Surjective Map Research Articles

Some algebraic and analytical properties of a class of two-place functions

This article deals with the formula f(−1)(F(f(x),f(y))) generated by a one-place function f:[0,1]→[0,1] and a binary function F:[0,1]2→[0,1]. When the f is a strictly increasing function and F is a continuous, non-decreasing and associative function with neutral element in [0,1], the following algebraic and analytical properties of the formula are studied: idempotent elements, the continuity (resp. left-continuity/right-continuity), the associativity and the limit property. Relationship among these properties is investigated. Some necessary conditions and some sufficient conditions are given for the formula being a triangular norm (resp. triangular conorm). In particular, a necessary and sufficient condition are expressed for obtaining a continuous Archimedean triangular norm (resp. triangular conorm). When the f is a non-decreasing surjective function and F is a non-decreasing associative function with neutral element in [0,1], we investigate the associativity of the formula.

Bi-monotone maps on the set of all variance-covariance matrices with respect to minus partial order

Let Hn+(R) be the cone of all positive semidefinite n×n real matrices. We describe the form of all surjective maps on Hn+(R), n≥3, that preserve the minus partial order in both directions.

Images of dominant endomorphisms of affine space

A basic problem in algebraic morphisms is which sets can be the image of an endomorphism of affine space. This paper extends the results previously obtained by the first author on the question of existence of surjective maps F : A n → A n ∖ Z , where Z is an algebraic subvariety of A n of codimension at least 2. In particular, we show that for any (affine) algebraic variety Z of dimension at most n − 2 , there is an algebraic variety W ⊂ A n birational to Z and a surjective algebraic morphism A n → A n ∖ W . We also propose a conjectural approach toward resolving unknown cases.

Nodecness of Soft Generalized Topological Spaces

In this work, we define a new class of soft generalized topological spaces, namely strongly soft nodec, with the use of strongly soft nowhere dense sets. Then, we study the basic properties of these spaces and show that if the product of two soft generalized topological spaces is a strongly soft nodec space, then each one is a strongly soft nodec space. Then, we extend these notions to T0-strongly soft nodec generalized topological spaces by using the soft quotient functions and discussing their main properties. We also show the inverse of a surjective soft quotient function preserves the soft closure and soft interior of a soft subset of a codomain soft set in soft generalized topological space. Further, we use soft quasi-homeomorphism and soft quotient functions to make comparisons and connections between these spaces with the support of appropriate counterexamples. Then, we successfully determine a condition under which the soft generalized topological space is a soft weak Baire space and hence a strongly soft second category.

Maps preserving ascent or descent of triple Jordan product

Let X be a real or complex Banach space of dimension at least 3. We give a complete description of surjective mappings on B(X) that preserve the ascent of Jordan triple product of operators or, preserve the descent of Jordan triple product of operators.

On the diameter of semigroups of transformations and partitions

AbstractFor a semigroup whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right‐), the right diameter of is a parameter that expresses how ‘far apart’ elements of can be from each other, in a certain sense. To be more precise, for each finite generating set for the universal right congruence on , we have a metric space where is the minimum length of derivations for as a consequence of pairs in ; the right diameter of with respect to is the diameter of this metric space. The right diameter of is then the minimum of the set of all right diameters with respect to finite generating sets. We develop a theoretical framework for establishing whether a semigroup of transformations or partitions on an arbitrary infinite set has a finitely generated universal right/left congruence, and, if it does, determining its right/left diameter. We apply this to prove results such as the following. Each of the monoids of all binary relations on , of all partial transformations on , and of all full transformations on , as well as the partition and partial Brauer monoids on , have right diameter 1 and left diameter 1. The symmetric inverse monoid on has right diameter 2 and left diameter 2. The monoid of all injective mappings on has right diameter 4, and its minimal ideal (called the Baer–Levi semigroup on ) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence.

Limits and Difficulties in the Design of Under-Approximation Abstract Domains

The main goal of most static analyses is to prove the absence of bugs : if the analysis reports no alarms then the program will not exhibit any unwanted behaviours. For this reason, they are designed to over-approximate program behaviours and, consequently, they can report some false alarms. O’Hearn’s recent work on incorrectness has renewed the interest in the use of under-approximations for bug finding , because they only report true alarms. In principle, Abstract Interpretation techniques can handle under-approximations as well as over-approximations, but, in practice, few attempts were developed for the former, notwithstanding the much wider literature on the latter. In this paper we investigate the possibility of exploiting under-approximation abstract domains for bug-finding analyses. First we restrict to consider concrete powerset domains and highlight some intuitive asymmetries between over- and under-approximations. Then, we prove that the effectiveness of abstract domains defined by Under-approximation Galois connection is limited, because the analysis is likely to return trivial results whenever common transfer functions are encoded in the program. To this aim, we introduce the original concepts of non-emptying functions and highly surjective function family and we prove the nonexistence of abstract domains able to under-approximate such functions in a non-trivial way. We show many examples of finite and infinite numerical domains, as well as other generic domains. In all such cases, we prove the impossibility of performing nontrivial analyses via under-approximating Galois connections.

Онтологизация познания: уровни, направления и средства реализации

The article investigates the problem of ontologization of cognition, i.e. increasing the level of conformity between the knowledge system and being. Various levels of ontologization of knowledge are examined depending on the nature of the reflection of reality in the system of knowledge. Four levels of ontology are distinguished: the first level corresponds to the injective and bijective mappings of reality onto the knowledge system; the second level involves surjective mapping; the third level encompasses knowledge formed through isomorphic mapping of relationships between elements from reality onto the knowledge system; the fourth level involves knowledge formed through isomorphic mapping of relationships between elements within the knowledge system. Prospects for the ontologization of cognition are related to the expansion of metaphysical and consequent knowledge, the elevation to the fourth level of knowledge domains that lack ontologicality, and the expansion of knowledge linking secondary laws and properties of objects to patterns. The main tool for ontologizing cognition should be a general theory of systems, extended with a metaphysical component and supplemented with extensive collections of patterns, primitives, secondary properties and laws. The author is working on such a general systems theory, referred to as the “empirico-metaphysical general theory of systems”. In conclusion, development of an empirico-metaphysical theory of systems promises to provide a robust framework for addressing the challenges of cognition representation, synthesis, and application across diverse domains. As researchers continue to delve into the nuances of ontological alignment, the proposed framework offers a promising avenue for fostering interdisciplinary collaboration and advancing our understanding of the fundamental nature of cognition.

On Wigner's theorem in complex smooth normed spaces

In this note, we present a generalization of Wigner's theorem. Let X and Y be complex normed spaces with Y being smooth. We show that a surjective mapping f:X→Y satisfies{‖f(x)+βf(y)‖:β∈Tn}={‖x+βy‖:β∈Tn},x,y∈X, where n≥3 is a positive integer and Tn is the set of the nth roots of unity, if and only if there exists a phase function σ:X→Tn such that σ⋅f is a linear or an anti-linear isometry.

Banach algebra mappings preserving the invertibility of linear pencils

Let A and B be complex unital Banach algebras, and let φ,ψ:A→B be surjective mappings. If A is semisimple with an essential socle and φ and ψ together preserve the invertibility of linear pencils in both directions, that is, for any x,y∈A and λ∈C, λx+y is invertible in A if and only if λφ(x)+ψ(y) is invertible in B, then we show that there exists an invertible element u in B and a Jordan isomorphism J:A→B such that φ(x)=ψ(x)=uJ(x) for all x∈A.

Adjacency preserving maps on symmetric tensors

Let r and s be positive integers such that r⩾2. Let U and V be vector spaces over fields F and K, respectively, such that dim⁡U⩾3 and F has at least r+1 elements. In this paper, we characterize surjective maps ψ:SrU→SsV preserving adjacency in both directions on symmetric tensors of finite order, which generalizes Hua's fundamental theorem of geometry of symmetric matrices. We give examples showing the indispensability of the assumptions dim⁡U⩾3 and the cardinality |F|⩾r+1 in our result.

Rank one quaternionic operators and additive preservers

In this paper, we completely describe all additive surjective maps, on the set of all bounded finite rank right linear operators acting on a right quaternionic Banach space, that preserve the set of all bounded rank one (resp. rank one idempotent, rank one nilpotent) right linear operators, in both directions.

The virtual spectrum of linkoids and open curves in 3-space

The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of collections of open curves in 3-space using the theory of knotoids and linkoids, which are equivalence classes of diagrams with open arcs. In this paper, new invariants of linkoids are introduced via a surjective map between linkoids and virtual knots. This leads to a new collection of strong invariants of linkoids that are independent of any given virtual closure. This gives rise to a collection of novel measures of entanglement of open curves in 3-space, which are continuous functions of the curve coordinates and tend to their corresponding classical invariants when the endpoints of the curves tend to coincide.

Adjacency preserving maps between exterior powers of vector spaces

In this paper, a structural characterization of adjacency preserving surjective maps between exterior powers of arbitrary vector spaces is established, which generalizes the fundamental theorem of the geometry of alternate matrices.

Classification aggregation without unanimity

A classification is a surjective mapping from a set of objects to a set of categories. A classification aggregation function aggregates every vector of classifications into a single one. We show that every citizen sovereign and independent classification aggregation function is essentially a dictatorship. This impossibility implies an earlier result of Maniquet and Mongin (2016), who show that every unanimous and independent classification aggregation function is a dictatorship. The relationship between the two impossibilities is reminiscent to the relationship between Wilson’s (1972) and Arrow’s (1951) impossibilities in preference aggregation. Moreover, while the Maniquet-Mongin impossibility rests on the existence of at least three categories, we propose an alternative proof technique that covers the case of two categories, except when the number of objects is also two. We also identify all independent and unanimous classification aggregation functions for the case of two categories and two objects.

A Note on Stronger Forms of Sensitivity for Non-Autonomous Dynamical Systems on Uniform Spaces.

This paper introduces the notion of multi-sensitivity with respect to a vector within the context of non-autonomous dynamical systems on uniform spaces and provides insightful results regarding N-sensitivity and strongly multi-sensitivity, along with their behaviors under various conditions. The main results established are as follows: (1) For a k-periodic nonautonomous dynamical system on a Hausdorff uniform space (S,U), the system (S,fk∘⋯∘f1) exhibits N-sensitivity (or strongly multi-sensitivity) if and only if the system (S,f1,∞) displays N-sensitivity (or strongly multi-sensitivity). (2) Consider a finitely generated family of surjective maps on uniform space (S,U). If the system (S,f1,∞) is N-sensitive, then the system (S,fk,∞) is also N-sensitive. When the family f1,∞ is feebly open, the converse statement holds true as well. (3) Within a finitely generated family on uniform space (S,U), N-sensitivity (and strongly multi-sensitivity) persists under iteration. (4) We present a sufficient condition under which an nonautonomous dynamical system on infinite Hausdorff uniform space demonstrates N-sensitivity.

Buildings, valuated matroids, and tropical linear spaces

AbstractAffine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of parameterizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite‐dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise‐linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.

Maps preserving maximal numerical range of operator products

Let be a complex Hilbert space and denote by the algebra of all linear bounded operators on . For any , denote by V 0(T) its maximal numerical range. We prove that if is a surjective map such that for any then there exists a unitary operator such that ϕ(T ) = ±UTU* (resp. ϕ(T) = λUTU* with |λ| = 1) for any .

Unipotent character sheaves and strata of a reductive group

Let H H be a connected reductive group over an algebraically closed field. We define a surjective map from the set C S ( H ) CS(H) of unipotent character sheaves on H H (up to isomorphism) to the set of strata of H H . To do this we use the generalized Springer correspondence. We also give a new parametrization of C S ( H ) CS(H) in terms of data coming from bad characteristic.

Bi-asymptotic c-expansivity

In this paper, we define bi-asymptotically c-expansive maps on metric spaces and study its relationship with other variants of expansivity such as bi-asymptotically expansive maps and N-expansive maps. We also provide an example to establish that expansive homeomorphisms need not be bi-asymptotically expansive. Finally, we prove a spectral decomposition theorem for bi-asymptotically c-expansive continuous surjective maps with the shadowing property on compact metric spaces.