Terminology about trees












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In set theory, a tree 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







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    $begingroup$


    In set theory, a tree 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











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      $begingroup$


      In set theory, a tree 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 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






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      share|cite|improve this question













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      edited 2 hours ago









      YCor

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      asked 2 hours ago









      Monroe EskewMonroe Eskew

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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.






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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.






            share|cite|improve this answer









            $endgroup$


















              3












              $begingroup$

              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









              $endgroup$
















                3












                3








                3





                $begingroup$

                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 2 hours ago









                Joel David HamkinsJoel David Hamkins

                164k25502869




                164k25502869






























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