scholarly journals A New Relationship between Block Designs

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Alexander Shramchenko ◽  
Vasilisa Shramchenko
Keyword(s):  
Partial Order ◽  
Partial Ordering ◽  
Block Designs ◽  
Partially Ordered ◽  
Power Set ◽  
Relationship Of

We propose a procedure of constructing new block designs starting from a given one by looking at the intersections of its blocks with various sets and grouping those sets according to the structure of the intersections. We introduce a symmetric relationship of friendship between block designs built on a set V and consider families of block designs where all designs are friends of each other, the so-called friendly families. We show that a friendly family admits a partial ordering. Furthermore, we exhibit a map from the power set of V, partially ordered by inclusion, to a friendly family of a particular type which preserves the partial order.

scholarly journals Retaining the double dimension of species diversity: Application of partially ordered set theory and Hasse diagrams

2014 ◽  
Author(s):  
Christian Kampichler ◽  
Ralf Wieland
Keyword(s):  
Species Richness ◽  
Species Diversity ◽  
Partial Order ◽  
Biological Diversity ◽  
Diversity Index ◽  
Order Theory ◽  
Partial Ordering ◽  
Attractive Alternative ◽  
Partially Ordered ◽  
Hasse Diagrams

The measurement of species diversity has been a central task of community ecology from the mid 20th century onward. The conventional method of designing a diversity index is to combine values for species richness and assemblage evenness into a single composite score. The literature abounds with such indices. Each index weights richness and evenness in a different fashion. The conventional approach has repeatedly been criticized since there is an infinite number of potential indices which have a minimum value when S (species richness) = 1 and a maximum value when S = N (number of individuals). We argue that partial order theory is a sound mathematical fundament and demonstrate that it is an attractive alternative for comparing and ranking biological diversity without the necessity of combining values for species richness and evenness into an ambiguous diversity index. The general principle of partial ordering is simple: one particular assemblage is regarded as more diverse than another when both its species richness and its evenness are higher. Assemblages are not comparable with each other when one has a higher value for species richness and a lower value for evenness. Hasse diagrams can graphically represent partially ordered communities. Linear extensions and rank-frequency distributions reveal the potential of partial order theory as a means to support decisions when assemblage ranking is desired.

Partial Orders on the 2-Cell

1975 ◽  
Vol 18 (3) ◽  
pp. 411-416
Author(s):  
E. D. Tymchatyn
Keyword(s):  
Partial Order ◽  
Metric Space ◽  
Closed Subset ◽  
Cartesian Product ◽  
Partial Orders ◽  
Partial Ordering ◽  
Compact Metric Space ◽  
Partially Ordered ◽  
Ordered Space

A partially ordered space is an ordered pair (X, ≤) where X is a compact metric space and ≤ is a partial ordering on X such that ≤ is a closed subset of the Cartesian product X×X. ≤ is said to be a closed partial order on X.

scholarly journals Retaining the double dimension of species diversity: Application of partially ordered set theory and Hasse diagrams

2014 ◽  
Author(s):  
Christian Kampichler ◽  
Ralf Wieland
Keyword(s):  
Species Richness ◽  
Species Diversity ◽  
Partial Order ◽  
Biological Diversity ◽  
Diversity Index ◽  
Order Theory ◽  
Partial Ordering ◽  
Attractive Alternative ◽  
Partially Ordered ◽  
Hasse Diagrams

The measurement of species diversity has been a central task of community ecology from the mid 20th century onward. The conventional method of designing a diversity index is to combine values for species richness and assemblage evenness into a single composite score. The literature abounds with such indices. Each index weights richness and evenness in a different fashion. The conventional approach has repeatedly been criticized since there is an infinite number of potential indices which have a minimum value when S (species richness) = 1 and a maximum value when S = N (number of individuals). We argue that partial order theory is a sound mathematical fundament and demonstrate that it is an attractive alternative for comparing and ranking biological diversity without the necessity of combining values for species richness and evenness into an ambiguous diversity index. The general principle of partial ordering is simple: one particular assemblage is regarded as more diverse than another when both its species richness and its evenness are higher. Assemblages are not comparable with each other when one has a higher value for species richness and a lower value for evenness. Hasse diagrams can graphically represent partially ordered communities. Linear extensions and rank-frequency distributions reveal the potential of partial order theory as a means to support decisions when assemblage ranking is desired.

scholarly journals Primitive idempotent measures on compact semitopological semigroups

1972 ◽  
Vol 13 (4) ◽  
pp. 451-455 ◽  
Author(s):  
Stephen T. L. Choy
Keyword(s):  
Partial Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Zero Element ◽  
Primitive Idempotent ◽  
Natural Partial Order ◽  
Partially Ordered

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

Monotone but not positive subsets of the Cantor space

1987 ◽  
Vol 52 (3) ◽  
pp. 817-818 ◽  
Author(s):  
Randall Dougherty
Keyword(s):  
Partial Ordering ◽  
Background Information ◽  
Countable Union ◽  
Abstract Form ◽  
Cantor Space ◽  
Countable Intersection ◽  
Power Set ◽  
Countably Infinite ◽  
Infinite Set

A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive (-positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive.First we note a few isomorphisms. The space (ω2, ≤) is isomorphic to the space (ω2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, (ω2, ≤) is isomorphic to ((ω), ⊆), where denotes the power set operator, or to ((S), ⊆) for any countably infinite set S.In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).

Partial orders on the power sets of Baer rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050011 ◽  
Author(s):  
B. Ungor ◽  
S. Halicioglu ◽  
A. Harmanci ◽  
J. Marovt
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Baer Ring ◽  
Power Set ◽  
Von Neumann Regular ◽  
Baer Rings ◽  
The Right

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

Jensen's ⃞ principles and the Novák number of partially ordered sets

1986 ◽  
Vol 51 (1) ◽  
pp. 47-58 ◽  
Author(s):  
Boban Veličković
Keyword(s):  
Partial Order ◽  
Partially Ordered Sets ◽  
Large Cardinals ◽  
Ordered Sets ◽  
Weakly Compact ◽  
Cardinal Function ◽  
Partially Ordered ◽  
Suslin Tree

In this paper we consider various properties of Jensen's □ principles and use them to construct several examples concerning the so-called Novák number of partially ordered sets.In §1 we give the relevant definitions and review some facts about □ principles. Apart from some simple observations most of the results in this section are known.In §2 we consider the Novák number of partially ordered sets and, using □ principles, give counterexamples to the productivity of this cardinal function. We also formulate a principle, show by forcing that it is consistent and use it to construct an ℵ2-Suslin tree T such that forcing with T × T collapses ℵ1.In §3 we briefly consider games played on partially ordered sets and relate them to the problems of the previous section. Using a version of □ we give an example of a proper partial order such that the game of length ω played on is undetermined.In §4 we raise the question of whether the Novák number of a homogenous partial order can be singular, and show that in some cases the answer is no.We assume familiarity with the basic techniques of forcing. In §1 some facts about large cardinals (e.g. weakly compact cardinals are -indescribable) and elementary properties of the constructible hierarchy are used. For this and all undefined terms we refer the reader to Jech [10].

scholarly journals Bringing Order to Chaos – A Compact Representation of Partial Order in SAT-Based HTN Planning

2019 ◽  
Vol 33 ◽  
pp. 7520-7529 ◽  
Author(s):  
Gregor Behnke ◽  
Daniel Höller ◽  
Susanne Biundo
Keyword(s):  
Partial Order ◽  
Propositional Logic ◽  
State Of The Art ◽  
Partial Ordering ◽  
Compact Representation ◽  
Planning Techniques ◽  
Model Complex ◽  
Htn Planning ◽  
Sat Solver ◽  
Planning Problems

HTN planning provides an expressive formalism to model complex application domains. It has been widely used in realworld applications. However, the development of domainindependent planning techniques for such models is still lacking behind. The need to be informed about both statetransitions and the task hierarchy makes the realisation of search-based approaches difficult, especially with unrestricted partial ordering of tasks in HTN domains. Recently, a translation of HTN planning problems into propositional logic has shown promising empirical results. Such planners benefit from a unified representation of state and hierarchy, but until now require very large formulae to represent partial order. In this paper, we introduce a novel encoding of HTN Planning as SAT. In contrast to related work, most of the reasoning on ordering relations is not left to the SAT solver, but done beforehand. This results in much smaller formulae and, as shown in our evaluation, in a planner that outperforms previous SAT-based approaches as well as the state-of-the-art in search-based HTN planning.

scholarly journals Global solutions to nonlinear second order interval integrodifferential equations by fixed point in partially ordered sets

2018 ◽  
Vol 37 (4) ◽  
pp. 153-172
Author(s):  
Robab Alikhani ◽  
Fariba Bahrani
Keyword(s):  
Fixed Point ◽  
Global Solution ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Partially Ordered Sets ◽  
Second Order ◽  
Partial Ordering ◽  
Order Interval ◽  
Ordered Sets ◽  
Partially Ordered

In this paper, we prove the existence and uniqueness of global solution for second order interval valued integrodifferential equation with initial conditions admitting only the existence of a lower solution or an upper solution. In this study, in order to make the global solution on entire $[0,b]$, we use a fixed point in partially ordered sets on the subintervals of $[0,b]$ and obtain local solutions. Also, under weak conditions we show being well-defined a special kind of  H-difference involved in this work. Moreover, we compare the results of existence and uniqueness under consideration of two kind of partial ordering on fuzzy numbers.

Injective Hulls of Semilattices

1970 ◽  
Vol 13 (1) ◽  
pp. 115-118 ◽  
Author(s):  
G. Bruns ◽  
H. Lakser
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Binary Operation ◽  
Ordered Set ◽  
Upper Bounds ◽  
Partial Ordering ◽  
Partially Ordered ◽  
Image Position ◽  
Injective Hulls

A (meet-) semilattice is an algebra with one binary operation ∧, which is associative, commutative and idempotent. Throughout this paper we are working in the category of semilattices. All categorical or general algebraic notions are to be understood in this category. In every semilattice S the relationdefines a partial ordering of S. The symbol "∨" denotes least upper bounds under this partial ordering. If it is not clear from the context in which partially ordered set a least upper bound is taken, we add this set as an index to the symbol; for example, ∨AX denotes the least upper bound of X in the partially ordered set A.

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