Displaying similar documents to “Selfduality of the system of convex subsets of a partially ordered set”

On the jump number of lexicographic sums of ordered sets

Hyung Chan Jung, Jeh Gwon Lee (2003)

Czechoslovak Mathematical Journal

Similarity:

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 ) + ∑ x ∈ P s ( Q x ) , where s ( X ) denotes the jump number of an ordered set X . We first show that w ( P ) - 1 + ∑ x ∈ P s ( Q x ) ≤ s ( Q ) ≤ s ( P ) + ∑ x ∈ 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...

Characterization of posets of intervals

Judita Lihová (2000)

Archivum Mathematicum

Similarity:

If A is a class of partially ordered sets, let P ( A ) denote the system of all posets which are isomorphic to the system of all intervals of A for some A ∈ A . We give an algebraic characterization of elements of P ( A ) for A being the class of all bounded posets and the class of all posets A satisfying the condition that for each a ∈ A there exist a minimal element u and a maximal element v with u ≤ a ≤ v , respectively.

On monotone permutations of ℓ -cyclically ordered sets

Ján Jakubík (2006)

Czechoslovak Mathematical Journal

Similarity:

For an ℓ -cyclically ordered set M with the ℓ -cyclic order C let P ( M ) be the set of all monotone permutations on M . We define a ternary relation C ¯ on the set P ( M ) . Further, we define in a natural way a group operation (denoted by · ) on P ( M ) . We prove that if the ℓ -cyclic order C is complete and C ¯ ≠ ∅ , then ( P ( M ) , · , C ¯ ) is a half cyclically ordered group.

Unique a -closure for some ℓ -groups of rational valued functions

Anthony W. Hager, Chawne M. Kimber, Warren W. McGovern (2005)

Czechoslovak Mathematical Journal

Similarity:

Usually, an abelian ℓ -group, even an archimedean ℓ -group, has a relatively large infinity of distinct a -closures. Here, we find a reasonably large class with unique and perfectly describable a -closure, the class of archimedean ℓ -groups with weak unit which are “ ℚ -convex”. ( ℚ is the group of rationals.) Any C ( X , ℚ ) is ℚ -convex and its unique a -closure is the Alexandroff algebra of functions on X defined from the clopen sets; this is sometimes C ( X ) .

Orthomodular lattices and closure operations in ordered vector spaces

Jan Florek (2010)

Banach Center Publications

Similarity:

On a non-trivial partially ordered real vector space (V,≤) the orthogonality relation is defined by incomparability and ζ(V,⊥) is a complete lattice of double orthoclosed sets. We say that A ⊆ V is an orthogonal set when for all a,b ∈ A with a ≠ b, we have a ⊥ b. In our earlier papers we defined an integrally open ordered vector space and two closure operations A → D(A) and A → A ⊥ ⊥ . It was proved that V is integrally open iff D ( A ) = A ⊥ ⊥ for every orthogonal set A ⊆ V. In this paper we generalize this...