Order-preserving Maps Research Articles

Three-way concept lattice from adjunctive positive and negative concepts

Three-way concept lattices (TCLs) have been widely explored due to their clear hierarchical structures, concise visual description and good interpretability. In contrast to classic formal contexts, lattice-valued fuzzy contexts exhibit great capability in describing and representing concepts with uncertainty. Different from conventional approaches to research of TCLs, this paper focuses on investigating the algebraic structure and properties of three-way concept lattice (TCL) stemmed from the positive concept lattice and negative concept lattice in a lattice-valued formal context. Several associated concept lattices such as the Cartesian product of positive concept lattice and negative lattice (i.e., pos-neg lattice), lattices induced from the partition of the pos-neg lattice, and their relationship are explored. Specifically, the isomorphism, embedding and order-preserving mappings between them are built. The quotient set of pos-neg lattice when being defined a specific equivalence relation on it is a complete lattice and each equivalence class is a lower semi-lattice. It is further declared that the structure of TCL is intrinsically and determined wholly by the pos-neg lattice. A practical application of the built theory of TCL is provided to sort alternatives in multi-criteria decision making.

Connecting ideals in evolution algebras with hereditary subsets of its associated graph

AbstractIn this article, we introduce a relation including ideals of an evolution algebra A and hereditary subsets of vertices of its associated graph and establish some properties among them. This relation allows us to determine maximal ideals and ideals having the absorption property of an evolution algebra in terms of its associated graph. In particular, the maximal ideals can be determined through maximal hereditary subsets of vertices except for those containing $$A^2$$ A 2 . We also define a couple of order-preserving maps, one from the sets of ideals of an evolution algebra to that of hereditary subsets of the corresponding graph, and the other in the reverse direction. Conveniently restricted to the set of absorption ideals and to the set of hereditary saturated subsets, this is a monotone Galois connection. According to the graph, we characterize arbitrary dimensional finitely-generated (as algebras) evolution algebras under certain restrictions of its graph. Furthermore, the simplicity of finitely-generated perfect evolution algebras is described on the basis of the simplicity of the graph.

The quantale of order-preserving maps of a completely distributive lattice

The quantale of order-preserving maps of a completely distributive lattice

A note on the representation theorem for nullnorms on bounded lattices

Recently, we presented a complete representation for nullnorms on bounded lattices, a class of ordered semigroups, in terms of two order-preserving maps, a triangular conorm, a triangular norm and a conditionally associative function, which together satisfy seven conditions. In this paper, we illustrate that each of these seven conditions is independent of the other ones by rewriting the seventh condition. We also discuss in which cases the sixth or the seventh condition becomes dependent.

Monotones in General Resource Theories

A central problem in the study of resource theories is to find functions that are nonincreasing under resource conversions — termed monotones — in order to quantify resourcefulness. Various constructions of monotones appear in many different concrete resource theories. How general are these constructions? What are the necessary conditions on a resource theory for a given construction to be applicable? To answer these questions, we introduce a broad scheme for constructing monotones. It involves finding an order-preserving map from the preorder of resources of interest to a distinct preorder for which nontrivial monotones are previously known or can be more easily constructed; these monotones are then pulled back through the map. In one of the two main classes we study, the preorder of resources is mapped to a preorder of sets of resources, where the order relation is set inclusion, such that monotones can be defined via maximizing or minimizing the value of a function within these sets. In the other class, the preorder of resources is mapped to a preorder of tuples of resources, and one pulls back monotones that measure the amount of distinguishability of the different elements of the tuple (hence its information content). Monotones based on contractions arise naturally in the latter class, and, more surprisingly, so do weight and robustness measures. In addition to capturing many standard monotone constructions, our scheme also suggests significant generalizations of these. In order to properly capture the breadth of applicability of our results, we present them within a novel abstract framework for resource theories in which the notion of composition is independent of the types of the resources involved (i.e., whether they are states, channels, combs, etc.).

The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders

We prove the conjecture that the higher Tamari orders of Dimakis and Müller-Hoissen coincide with the first higher Stasheff-Tamari orders. To this end, we show that the higher Tamari orders may be conceived as the image of an order-preserving map from the higher Bruhat orders to the first higher Stasheff-Tamari orders. This map is defined by taking the first cross-section of a cubillage of a cyclic zonotope. We provide a new proof that this map is surjective and show further that the map is full, which entails the aforementioned conjecture. We explain how order-preserving maps which are surjective and full correspond to quotients of posets. Our results connect the first higher Stasheff-Tamari orders with the literature on the role of the higher Tamari orders in integrable systems.

Construction of semi-t-operators on bounded lattices by order-preserving mappings

This article investigates the new construction of semi-t-operators on bounded lattices. It first gives five methods to construct semi-t-operators on bounded lattices by semi-t-norms, semi-t-conorms, and order-preserving mappings. It then discusses the relationships between these methods and well-known ones, which show that the known methods in literature are special cases of these ones.

Order-Preserving and Order-Reversing Mappings between Consumption Bundles and Utilities

Economics and management believe that income and price play decisive roles in consumer demand for commodities. However, there is no analysis of the order-preserving or order-reversing mapping between consumer demand and utility when income and price are not constrained. Customer utility is always determined by the amount of commodities in real life. This research analyzes the mapping relationship between the amount of commodities and utility. The set which consists of consumption bundles with a binary relation is a poset. The set of utility values with the binary relation is a chain. Furthermore, the relationship between consumption bundles and utilities is illustrated by figures. If the increasing directions of utility and the amount of commodities are consistent, the utility function is order-preserving. If they are opposite, the utility function is order-reversing. Finally, examples are included to illustrate the mapping relationship between consumption bundles and utilities.

Real subset sums and posets with an involution

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

A complete representation theorem for nullnorms on bounded lattices with ample illustrations

We present a complete representation theorem for nullnorms on bounded lattices. Explicitly, any nullnorm on a bounded lattice can be represented in terms of two order-preserving maps, a triangular conorm, a triangular norm and a conditionally associative function. As a particular case, we retrieve the known representation of nullnorms only taking values that are comparable with the annihilator (in terms of two order-preserving maps, a triangular conorm and a triangular norm). As an illustration of the representation theorem, we generate construction methods for nullnorms, especially those taking at least one value that is incomparable with the annihilator.

On order-preserving and order-reversing mappings defined on cones of convex functions

In this paper, we first show that for a Banach space X there is a fully order-reversing mapping T from conv(X) (the cone of all extended real-valued lower semicontinuous proper convex functions defined on X) onto itself if and only if X is reflexive and linearly isomorphic to its dual X*. Then we further prove the following generalized Artstein-Avidan-Milman representation theorem: For every fully order-reversing mapping T: conv(X) → conv(X) there exist a linear isomorphism U: X → X*, x *0 , φ0 ∈ X*, α > 0 and r0 ∈ ℝ so that $$(Tf)(x) = \alpha ({\cal F}f)(Ux + x_0^ *) + \left\langle {{\varphi _0},x} \right\rangle + {r_0},\;\;\;\;\;\forall x \in X,$$ where \({\cal F}\): conv(X) → conv(X*) is the Fenchel transform. Hence, these resolve two open questions. We also show several representation theorems of fully order-preserving mappings defined on certain cones of convex functions. For example, for every fully order-preserving mapping S: semn(X) → semn(X) there is a linear isomorphism U: X → X so that $$(Sf)(x) = f(Ux),\;\;\;\;\;\forall f \in {\rm{semn}}(X),\;\;\;\;\;x \in X,$$ where semn(X) is the cone of all lower semicontinuous seminorms on X.

Some Common Fixed Point Theorems in Partially Ordered Sets

The purpose of this paper is to prove some new fixed point theorem and common fixed point theorems of a commuting family of order-preserving mappings defined on an ordered set, which unify and generalize some relevant fixed point theorems.

P-Partitions and Quasisymmetric Power Sums

Abstract The $(P, \omega )$-partition generating function of a labeled poset $(P, \omega )$ is a quasisymmetric function enumerating certain order-preserving maps from $P$ to ${\mathbb{Z}}^+$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $\{\psi _{\alpha }\}$. Using this expansion, we show that connected, naturally labeled posets have irreducible $P$-partition generating functions. We also show that series-parallel posets are uniquely determined by their partition generating functions. We conclude by giving a combinatorial interpretation for the coefficients of the $\psi _{\alpha }$-expansion of the $(P, \omega )$-partition generating function akin to the Murnaghan–Nakayama rule.

P-partition generating function equivalence of naturally labeled posets

The P-partition generating function of a (naturally labeled) poset P is a quasisymmetric function enumerating order-preserving maps from P to Z+. Using the Hopf algebra of posets, we give necessary conditions for two posets to have the same generating function. In particular, we show that they must have the same number of antichains of each size, as well as the same shape (as defined by Greene). We also discuss which shapes guarantee uniqueness of the P-partition generating function and give a method of constructing pairs of non-isomorphic posets with the same generating function.

A Unifying Perron--Frobenius Theorem for Nonnegative Tensors via Multihomogeneous Maps

We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a nonnegative tensor $T$ in terms of the associated shape partition. We recast the eigenvalue problem for $T$ as a fixed point problem on a suitable product of projective spaces. This allows us to use the theory of multi-homogeneous order-preserving maps to derive a new and unifying Perron-Frobenius theorem for nonnegative tensors which either implies earlier results of this kind or improves them, as weaker assumptions are required. We introduce a general power method for the computation of the dominant tensor eigenpair, and provide a detailed convergence analysis.

On categorical aspects of S -quantales

AbstractS-quantales are characterized as injective objects in the category of S-posets with respect to certain class of homomorphisms that are order-preserving mappings. This paper is devoted to exhibitions of categorical structures on S-quantales.

Cometic functors and representing order-preserving maps by principal lattice congruences

Let $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ and $$\mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ denote the category of selfdual bounded lattices of length 5 with $$\{0,1\}$$ -preserving lattice homomorphisms and that of bounded ordered sets with $$\{0,1\}$$ -preserving isotone maps, respectively. For an object L in $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ , the ordered set of principal congruences of the lattice L is denoted by $$\mathrm{Princ}(L)$$ . By means of congruence generation, $$\mathrm{Princ}:\mathbf {Lat}^{\mathrm{sd}}_{5}\rightarrow \mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ is a functor. We prove that if $$\mathbf {A}$$ is a small subcategory of $$\mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ such that every morphism of $$\mathbf {A}$$ is a monomorphism, understood in $$\mathbf {A}$$ , then $$\mathbf {A}$$ is the $$\mathrm{Princ}$$ -image of an appropriate subcategory of $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ . This result extends G. Gratzer’s earlier theorems where $$\mathbf {A}$$ consisted of one or two objects and at most one non-identity morphism, and the author’s earlier result where all morphisms of $$\mathbf {A}$$ were 0-separating and no hom-set had more the two morphisms. Furthermore, as an auxiliary tool, we derive some families of maps, also known as functions, from injective maps and surjective maps; this can be useful in various fields of mathematics, not only in lattice theory. Namely, for every small concrete category $$\mathbf {A}$$ , we define a functor $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ , called cometic functor, from $$\mathbf {A}$$ to the category $$\mathbf Set $$ of sets and a natural transformation $${{\varvec{\pi }}^{{\scriptscriptstyle \mathrm{{com}}}}}$$ , called cometic projection, from $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ to the forgetful functor of $$\mathbf {A}$$ into $$\mathbf Set $$ such that the $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ -image of every monomorphism of $$\mathbf {A}$$ is an injective map and the components of $${{\varvec{\pi }}^{{\scriptscriptstyle \mathrm{{com}}}}}$$ are surjective maps.

Discreteness of silting objects and t-structures in triangulated categories

We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite-dimensional algebra. For an ST-pair $(\C,\D)$, we construct an order-preserving map from silting objects in $\C$ to bounded $t$-structures on $\D$ and show that the map is bijective if and only if $\C$ is silting-discrete if and only if $\D$ is $t$-discrete. Based on a work of Qiu and Woolf, the above result is applied to show that if $\C$ is silting-discrete then the stability space of $\D$ is contractible. This is used to obtain the contractibility of the stability spaces of some Calabi--Yau triangulated categories associated to Dynkin quivers.

Ehresmann monoids: Adequacy and expansions

It is known that an Ehresmann monoid P(T,Y) may be constructed from a monoid T acting via order-preserving maps on both sides of a semilattice Y with identity, such that the actions satisfy an appropriate compatibility criterion. Our main result shows that if T is cancellative and equidivisible (as is the case for the free monoid X⁎), the monoid P(T,Y) not only is Ehresmann but also satisfies the stronger property of being adequate.Fixing T, Y and the actions, we characterise P(T,Y) as being unique in the sense that it is the initial object in a suitable category of Ehresmann monoids. We also prove that the operator P defines an expansion of Ehresmann monoids.

Order and Spectrum Preserving Maps on Positive Operators

AbstractWe describe the general form of surjective maps on the cone of all positive operators that preserve order and spectrum. The result is optimal as shown by counterexamples. As an easy consequence, we characterize surjective order and spectrum preserving maps on the set of all self-adjoint operators.