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Orders on Subsets Rationalised by Abstract Convex Geometries
Order - Tập 26 Số 3 - Trang 237-244 - 2009
κ-Complete Uniquely Complemented Lattices
Order - - 2008
We show that for any infinite cardinal κ, every complete lattice where each
element has at most one complement can be regularly embedded into a uniquely
complemented κ-complete lattice. This regular embedding preserves all joins and
meets, in particular it preserves the bounds of the original lattice. As a
corollary, we obtain that every lattice where each element has at most one
complement can be...
Shrinkability of Minimal Elements in Sphere Representations of Posets
Order - Tập 14 - Trang 59-66 - 1997
Let k be a positive integer and let P = (X,≤) be a poset. We call P a k-sphere
order provided there is a mapping f which assigns to each element x $$ \in$$ X a
ball f(x) in Rk so that x ≤ y in P if and only if f(x) $$ \subseteq$$ f(y). We
ask: Given that P is a k-sphere order, does there necessarily exist a
representation f with the property that every minimal element of P is assigned a
ball of ra...
The number of orders with thirteen elements
Order - Tập 9 - Trang 203-204 - 1992
The number of non-isomorphic posets on 13 elements is P13=33,823,827,452 An
announcement of this result was sent to the on July 4th, 1992. . This extends
our previous result P12 which constituted the greatest known value. A table
enumerates the posets according to their number of relations.
The Relationship of Partial Metric Varieties and Commuting Powers Varieties
Order - Tập 30 - Trang 403-414 - 2012
Holland et al. (Algebra Univers 67:1–18, 2012) considered varieties ${\mathcal
E}_n$ of lattice-ordered groups defined by partial metrics, and showed for all n
that ${\mathcal E}_n$ is contained within the variety ${\mathcal L}_n$ defined
by x n y n = y n x n . They also showed that if n were prime, then ${\mathcal
E}_n = {\mathcal L}_n$ . Letting ${\mathcal A}^2$ denote the metabelian variety
(d...
Scheduling rooted forests with communication delays
Order - - 1994
We show that a greedy algorithm for scheduling unit time jobs on two machines
with unit communication delays produces an optimal schedule when the precedence
constraints are given by a rooted forest. We also give a min/max relationship
for the length of such a schedule. The min/max result (for forests and two
machines) shows that the addition of unit communication delays increases the
optimal sche...
Order-Sensitive Domination in Partially Ordered Sets and Graphs
Order - Tập 40 - Trang 157-172 - 2022
For a (finite) partially ordered set (poset) P, we call a dominating set D in
the comparability graph of P, an order-sensitive dominating set in P if either x
∈ D or else a < x < b in P for some a,b ∈ D for every element x in P which is
neither maximal nor minimal, and denote by γos(P), the least size of an
order-sensitive dominating set of P. For every graph G and integer $$k\geqslant
2$$ , we as...
Cartesian Lattice Counting by the Vertical 2-sum
Order - Tập 39 - Trang 113-141 - 2021
A vertical 2-sum of a two-coatom lattice L and a two-atom lattice U is obtained
by removing the top of L and the bottom of U, and identifying the coatoms of L
with the atoms of U. This operation creates one or two nonisomorphic lattices
depending on the symmetry case. Here the symmetry cases are analyzed, and a
recurrence relation is presented that expresses the number of nonisomorphic
vertical 2-...
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