Natural Partial Order Research Articles

A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps

When a homogeneous convex cone is given, a natural partial order is introduced in the cone. We shall show that a homogeneous convex cone is a symmetric cone if and only if Vinberg´s ∗ -map and its inverse reverse the order. Actually our theorem is formulated in terms of the family of pseudoinverse maps including the ∗ -map, and states that the above order-reversing property is typical of the ∗ -map of a symmetric cone which coincides with the inverse map of the Jordan algebra associated with the symmetric cone.

NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH KESTEN DISTRIBUTIONS AND RELATED POISSON PROCESSES

We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition P we assign the weight w(P) = pe(P)qe'(P), where e(P) and e'(P) are, respectively, the numbers of disorders and orders in P related to the natural partial order on the set of blocks of P implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.

Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras

We say that an algebra $\Lambda$ over a commutative noetherian ring $R$ is Calabi-Yau of dimension $d$ ($d$-CY) if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length $\Lambda$-modules. We show that when $R$ is $d$-dimensional local Gorenstein the $d$-CY algebras are exactly the symmetric $R$-orders of global dimension $d$. We give a complete description of all tilting modules of projective dimension at most one for 2-CY algebras, and show that they are in bijection with elements of affine Weyl groups, preserving various natural partial orders. We show that there is a close connection between tilting theory for 3-CY algebras and the Fomin-Zelevinsky mutation of quivers (or matrices). We prove a conjecture of Van den Bergh on derived equivalence of noncommutative crepant resolutions.

Orbit closures in the enhanced nilpotent cone

We study the orbits of G = GL ( V ) in the enhanced nilpotent cone V × N , where N is the variety of nilpotent endomorphisms of V. These orbits are parametrized by bipartitions of n = dim V , and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato's exotic nilpotent cone in type C, proving that the closure ordering is the same, and conjecturing that the intersection cohomology is the same but with degrees doubled.

The absolute order on the symmetric group, constructible partially ordered sets and Cohen–Macaulay complexes

The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen–Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed.

First Observations on Prefab Posets’ Whitney Numbers

We introduce a natural partial order ≤ in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers’ triangular array — are then calculated and the explicit formula for them is provided. Next — in the second construction — we endow the set sums of prefabiants with such an another partial order that their Bell-like numbers include Fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial Newton coefficients and Fibonacci numbers onto the infinity of their relatives among whom there are also the Fibonacci triad sequences and binomial-like coefficients (incidence coefficients included). The first partial order is F-sequence independent while the second partial order is F-sequence dependent where F is the so-called admissible sequence determining cobweb poset by construction. An F-determined cobweb poset’s Hasse diagram becomes Fibonacci tree sheathed with specific cobweb if the sequence F is chosen to be just the Fibonacci sequence. From the stand-point of linear algebra of formal series these are generating functions which stay for the so-called extended coherent states of quantum physics. This information is delivered in the last section.

F-Monoids

A semigroup S is called F-monoid if S has an identity and if there exists a group congruence ρ on S such that each ρ-class of S contains a greatest element with respect to the natural partial order of S (see Mitsch, 1986). Generalizing results given in Giraldes et al. (2004) and specializing some of Giraldes et al. (Submitted) five characterizations of such monoids S are provided. Three unary operations “*”, “○”, and “ − ” on S defined by means of the greatest elements in the different ρ-classes of S are studied. Using their properties a charaterization of F-monoids S by their regular part S° = {a°|a ∈ S} and the associates of elements in S° is given. Under the hypothesis that S * = {a *|a ∈ S} is a subsemigroup it is shown that S is regular, whence of a known structure (see Giraldes et al., 2004).

Characterizing cryptogroups with a finite number of $${\mathcal{H}}$$ -classes in each $${\mathcal{D}}$$ -class by their subsemigroups

We construct a family \({\mathcal{F}}\) of completely regular semigroups with the property that each completely regular semigroup S with a finite number of \({\mathcal{H}}\) -classes in each \({\mathcal{D}}\) -class is non-cryptic if and only if S contains an isomorphic image of a member of \({\mathcal{F}}\) . Each member F of \({\mathcal{F}}\) is an ideal extension of a Rees matrix semigroup J by a cyclic group B with a zero adjoined and the identity of B is the identity of F. Here \(J={\mathcal{M}}\left( I,G,\Lambda;P\right) \) with I and Λ finite, G is given by generators and relations, and P is given explicitly. Within completely regular semigroups, the cryptic property is equivalent to \({\mathcal{N}=\mathcal{S}},\) where \({\mathcal{N}}\) is the natural partial order and a\({\mathcal{S}} b\) if and only if a2 = ab = ba. Hence the above result can be formulated in terms of \({\mathcal{N}}\) and \({\mathcal{S}}\) .

Addition and multiplication of sets

AbstractOrdinal addition and multiplication can be extended in a natural way to all sets. I survey the structure of the sets under these operations. In particular, the natural partial ordering associated with addition of sets is shown to be a tree. This allows us to prove that any set has a unique representation as a sum of additively irreducible sets, and that the non‐empty elements of any model of set theory can be partitioned into infinitely many submodels, each isomorphic to the original model. Also any model of set theory has an isomorphic extension in which the empty set of the original model is non‐empty. Among other results, the relations between the arithmetical operations and the transitive closure and the adductive hierarchy are elucidated. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Geometry of contact transformations and domains: orderability versus squeezing

Gromov’s famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact nonsqueezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a contact manifold. In contrast to our earlier beliefs, we show that the answer to this question is very sensitive to the topology of the manifold. For instance, we prove that the standard contact sphere is non-orderable while the real projective space is known to be orderable. Our methods include a new embedding technique in contact geometry as well as a generalized Floer homology theory which contains both cylindrical contact homology and Hamiltonian Floer homology. We discuss links to a number of miscellaneous topics such as topology of free loops spaces, quantum mechanics and semigroups.

Regular Semigroups Each of Whose Least Completely Simple Congruence Classes Has a Greatest Element

A regular semigroup S is termed locally F-regular, if each class of the least completely simple congruence ξ contains a greatest element with respect to the natural partial order. It is shown that each locally F-regular semigroup S admits an embedding into a semidirect product of a band by S/ξ. Further, if ξ satisfies the additional property that for each s ∈ S and each inverse (sξ)′ of sξ in S/ξ the set (sξ)′ ∩ V(s) is not empty, we represent S both as a Rees matrix semigroup over an F-regular semigroup as well as a certain subsemigroup of a restricted semidirect product of a band by S/ξ.

Inverse semigroups determined by their partial automorphism monoids

AbstractThe partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup S with no isolated nontrivial subgroups is lattice determined ‘modulo semilattices’ and if T is an inverse semigroup whose partial automorphism monoid is isomorphic to that of S, then either S and T are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if T is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of S and T, respectively, are isomorphic. Moreover, for these results to hold, the conditions that S be tightly connected and have no isolated nontrivial subgroups are essential.

A PARTIAL ORDER ON THE ORTHOGONAL GROUP

ABSTRACT We define a natural partial order on the orthogonal group and completely describe the intervals in this partial order. The main technical ingredient is that an orthogonal transformation induces a unique orthogonal transformation on each subspace of the orthogonal complement of its fixed subspace.

Comparing probability measures using possibility theory: A notion of relative peakedness

Deciding whether one probability distribution is more informative (in the sense of representing a less indeterminate situation) than another one is typically done using well-established information measures such as, e.g., the Shannon entropy or other dispersion indices. In contrast, the relative specificity of possibility distributions is evaluated by means of fuzzy set inclusion. In this paper, we propose a technique for comparing probability distributions from the point of view of their relative dispersion without resorting to a numerical index. A natural partial ordering in terms of relative “peakedness” of probability functions is proposed which is closely related to order-1 stochastic dominance. There is also a close connection between this ordering on probability distributions and the standard specificity ordering on possibility distributions that can be derived by means of a known probability–possibility transformation. The paper proposes a direct proof showing that the (total) preordering on probability measures defined by probabilistic entropy refines the (partial) ordering defined by possibilistic specificity. This result, also valid for other dispersion indices, is discussed against the background of related work in statistics, mathematics (inequalities on convex functions), and the social sciences. Finally, an application of the possibilistic specificity ordering in the field of machine learning or, more specifically, the induction of decision forests is proposed.

Equivalence Classes of Ideals in the Nilradical of a Borel Subalgebra

AbstractAn equivalence relation is defined and studied on the set of B-stable ideals in the nilradical of the Lie algebra of a Borel subgroup B. Techniques are developed to compute the equivalence relation and these are carried out in the exceptional groups. There is a natural partial order on equivalence classes coming from inclusion of one ideal in another. A main theorem is that this partial order is a refinement of the closure ordering on nilpotent orbits.

A dichotomy in classifying quantifiers for finite models

AbstractWe consider a family of finite universes. The second order existential quantifier Qℜ means for each U Є quantifying over a set of n(ℜ)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Qℜ, either Qℜ is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Qℜ) (first order logic plus the quantifier Qℜ) is undecidable.

On stable embeddability of partitions

Several natural partial orders on integral partitions, such as the embeddability, the stable embeddability, the bulk embeddability and the supermajorization, arise in quantum computation, bin-packing and matrix analysis. We find the implications of these partial orders. For integral partitions whose entries are all powers of a fixed number p , we show that the embeddability is completely determined by the supermajorization order and we find an algorithm for determining the stable embeddability.

Finding optimal satisficing strategies for and-or trees

Many tasks require evaluating a specified Boolean expression φ over a set of probabilistic tests whose costs and success probabilities are each known. A strategy specifies when to perform which test, towards determining the overall outcome of φ. We are interested in finding the strategy with the minimum expected cost. As this task is typically NP-hard—for example, when tests can occur many times within φ, or when there are probabilistic correlations between the test outcomes—we consider those cases in which the tests are probabilistically independent and each appears only once in φ. In such cases, φ can be written as an and-or tree, where each internal node corresponds to either the “and” or “or” of its children, and each leaf node is a probabilistic test. In this paper we investigate “probabilistic and-or tree resolution” ( PAOTR), namely the problem of finding optimal strategies for and-or trees. We first consider a depth-first approach: evaluate each penultimate rooted subtree in isolation, replace each such subtree with a single “mega-test”, and recurse on the resulting reduced tree. We show that the strategies produced by this approach are optimal for and-or trees with depth at most two but can be arbitrarily sub-optimal for deeper trees. Each depth-first strategy can be described by giving the linear relative order in which tests are to be executed, with the understanding that any test whose outcome becomes irrelevant is skipped. The class of linear strategies is strictly larger than depth-first strategies. We show that even the best linear strategy can also be arbitrarily sub-optimal. We next prove that an optimal strategy honors a natural partial order among tests with a common parent node (“leaf-sibling tests”), and use this to produce a dynamic programming algorithm that finds the optimal strategy in time O ( d 2 ( r + 1 ) d ) , where r is the maximum number of leaf-siblings and d is the number of leaf-parents; hence, for trees with a bounded number of internal nodes, this run-time is polynomial in the tree size. We also present another special class of and-or trees for which this task takes polynomial time. We close by presenting a number of other plausible approaches to PAOTR, together with counterexamples to show their limitations.

Jordan Blocks of Richardson Classes in the Classical Groups and the Bala-Carter Theorem#

This article provides new, relatively simple proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive the description of the natural partial order of unipotent classes in the general linear group in terms of their Jordan blocks. This result was originally due to Gerstenhaber. Finally, we obtain a proof of the Bala-Carter Theorem, which holds even in certain bad characteristics. This proof requires the prior knowledge of the number of unipotent classes, unlike the original proofs due to Bala, Carter and Pommerening.

Partial orders on linear transformation semigroups

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear α: A → B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by ⊆ (that is, α ⊆ β if and only if dom α ⊆ dom β and α = β | dom α). We compare this order with the so-called 'natural partial order' ≤ on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all α ∈ P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below ≤, and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned