Order-preserving Transformations Research Articles

Nash bargaining in ordinal environments

We analyze the implications of Nash’s (Econometrica 18:155–162, 1950) axioms in ordinal bargaining environments; there, the scale invariance axiom needs to be strenghtened to take into account all order-preserving transformations of the agents’ utilities. This axiom, called ordinal invariance, is a very demanding one. For two-agents, it is violated by every strongly individually rational bargaining rule. In general, no ordinally invariant bargaining rule satisfies the other three axioms of Nash. Parallel to Roth (J Econ Theory 16:247–251, 1977), we introduce a weaker independence of irrelevant alternatives (IIA) axiom that we argue is better suited for ordinally invariant bargaining rules. We show that the three-agent Shapley–Shubik bargaining rule uniquely satisfies ordinal invariance, Pareto optimality, symmetry, and this weaker IIA axiom. We also analyze the implications of other independence axioms.

Novel graphical representation of genome sequence and its applications in similarity analysis

In order to compare different genome sequences, an alignment-free method has been proposed. Considering the essential property of sequence is sequentiality, we define a compound transformation which transforms a genome sequence into a sparse 16 by L−1 matrix M based on 16 kinds of 2-mer (dinucleotides). Furthermore, we found the transformation above-mentioned is an order-preserving transformation (OPT). Based on the theory of matrix analysis, we derive a 16-dimensional vector to characterize a genome sequence via singular value decomposition (SVD) of M. Finally, we analyze the similarities among multiple sequences from 20 eutherian species. The experiment results show that our approach performs well in the field of sequence analysis.

On the Monoid of All Partial Order-Preserving Extensive Transformations

A partial transformation α on an n-element chain X n is called order-preserving if x ≤ y implies xα ≤yα for all x, y in the domain of α and it is called extensive if x ≤ xα for all x in the domain of α. The set of all partial order-preserving extensive transformations on X n forms a semiband POE n . We determine the maximal subsemigroups as well as the maximal subsemibands of POE n .

Maximal regular subsemibands of finite order-preserving transformation semigroups K(n,r)

Let O n be the order-preserving transformation semigroup on X n . For an arbitrary integer r such that 1≤r≤n−2, we completely describe the maximal regular subsemibands of the semigroup K(n,r)={α∈O n :|im(α)|≤r}. We also formulate the cardinal number of such subsemigroups.

Green’s relations and regularity for semigroups of transformations that preserve reverse direction equivalence

Let TX be the full transformation semigroup on a set X, $$T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\}$$ be the subsemigroup of TX determined by an equivalence E on X. In this paper the set X under consideration is a totally ordered set with n points. The set of all order preserving transformations in \(T_{E^{*}}(X)\) forms a subsemigroup of \(T_{E^{*}}(X)\) denoted by $$O_{E^*}(X)=\{\alpha\in T_{E^*}(X): \forall x,y\in X, x\leq y\Rightarrow x\alpha\leq y\alpha\}.$$ In this paper, we discuss Green’s relations for \(O_{E^{*}}(X)\) and prove that \(O_{E^{*}}(X)\) is a regular semigroup.

A classification of maximal idempotent-generated subsemigroups of finite orientation-preserving singular partial transformation semigroups

In this paper we describe the maximal idempotent-generated subsemigroups of the finite orientation-preserving singular partial transformation semigroup SPOPn and obtain their complete classification. We also obtain a classification of the maximal idempotent-generated subsemigroups of the finite order-preserving singular partial transformation semigroup \(\mathit{PO}^{k}_{n}\) with respect to ≤k.

On multivariate quantiles under partial orders

This paper focuses on generalizing quantiles from the ordering point of view. We propose the concept of partial quantiles, which are based on a given partial order. We establish that partial quantiles are equivariant under order-preserving transformations of the data, robust to outliers, characterize the probability distribution if the partial order is sufficiently rich, generalize the concept of efficient frontier, and can measure dispersion from the partial order perspective. We also study several statistical aspects of partial quantiles. We provide estimators, associated rates of convergence, and asymptotic distributions that hold uniformly over a continuum of quantile indices. Furthermore, we provide procedures that can restore monotonicity properties that might have been disturbed by estimation error, establish computational complexity bounds, and point out a concentration of measure phenomenon (the latter under independence and the componentwise natural order). Finally, we illustrate the concepts by discussing several theoretical examples and simulations. Empirical applications to compare intake nutrients within diets, to evaluate the performance of investment funds, and to study the impact of policies on tobacco awareness are also presented to illustrate the concepts and their use.

Statistical Change Detection by the Pool Adjacent Violators Algorithm

In this paper, we present a statistical change detection approach aimed at being robust with respect to the main disturbance factors acting in real-world applications such as illumination changes, camera gain and exposure variations, noise. We rely on modeling the effects of disturbance factors on images as locally order-preserving transformations of pixel intensities plus additive noise. This allows us to identify within the space of all of the possible image change patterns the subspace corresponding to disturbance factors effects. Hence, scene changes can be detected by a-contrario testing the hypothesis that the measured pattern is due to disturbance factors, that is, by computing a distance between the pattern and the subspace. By assuming additive Gaussian noise, the distance can be computed within a maximum likelihood nonparametric isotonic regression framework. In particular, the projection of the pattern onto the subspace is computed by an O(N) iterative procedure known as Pool Adjacent Violators algorithm.

Combinatorial results for order-preserving and order-decreasing transformations

Let O_n and C_n be the semigroup of all order-preserving transformations and of all order-preserving and order-decreasing transformations on the finite set X_n={1,2,\ldots ,n}, respectively. Let \fix (\alpha )={x\in X_n:x\alpha =x} for any transformation \alpha. In this paper, for any Y\subseteq X_n, we find the cardinalities of the sets O_{n,Y}={\alpha\in O_n:\fix (\alpha)=Y} and C_{n,Y}={\alpha\in C_n: \fix (\alpha )=Y}. Moreover, we find the numbers of transformations of O_n and C_n with r fixed points.

On the maximal regular subsemigroups of ideals of order-preserving or order-reversing transformations

We characterize the maximal regular subsemigroups of the ideals of the semigroup of all order-preserving transformations as well as of the semigroup of all order-preserving or order-reversing transformations on a finite ordered set.

ORDERED SEMIGROUPS, UPPER-TRIANGULAR REFLEXIVE RELATIONS AND SEMIGROUPS OF LANGUAGES

We study the connection between the classes of semigroups of positive languages, upper-triangular reflexive relations over posets and semigroups of order-preserving and extensive transformations of sets. From our constructions we derive some interesting consequences also for the finite case. For example, we prove that the class of finite semigroups of languages coincides with the class of semigroups of reflexive upper-triangular binary relations on finite sets.

Perfection for pomonoids

A pomonoid S is a monoid equipped with a partial order that is compatible with the binary operation. In the same way that M-acts over a monoid M correspond to the representation of M by transformations of sets, S-posets correspond to the representation of a pomonoid S by order preserving transformations of posets.

Econometric Effects of Utility Order-Preserving Transformations in Discrete Choice Models

In the random utility modelling context, choice probabilities are unaffected by increasing linear transformations of the systematic utility; hence its empirical specification is derived on the basis that only differences in utility matters and that the scale of utility is arbitrary. We argue that choice probabilities remain unchanged if these linear transformations are made under the deterministic perspective of a single individual choosing several times. But, in the random utility setting, parameter estimates might be significantly affected by these transformations. In particular we focus on the effect of two order-preserving transformations usually applied in the derivation of the representative utility from the conditional indirect utility function: adding a constant to the utility of all alternatives and multiplying each alternative utility by a constant. We concentrate on the two most popular specifications in transport mode choice: the “wage rate” (Train and McFadden Transport Res 12:349–353, 1978) and the “expenditure rate” (Jara-Diaz and Farah Transport Res 22B:159–171, 1987) specifications. Using a collection of synthetic datasets generated in a new fashion directly from the conditional indirect utility function, i.e. before applying any expansion or transformation, we demonstrate how taking this class of order-preserving transformations could lead to misinterpretation of the econometric results, such as detecting randomly distributed and correlated parameters and/or income and time effects which are in fact not present.

Hereditarily finitely based monoids of extensive transformations

It is shown that the monoid En of extensive transformations of a chain of order n is hereditarily finitely based if and only if n ≤ 3. It follows that the submonoid OEn of order-preserving transformations in En is also hereditarily finitely based if and only if n ≤ 3.

On semigroups of order-preserving transformations of countable linearly ordered sets

We consider semigroups of endomorphisms of linearly ordered sets ℕ and ℤ and their subsemigroups of cofinite endomorphisms. We study the Green relations, groups of automorphisms, conjugacy, centralizers of elements, growth, and free subsemigroups in these subgroups.

Regularity conditions for semigroups of isotone transformations of countable chains

Let Γ be a linearly ordered set (a chain), O(Γ) be the semigroup of all isotone transformations of Γ (i.e., order-preserving transformations). We find some necessary and some sufficient conditions on the chain Γ for the semigroup O(Γ) to be regular. For example, if Γ is a complete chain with a maximal element and a minimal one, then O(Γ) is regular. In particular, O(Γ) is regular if Γ is finite. We find necessary and sufficient conditions for the regularity of O(Γ) in the case where Γ is countable.

Order preserving transformations of the Hilbert Grassmannian

Let H be a separable real Hilbert space. Denote by \({\mathcal{G}}_{\infty}\) (H) the Grassmannian consisting of closed subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every order preserving transformation of \({\mathcal{G}}_{\infty}\) (H) can be extended to an automorphism of the lattice of closed subspaces of H. It follows from Mackey’s result [8] that automorphisms of this lattice are induced by invertible bounded linear operators.

Asymptotic Results for Semigroups of Order-Preserving Partial Transformations

ABSTRACT Let 𝒫 𝒞 n be the semigroup of all decreasing and order-preserving partial transformations of an n-element chain, and let E(𝒫 𝒞 n ) be its set of idempotents. Among other results, asymptotic formulae for |𝒫 𝒞 n | and |E(𝒫 𝒞 n )|/|𝒫 𝒞 n | are obtained. Similar results for 𝒫 𝒪 n the (larger) semigroup of all order-preserving partial transformations of an n-element chain are also obtained.

Green's Equivalences on Semigroups of Transformations Preserving Order and an Equivalence Relation

Let ${\cal T}_X$ be the full transformation semigroup on the set $X$, \[ T_{E}(X)=\{f\in {\cal T}_X\colon \ \forall(a,b)\in E,(f(a),f(b))\in E\} \] be the subsemigroup of ${\cal T}_X$ determined by an equivalence $E$ on $X$. In this paper the set $X$ under consideration is a totally ordered set with $mn$ points where $m\geq 2$ and $n\geq 3$. The equivalence $E$ has $m$ classes each of which contains $n$ consecutive points. The set of all order preserving transformations in $T_{E}(X)$ forms a subsemigroup of $T_E(X)$ denoted by \[ {\cal O}_{E}(X)=\{f\in T_{E}(X)\colon \ \forall\, x, y\in X, \ x\leq y \mbox{ implies } f(x)\leq f(y)\}. \] The nature of regular elements in ${\cal O}_{E}(X)$ is described and the Green's equivalences on ${\cal O}_{E}(X)$ are characterized completely.

A family of ordinal solutions to bargaining problems with many players

A solution to bargaining problems is ordinal when it is covariant with respect to order-preserving transformations of utility. Shapley has constructed an ordinal, symmetric, efficient solution to three-player problems. Here, we extend Shapley's solution in two directions. First, we extend it to more than three players. Second, we show that this extension lends itself to the construction of a continuum of ordinal, symmetric, efficient solutions. The construction makes use of ordinal path-valued solutions that were suggested and studied by O'Neil et al. [Games Econ. Behav. 48 (2004) 139–153].