Natural Partial Order Research Articles

Sequences with increasing subsequence

We study analytic and Borel subsets defined similarly to the old example of analytic complete set given by Luzin. Luzin's example, which is essentially a subset of the Baire space, is based on the natural partial order on naturals, i.e. division. It consists of sequences which contain increasing subsequence in given order.We consider a variety of sets defined in a similar way. Some of them occurs to be Borel subsets of the Baire space, while others are analytic complete, hence not Borel.In particular, we show that an analogon of Luzin example based on the natural linear order on rationals is analytic complete. We also characterize all countable linear orders having such property.

Idempotent cellular automata and their natural order

Motivated by the search for idempotent cellular automata (CA), we study CA that act almost as the identity unless they read a fixed pattern p. We show that constant and symmetrical patterns always generate idempotent CA, and we characterize the quasi-constant patterns that generate idempotent CA. Our results are valid for CA over an arbitrary group G. Moreover, we study the semigroup theoretic natural partial order defined on idempotent CA. If G is infinite, we prove that there is an infinite independent set of idempotent CA, and if G has an element of infinite order, we prove that there is an infinite increasing chain of idempotent CA.

The natural partial order on semigroups of transformations with restricted range that preserve an equivalence

The natural partial order on semigroups of transformations with restricted range that preserve an equivalence

Green’s relations and natural partial order on Baer-Levi semigroups of partial transformations with restricted range

Green’s relations and natural partial order on Baer-Levi semigroups of partial transformations with restricted range

A Monotonous Intuitionistic Fuzzy TOPSIS Method Under General Linear Orders via Admissible Distance Measures

All intuitionistic fuzzy TOPSIS methods contain two key elements: (1) the order structure, which can affect the choices of positive ideal-points and negative ideal-points, and construction of admissible distance/similarity measures; (2) the distance/similarity measure, which is closely related to the values of the relative closeness degrees and determines the accuracy and rationality of decision-making. For the order structure, many efforts are devoted to constructing some score functions, which can strictly distinguish different intuitionistic fuzzy values (IFVs) and preserve the natural partial order for IFVs.This paper proves that such a score function does not exist, namely the application of a single monotonous and continuous function does not distinguish all IFVs. For the distance or similarity measure, some examples are given to show that classical similarity measures based on the normalized Euclidean distance and normalized Minkowski distance do not meet the axiomatic definition of intuitionistic fuzzy similarity measures. Moreover,some illustrative examples are given to show that classical intuitionistic fuzzy TOPSIS methods do not ensure the monotonicity with the natural partial order or linear orders, which may yield some counter-intuitive results. To overcome the limitation of non-monotonicity, we propose a novel intuitionistic fuzzy TOPSIS method,using three new admissible distances with the linear orders measured by a score degree/similarity function and accuracy degree, or two aggregation functions, and prove that the proposed TOPSIS method is monotonous under these three linear orders.} This is the first result with a strict mathematical proof on the monotonicity with the linear orders for the intuitionistic fuzzy TOPSIS method.

Approximation Relations on the Posets of Pseudoultrametrics

In this paper we study pseudoultrametrics, which are a natural mixture of ultrametrics and pseudometrics. They satisfy a stronger form of the triangle inequality than usual pseudometrics and naturally arise in problems of classification and recognition. The text focuses on the natural partial order on the set of all pseudoultrametrics on a fixed (not necessarily finite) set. In addition to the “way below” relation induced by a partial order, we introduce its version which we call “weakly way below”. It is shown that a pseudoultrametric should satisfy natural conditions closely related to compactness, for the set of all pseudoultrametric weakly way below it to be non-trivial (to consist not only of the zero pseudoultrametric). For non-triviality of the set of all pseudoultrametrics way below a given one, the latter must be compact. On the other hand, each compact pseudoultrametric is the least upper bound of the directed set of all pseudoultrametrics way below it, which are compact as well. Thus it is proved that the set CPsU(X) of all compact pseudoultrametric on a set X is a continuous poset. This shows that compactness is a crucial requirement for efficiency of approximation in methods of classification by means of ultrapseudometrics.

Function space of continuous maps from [0,1] to tree II

For a finite tree T and its endpoint ⊤, we can define a natural partial order ≤ such that ⊤ is the top element in T. For X=[0,1] and a continuous map f:X→T, let ↓f={(x,t)|t≤f(x)} and ↓C(X,T)={↓f|fis continuous from XtoT}. We investigate the subspace ↓C(X,T) of the family of all non-empty closed sets in X×T with the Hausdorff distance. Let ↓C(X,T)‾ be the closure of ↓C(X,T) in Cld(X×T). Let Ev be the family of all edges in T with the upper vertex v, Vv be the set of lower vertex of all edges in Ev. For a pair of vertices v and u∈Vv, define T[v,u]=vuˆ∪↓u. For A∈Cld(X×T) and x∈X, let A(x)={t∈T|(x,t)∈A}. In the paper, we show↓C(X,T)‾={A∈Cld(X×T)| for allx∈X, there exists a(x)∈A(x) such that either A(x)=↓a(x) or a(x) is a vertex and A(x)=T[a(x),u] for some u∈Va(x)}.

INITIATION OF FLUOXETINE IN A PEDIATRIC PATIENT WITH MUCOPOLYSACCHARIDOSIS IIIA: EARLY OBSERVATIONS

For a finite tree T and its endpoint ⊤, we can define a natural partial order ≤ such that ⊤ is the top element in T. For X=[0,1] and a continuous map f:X→T, let ↓f={(x,t)|t≤f(x)} and ↓C(X,T)={↓f|fis continuous from XtoT}. We investigate the subspace ↓C(X,T) of the family of all non-empty closed sets in X×T with the Hausdorff distance. Let ↓C(X,T)‾ be the closure of ↓C(X,T) in Cld(X×T). Let Ev be the family of all edges in T with the upper vertex v, Vv be the set of lower vertex of all edges in Ev. For a pair of vertices v and u∈Vv, define T[v,u]=vuˆ∪↓u. For A∈Cld(X×T) and x∈X, let A(x)={t∈T|(x,t)∈A}. In the paper, we show↓C(X,T)‾={A∈Cld(X×T)| for allx∈X, there exists a(x)∈A(x) such that either A(x)=↓a(x) or a(x) is a vertex and A(x)=T[a(x),u] for some u∈Va(x)}.

Frobenius Allowable Gaps of Generalized Numerical Semigroups

A generalized numerical semigroup is a submonoid $S$ of $\mathbb{N}^d$ for which the complement $\mathbb{N}^d\setminus S$ is finite. The points in the complement $\mathbb{N}^d\setminus S$ are called gaps. A gap $F$ is considered Frobenius allowable if there is some relaxed monomial ordering on $\mathbb{N}^d$ with respect to which $F$ is the largest gap. We characterize the Frobenius allowable gaps of a generalized numerical semigroup. A generalized numerical semigroup that has only one maximal gap under the natural partial ordering of $\mathbb{N}^d$ is called a Frobenius generalized numerical semigroup. We show that Frobenius generalized numerical semigroups are precisely those whose Frobenius gap does not depend on the relaxed monomial ordering. We estimate the number of Frobenius generalized numerical semigroup with a given Frobenius gap $F=(F^{(1)},\dots,F^{(d)})\in\mathbb{N}^d$ and show that it is close to $\sqrt{3}^{(F^{(1)}+1)\cdots (F^{(d)}+1)}$ for large $d$. We define notions of quasi-irreducibility and quasi-symmetry for generalized numerical semigroups. While in the case of $d=1$ these notions coincide with irreducibility and symmetry, they are distinct in higher dimensions.

Largest hyperbolic actions and quasi-parabolic actions in groups

The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the “best” hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a largest hyperbolic action. While hyperbolic groups admit the largest hyperbolic actions, we give evidence in this paper that this phenomenon is rare for non-hyperbolic groups. In particular, we prove that many families of groups of geometric origin do not have the largest hyperbolic actions, including for instance many 3-manifold groups and most mapping class groups. Our proofs use the quasi-trees of metric spaces of Bestvina–Bromberg–Fujiwara, among other tools. In addition, we give a complete characterization of the poset of hyperbolic actions of Anosov mapping torus groups, and we show that mapping class groups of closed surfaces of genus at least two have hyperbolic actions which are comparable only to the trivial action.

Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors

There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states {mathbf {P}}—i.e., the set of one-dimensional projections on a complex Hilbert space H– and the orthomodular lattice {mathbf {L}} of closed subspaces of H. These six groups are isomorphic when the dimension of H is ge 3. The latter hypothesis is absolutely necessary in this identification. For example, the automorphisms group of all bijections preserving orthogonality and the order on {mathbf {L}} identifies with the bijections on {mathbf {P}} preserving transition probabilities only if dim(H)ge 3. Despite of the difficulties caused by M_2({mathbb {C}}), rank two algebras are used for quantum mechanics description of the spin state of spin-frac{1}{2} particles. However, there is a counterexample for Uhlhorn’s version of Wigner’s theorem for such state space. In this note we prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and orthogonality, but we also need to preserve the partial order set of all tripotents and orthogonality among them (a set which strictly enlarges the lattice of projections). Concretely, let M and N be two atomic JBW^*-triples not containing rank–one Cartan factors, and let {mathcal {U}} (M) and {mathcal {U}} (N) denote the set of all tripotents in M and N, respectively. We show that each bijection Phi : {mathcal {U}} (M)rightarrow {mathcal {U}} (N), preserving the partial ordering in both directions, orthogonality in one direction and satisfying some mild continuity hypothesis can be extended to a real linear triple automorphism. This, in particular, extends a result of Molnár to the wider setting of atomic JBW^*-triples not containing rank–one Cartan factors, and provides new models to present quantum behavior.

A partial order on transformation semigroups with restricted range that preserve double direction equivalence

Abstract Let T ( X ) T\left(X) be the full transformation semigroup on a set X X . For an equivalence E E on X X , let T E ∗ ( X ) = { α ∈ T ( X ) : ∀ x , y ∈ X , ( x , y ) ∈ E ⇔ ( x α , y α ) ∈ E } . {T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X):\forall x,y\in X,\left(x,y)\in E\iff \left(x\alpha ,y\alpha )\in E\right\}. For each nonempty subset Y Y of X X , we denote the restriction of E E to Y Y by E Y {E}_{Y} . Let T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) be the intersection of the semigroup T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) with the semigroup of all transformations with restricted range Y Y under the condition that ∣ X / E ∣ = ∣ Y / E Y ∣ | X\hspace{-0.1em}\text{/}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| . Equivalently, T E ∗ ( X , Y ) = { α ∈ T E ∗ ( X ) : X α ⊆ Y } {T}_{{E}^{\ast }}\left(X,Y)=\left\{\alpha \in {T}_{{E}^{\ast }}\left(X):X\alpha \subseteq Y\right\} , where ∣ X / E ∣ = ∣ Y / E Y ∣ | X\hspace{-0.1em}\text{/}\hspace{-0.1em}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| . Then T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) is a subsemigroup of T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) . In this paper, we characterize the natural partial order on T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) . Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.

Degenerate involutive set-theoretic solutions to the Yang-Baxter equation

Degenerate solutions to the Yang-Baxter equation are studied by means of associated semibraces and groups. Using a characterization in terms of cycle sets, a non-degenerate part is separated from a purely degenerate one. More precisely, any cycle set X has a distinguished non-degenerate sub-cycle set X×. If X× is trivial, the permutation group G(X) of X admits a natural partial order, with a strong semibrace S(X) as its negative cone, so that G(X) is a group of left fractions of S(X). The semibrace S(X) is constructed exactly like the self-similar closure of an L-algebra. It is proved that each non-trivial Garside group (e.g., Artin's braid group) gives rise to a degenerate cycle set. With a graded algebra related to the first Weyl algebra, a negative answer to a recent problem of Bonatto et al. (2021) [2] is obtained.

A combinatorial approach to the structure of locally inverse semigroups

This paper introduces a notion of presentation for locally inverse semigroups and develops a graph structure to describe the elements of locally inverse semigroups given by these presentations. These graphs will have a role similar to the role that Cayley graphs have for group presentations or that Sch\"utzenberger graphs have for inverse monoid presentations. However, our graphs have considerable differences with the latter two, even though locally inverse semigroups generalize both groups and inverse semigroups. For example, the graphs introduced here are not `inverse word graphs'. Instead, they are bipartite graphs with both oriented and non-oriented edges, and with labels on the oriented edges only. A byproduct of the theory developed here is the introduction of a graphical method for dealing with general locally inverse semigroups. These graphs are able to describe, for a locally inverse semigroup given by a presentation, many of the usual concepts used to study the structure of semigroups, such as the idempotents, the inverses of an element, the Green's relations, and the natural partial order. Finally, the paper ends characterizing the semigroups belonging to some usual subclasses of locally inverse semigroups in terms of properties on these graphs.

Filtering germs: Groupoids associated to inverse semigroups

We investigate various groupoids associated to an arbitrary inverse semigroup with zero. We show that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoid of germs arising from the standard action of the inverse semigroup on the space of idempotent filters. We also investigate the restriction of this isomorphism to the groupoid of tight filters and to the groupoid of ultrafilters.

Gradedness of the set of rook placements in A n−1

Abstract A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in An− 1 with respect to a slightly different order and prove that this poset is graded.

Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes

Let G be a reductive group over an algebraically closed field and let W be its Weyl group. In a series of papers, Lusztig introduced a map from the set [W] of conjugacy classes of W to the set [Gu] of unipotent classes of G. This map, when restricted to the set of elliptic conjugacy classes [We] of W, is injective. In this paper, we show that Lusztig's map [We]→[Gu] is order-reversing, with respect to the natural partial order on [We] arising from combinatorics and the natural partial order on [Gu] arising from geometry.

Natural partial order on semigroup partial linear transformation with restricted range

The set with assosiative binary operations is semigroup. Of all set partial linear transformations with a composition function is semigroup which is called a partial linear transformation semigroup that is denoted by (PT (X), o). If Y is a subspace of X, with all images of PT (X) a subset of Y, so the semigroup which is formed is a semigroup of partial linear transformation with restricted range that is denoted by (PT(X,Y),o). A natural partial order is a relation which defined by a ≤ b if only if a = xb = by, xa = a for some x,y ∈PT(X,Y) in a semigroup partial linear transformation with restricted range. In this article using the literature study method, it discusses the characterization of natural partial order on partial linear transformations semigroups with restricted range resulting neccesary and sufficient condition for a natural partial order in semigroup partial linear transformation with restricted range.

Function space of continuous maps from Peano continuum to tree I

For a finite tree T and its endpoint v, we can define a natural partial order ≤ such that v is the greatest element in T. For a compact metric space X and a continuous map f:X→T, let ↓f={(x,t):t≤f(x)} and ↓C(X,T)={↓f:fis continuous from XtoT}. We investigate the subspace ↓C(X,T) of the family of all non-empty closed sets in X×T with the Hausdorff distance. The following result is proved: For every non-degenerate Peano continuum X and a tree T with n segments {S1,S2,⋯,Sn}, ↓C(X,T)≈⨁i=1n↓CUB(Si), where CUB(Si)={f∈C(X,T):max⁡f(X)∈Si}. Moreover, every ↓CUB(Si) is contractible space. As a conclusion, ↓C(X,T) has n connected component.

Natural partial order induced by a commutative, associative and idempotent function

We study the natural partial order of commutative, associative, idempotent functions, covering as a special case t-norms, t-conorms, uninorms and nullnorms on bounded lattices. Unlike other approaches known in the literature, where the order induced by a given function is defined with respect to the original order on the given bounded lattice, we use original ideas of Hartwig, Nambooripad and Mitsch and define the natural partial order on an arbitrary set without connection to any order. We show that the natural partial order induced by any commutative, associative, idempotent function F corresponds to a meet semi-lattice and that F(x,y) can be expressed by the meet of x and y with respect to the natural partial order. This result can be used for an easy verification of the associativity of a commutative, idempotent function. Further, in some special cases we show the necessary and the sufficient conditions for such a function to be non-decreasing in each coordinate. We show that the natural meet semi-lattice connects the ordinal sum in the sense of Clifford and the ordinal sum in the sense of Birkhoff. An example of an ordinal sum of functions (which need not to be idempotent) on a horizontal sum lattice is also introduced.