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Let I $I$ be a poset and for any i $i$ let Pi $P_i$ be a poset. Let P $P$ be the sum over I $I$ of the sets Pi, $P_i$, and let <P$<_P$ be the relation defined on P $P$ by $q<Pr<_Pr$ iff q $q$ and r $r$ are members of the same Pi $P_i$ and $q<ir<_ir$ or q $q$ is member of Pj r $P_j$ $r$ is member of Pk $P_k$ and $j<ik. <_ik$. Maybe that formally, I should have taken Qi $Q_i$ as the cartesian product of the singleton of i $i$ with Pi, $P_i$, and Q $Q$ as the union of the Qi's, $Q_i$'s, and compared couples as (i,q) $(i,q)$ and (j,r). $(j,r)$. Anyway, it is clear that P, $P$, equipped with our relation is a poset, that is the poset sum over I $I$ of the posets Pi. $P_i$. Particularly, in the case that I $I$ and Pi $P_i$ all are totally ordered sets , P $P$ is a totally ordered set. And, in this particular case, a theorem of Schoenfliess Schoenflies asserts that "every (totally) ordered set is the union of scattered sets over a densely ordrerd ordered indexing setset";Qestion

Question: does there exist a similar decomposition theorem in the general case of posets ?

Gérard Lang

3 inserted some backticks to supress the (unwanted) effect of < and making the question display in its entirety

Let $I$ I be a poset and for any $i$ i let $P_i$ Pi be a poset. Let $P$ P be the sum over $I$ I of the sets $P_i$, Pi, and let $<_P$ <P be the relation defined on $P$ P by $q<_Pr$ <Pr iff $q$ q and $r$ r are members of the same $P_i$ Pi and $q<_ir$ <ir or $q$ q is member of $P_j$ $r$ Pj r is member of $P_k$ Pk and $j<_ik$. <ik. Maybe that formally, I should have taken $Q_i$ Qi as the cartesian product of the singleton of $i$ i with $P_i$, Pi, and $Q$ Q as the union of the $Q_i$'s, Qi's, and compared couples as $(i,q)$ (i,q) and $(j,r)$. (j,r). Anyway, it is clear that $P$, P, equipped with our relation is a poset, that is the poset sum over $I$ I of the posets $P_i$. Pi. Particularly, in the case that $I$ I and $P_i$ Pi all are totally ordered sets , $P$ P is a totally ordered set. And, in this particular case, a theorem of Schoenflies Schoenfliess asserts that " every (totally) ordered set is the union of scattered sets over a densely ordered ordrerd indexing set"set;

QuestionQestion: does there exist a similar decomposition theorem in the general case of posets ? Gérard Lang

2 edit into latex, grammar

Let I $I$ be a poset and for any i $i$ let Pi $P_i$ be a poset. Let P $P$ be the sum over I $I$ of the sets Pi, $P_i$, and let $<_P$ be the relation defined on $P$ by $q<_Pr$ iff $q$ and $r$ are members of the same $P_i$ and $q<_ir$ or $q$ is member of $P_j$ $r$ is member of $P_k$ and $j<_ik$. Maybe that formally, I should have taken $Q_i$ as the cartesian product of the singleton of $i$ with $P_i$, and $Q$ as the union of the $Q_i$'s, and compared couples as $(i,q)$ and $(j,r)$. Anyway, it is clear that $P$, equipped with our relation is a poset, that is the poset sum over $I$ of the posets $P_i$. Particularly, in the case that $I$ and $P_i$ all are totally ordered sets , $P$ is a totally ordered set. And, in this particular case, a theorem of Schoenflies asserts that "every (totally) ordered set is the union of scattered sets over a densely ordered indexing set";

Question: does there exist a similar decomposition theorem in the general case of posets ?

Gérard Lang

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