-output .geo-default,.mw-parser-output .geo-dms,.mw-parser-output .geo-dec{display:inline}.mw-parser-output .geo-nondefault,.mw-parser-output .geo-multi-punct{display:none}.mw-parser-output .longitude,.mw-parser-output .latitude{white-space:nowrap} 33°03′S 71°37′W  /  33.05°S 71.617°W  / -33.05; -71.617 Vallis Paradisi Res apud Vicidata repertae: Civitas : Chilia Numerus incolarum : 296 655 Zona horaria : UTC-4 Situs interretialis Nomen officiale : Valparaíso Gubernium Consilium : municipal council of Valparaíso Geographia Superficies : 401 600 000 Metrum quadratum Coniunctiones urbium Urbes gemellae : Ovetum , Badalona , Santa Fe , Ora longa , Sciamhaevum , Quantunia , Callao , Bat Yam , Melaka , Novorossiysk , Busan , Corduba , Odessa , Rosario , Malacca , Barcino , Soteropolis , Franciscopolis (California) , Guanajuato, Silao Tabula aut despectus Despectus in portum sinumque et urbem Vallis Paradisi Vallis Para...
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            Tabula Rosettana

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            Tabula multilinguis Rosettana in Museo Britannico ostenditur. Tabula Rosettana, [1] etiam titulo OGIS 90 agnita, est stela decreto de rebus sacris in Aegypto anno 196 a.C.n. lato inscripta. Tabula iuxta Rosettam Aegypti, urbem in delta Nili et ad oram maris Mediterranei iacentem, anno 1799 a milite Francico reperta est. Inventio stelae, linguis duabus scripturisque tribus inscriptae, eruditis Instituti Aegypti statim nuntiata est; ibi enim iussu imperatoris Napoleonis eruditi omnium scientiarum (sub aegide Commissionis Scientiarum et Artium) properaverant cum expeditione Francica. Qua a Britannis mox debellata, tabula Rosettana Londinium missa hodie apud Museum Britannicum iacet. Textus Graecus cito lectus interpretationi textuum Aegyptiorum (in formis hieroglyphica et demotica expressorum) gradatim adiuvit. Denique textum plene interpretatus est Ioannes Franciscus Champollion. Ab opere eruditorum cumulativo coepit hodiernus scripturae hieroglyphicae linguaeque Aegyptiae a...
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