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Start Over Subject ordered set

2. Linear extensions of orderings

Creator:
Novák, Vítězslav and Novotný, Miroslav
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
ordered set, linear extension, natural representation, lexicographic sum, and dense subset
Language:
English
Description:
A construction is given which makes it possible to find all linear extensions of a given ordered set and, conversely, to find all orderings on a given set with a prescribed linear extension. Further, dense subsets of ordered sets are studied and a procedure is presented which extends a linear extension constructed on a dense subset to the whole set.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. On the jump number of lexicographic sums of ordered sets

Creator:
Jung, Hyung Chan and Lee, Jeh Gwon
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
ordered set, jump (setup) number, lexicographic sum, and jump-critical
Language:
English
Description:
Let $Q$ be the lexicographic sum of finite ordered sets $Q_x$ over a finite ordered set $P$. For some $P$ we can give a formula for the jump number of $Q$ in terms of the jump numbers of $Q_x$ and $P$, that is, $s(Q)=s(P)+ \sum _{x\in P} s(Q_x)$, where $s(X)$ denotes the jump number of an ordered set $X$. We first show that $w(P)-1+\sum _{x\in P} s(Q_x)\le s(Q) \le s(P)+ \sum _{x\in P} s(Q_x)$, where $w(X)$ denotes the width of an ordered set $X$. Consequently, if $P$ is a Dilworth ordered set, that is, $s(P) = w(P)-1$, then the formula holds. We also show that it holds again if $P$ is bipartite. Finally, we prove that the lexicographic sum of certain jump-critical ordered sets is also jump-critical.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

4. Relative polars in ordered sets

Creator:
Halaš, Radomír
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
ordered set, distributive set, ideal, prime ideal, $R$-polar, and annihilator
Language:
English
Description:
In the paper, the notion of relative polarity in ordered sets is introduced and the lattices of $R$-polars are studied. Connections between $R$-polars and prime ideals, especially in distributive sets, are found.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public