In this paper, the widely used ordering and maximality principles of Haim Brézis and Felix Browde... more In this paper, the widely used ordering and maximality principles of Haim Brézis and Felix Browder, together with their proofs, are slightly generalized in content and greatly improved in format. For this, the concept of preorders is used instead of the concept of partial orders. Also, the more convenient notation S is used instead of "≤ ". Moreover, the importance of the supremal composition and a theorem on the existence of a particular increasing sequence is highlighted. Furthermore, instead of an ordering principle, a maximality principle is proved first and it is shown that the arguments applied by Brézis and Browder allow some more general results, even in much more attractive formulations.
The infimal convolution can be used to derive extension theorems from the sandwich ones
Acta Scientiarum Mathematicarum, Dec 1, 2010
ABSTRACT A nonvoid set X equipped with a binary operation + and a relation ≤ is called a gogroupo... more ABSTRACT A nonvoid set X equipped with a binary operation + and a relation ≤ is called a gogroupoid (generalized ordered groupoid). Let f,g:X→ℝ∪{-∞} be two functions having the domains D f , D g . The (general) infimal convolution of f and g is the function f*g:X→ℝ ¯ defined by (f*g)(x)=inf{f(u)+g(v):u∈D f ,v∈D g , x≤u+v}. In Section 1 (Theorem 1.12 and Corollaries 1.13 and 1.14) and Section 2 (Theorems 2.1–2.5), the author presents some properties of infimal convolutions, depending on the properties of the functions f and g, and also of supplementary properties imposed on the gogroupoid X. Using the infimal convolution defined above, the author obtains simple proofs and improvements of a sandwich theorem by B. Fuchssteiner and W. Lusky [Convex cones. Amsterdam etc.: North-Holland Publishing Company (1981; Zbl 0478.46002)] and of an extension theorem derived from it.
ABSTRACT The paper presents a Riemann-type integration for vector valued functions. If (Ω,S) is a... more ABSTRACT The paper presents a Riemann-type integration for vector valued functions. If (Ω,S) is a premeasurable space, then a defining net for integration means a family ((s α ,t α )) α∈G where G is a directed set and s α =(s αi ) i∈I α , t α =(t αi ) i∈I α are finite families in S and Ω, respectively. Let (X,Y,Z) be a multiplication system of Banach spaces, f a function from Ω into X and m from S into Y. The net integral of f with respect to m is defined as the limit ∫fdm=lim α ∑ i∈I α f(t αi )m(s αi ) whenever it exists (the integral is an element of Z). It is shown that the most important linearity and continuity properties of the usual integrals remain true provided the function spaces X Ω and Y S are equipped with so called conjugate seminorms.
Acta Mathematica Universitatis Comenianae. New Series, 2009
For a function f of one preordered set X to another Y , we shall establish several consequences o... more For a function f of one preordered set X to another Y , we shall establish several consequences of the following two definitions: (a) f is increasingly ϕ-regular, for some function ϕ of X to itself, if for any for some function g of Y to X , if for any x ∈ X and y ∈ Y we have f (x) ≤ y if and only if x ≤ g (y). These definitions have been mainly suggested to us by a recent theory of relators (families of relations) worked out by Á. Száz and G. Pataki and the extensive literature on Galois connections and residuated mappings.
Acta Universitatis Sapientiae: Mathematica, Jul 1, 2020
Improving and extending some ideas of Gottlob Frege from 1874 (on a generalization of the notion ... more Improving and extending some ideas of Gottlob Frege from 1874 (on a generalization of the notion of the composition iterates of a function), we consider the composition iterates ϕ n of a relation ϕ on X, defined by In particular, by using the relational inclusion ϕ n •ϕ m ⊆ ϕ n+m with n, m ∈ N 0 = {0} ∪ N ∪ {∞}, we show that the function α, defined by Moreover, the function f, defined by for n ∈ N 0 and A ⊆ X,
Birelator Spaces Are Natural Generalizations of Not Only Bitopological Spaces, But Also Ideal Topological Spaces
Springer optimization and its applications, 2019
In 1962, W. J. Pervin proved that every topology \(\mathscr {T}\) on a set X can be derived from ... more In 1962, W. J. Pervin proved that every topology \(\mathscr {T}\) on a set X can be derived from the quasi-uniformity \(\mathscr {U}\) on X generated by the preorder relations Open image in new window with \(A\in \mathscr {T}\).
A relation F on a groupoid X to a set Y is called a pointwise translation relation if F ( y )⊂ F ... more A relation F on a groupoid X to a set Y is called a pointwise translation relation if F ( y )⊂ F ( x+y ) for all x, y ∈ X. Moreover, a relation F on a groupoid X is called a global translation relation if x + F (y) ⊂ F ( x + y ) for all x, y ∈ X. After establishing some basic properties of the translation relations, we prove some simple theorems about the pointwise and global sums and negatives of translation relations. For instance, we show that if F is a normal and G is an arbitrary global translation relation on a group, then the global sum of F and G coincides with the composition of G and F . Ȧnd the global negative of F coincides with the inverse of F . Global translation functions and relations play important roles in the extensions and uniformizations of semigroups and groups, respectively. While the pointwise ones seem to have no such applications despite that they are also closely related to additive relations.
Semicontinuity and closedness properties of relations in relator spaces
ABSTRACT We extend some basic theorems on semicontinuity and closedness properties of set-valued ... more ABSTRACT We extend some basic theorems on semicontinuity and closedness properties of set-valued functions of topological spaces to relations on relator (generalized uniform) spaces. Moreover, in addition, we prove that a closed-valued and uniformly lower semicontinuous relation on a topologically semisymmetric relator space to a uniformly topologically transitive relator space is closed.
Having proved some basic characterizations of nonexpansive multipliers on partially ordered sets,... more Having proved some basic characterizations of nonexpansive multipliers on partially ordered sets, we establish some intimate connections between nonexpansive multipliers and interior (quasi-interior) operators. The results obtained naturally extend and supplement some of the former statements of G. Szasz, J. Szendrei, M. Kolibiar, W. H. Cornish and the second author on some particular multipliers on semilattices and partially ordered sets.
In this paper, in a purely algebraic way, Schwartz distributions in several variables are general... more In this paper, in a purely algebraic way, Schwartz distributions in several variables are generalized in accordance with their homomorphism interpretation proposed by R. A. Struble. 0. Introduction. R. A. Struble in [10] has shown that Schwartz distributions can be characterized simply as mappings, from the space 3) of test functions into the space % of smooth functions, which commute with ordinary convolution. This new view of distributions has turned out to be very useful [11,12] and motivated us to give a simple generalization for distributions which is closely related to Mikusiήski operators and convolution quotients of other types [11, . The method employed here is an appropriate modification of a general algebraic method . Mappings which commute with convolution are called convolution multipliers here. (Distributions can be characterized as convolution multipliers, Mikusiήski operators themselves are convolution multipliers.) In §1, convolution multipliers from various subsets of 3) into % are discussed. We are primarily concerned with their maximal extensions. In §2, a module Wl of certain maximal convolution multipliers is constructed and investigated from an algebraic point of view. In §3, Schwartz distributions are embedded and characterized in W. For example, we prove that distributions are the only continuous elements of 2)ϊ. Finally, we show that there are elements in W which are not distributions. To illustrate the appropriateness of our generalizations, we refer to the following facts: One of the difficulties in working with Schwartz distributions is that only distributions Λ satisfying Λ * 3) = 3 are invertible in 3)'. Whereas, distibutions Λ satisfying A*3) C3) such that Λ * 3) has no proper annihilators in % are invertible in 3ft. (The heat operator in two dimensions [1] seems to be a distribution which is not invertible in 3)\ but is invertible in 9K.) There are regular Mikusiήski operators [1] which are not distributions. Whereas, normal Mikusiήski operators [11] can be embedded in Wl. their maximal extensions. Let k be a fixed positive integer, R* be the kdimensional Euclidean space and C be the field of complex numbers.
First, we establish a useful characterization of effective sets in conditionally complete partial... more First, we establish a useful characterization of effective sets in conditionally complete partially ordered sets. Then, we prove that each maximal nonexpansive partial multiplier on a conditionally complete and infinitely distributive partially ordered set with upper bounded centre is inner. Finally, we show that some analogous results hold for T 1 -families of sets partially ordered by inclusion.
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Papers by Árpád Száz