Free Vector Lattices

1968 ◽  
Vol 20 ◽  
pp. 58-66 ◽  
Author(s):  
Kirby A. Baker
Keyword(s):  
Universal Algebra ◽  
Vector Lattice ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
Real Numbers ◽  
Vector Lattices ◽  
Explicit Characterization

This note presents a useful explicit characterization of the free vector lattice FVL(ℵ) on ℵ generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where ℵ is any cardinal and R is the set of real numbers. A transfinite construction of FVL(ℵ) has been given by Weinberg (14) and simplified by Holland (13, § 5). Weinberg's construction yields the fact that FVL(ℵ) is semi-simple; the present characterization is obtained by combining this fact with a theorem from universal algebra due to Garrett Birkhoff.

scholarly journals Countable vector lattices

1974 ◽  
Vol 10 (3) ◽  
pp. 371-376 ◽  
Author(s):  
Paul F. Conrad
Keyword(s):  
Vector Space ◽  
Vector Lattice ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Real Numbers ◽  
Vector Lattices ◽  
Direct Sum ◽  
Ordered Vector Spaces ◽  
Countable Dimension

In his paper “On the structure of ordered real vector spaces” (Publ. Math. Debrecen 4 (1955–56), 334–343), Erdös shows that a totally ordered real vector space of countable dimension is order isomorphic to a lexicographic direct sum of copies of the group of real numbers. Brown, in “Valued vector spaces of countable dimension” (Publ. Math. Debrecen 18 (1971), 149–151), extends the result to a valued vector space of countable dimension and greatly simplifies the proof. In this note it is shown that a finite valued vector lattice of countable dimension is order isomorphic to a direct sum of o–simple totally ordered vector spaces. One obtains as corollaries the result of Erdös and the applications that Brown makes to totally ordered spaces.

scholarly journals Regular variation for measures on metric spaces

2006 ◽  
Vol 80 (94) ◽  
pp. 121-140 ◽  
Author(s):  
Henrik Hult ◽  
Filip Lindskog
Keyword(s):  
Regular Variation ◽  
Metric Spaces ◽  
General Setting ◽  
Continuous Functions ◽  
Separable Space ◽  
Real Numbers ◽  
Space Of Continuous Functions ◽  
Relative Compactness ◽  
Borel Measures

The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regular variation is defined in this general setting and several statements that are equivalent to this definition are presented. This extends the notion of regular variation for Borel measures on the Euclidean space Rd to more general metric spaces. Some examples, including regular variation for Borel measures on Rd, the space of continuous functions C and the Skorohod space D, are provided.

scholarly journals Неограниченная порядковая сходимость и теорема Гордона

2019 ◽  
Author(s):  
E.Y. Emelyanov ◽  
S.G. Gorokhova ◽  
S.S. Kutateladze
Keyword(s):  
Elementary Proof ◽  
Vector Lattice ◽  
Order Convergence ◽  
Vector Lattices ◽  
Convergence Of Sequences ◽  
Universal Completion

The celebrated Gordons theorem is a natural tool for dealing with universal completions of Archimedean vector lattices. Gordons theorem allows us to clarify some recent results on unbounded order convergence. Applying the Gordon theorem, we demonstrate several facts on order convergence of sequences in Archimedean vector lattices. We present an elementary Boolean-Valued proof of the Gao--Grobler--Troitsky--Xanthos theorem saying that a sequence x_n in an Archimedean vector lattice X is uo-null (uo-Cauchy) in X if and only if x_n is o-null (o-convergent) in Xu. We also give elementary proof of the theorem, which is a result of contributions of several authors, saying that an Archimedean vector lattice is sequentially uo-complete if and only if it is sigma-universally complete. Furthermore, we provide a comprehensive solution to Azouzis problem on characterization of an Archimedean vector lattice in which every uo-Cauchy net is o-convergent in its universal completion.

scholarly journals Vector lattices over subfields of the reals

1990 ◽  
Vol 48 (3) ◽  
pp. 359-375 ◽  
Author(s):  
P. Bixler ◽  
P. Conrad ◽  
W. B. Powell ◽  
C. Tsinakis
Keyword(s):  
Vector Lattice ◽  
Scalar Multiplication ◽  
Real Numbers ◽  
The Real ◽  
Vector Lattices ◽  
Non Linear ◽  
Minimal Prime

AbstractIn this paper we consider classes of vector lattices over subfields of the real numbers. Among other properties we relate the archimedean condition of such a vector lattice to the uniqueness of scalar multiplication and the linearity of l-automorphisms. If a vector lattice in the classes considered admits an essential subgroup that is not a minimal prime, then it also admits a non-linear l-automorphism and more than one scalar multiplication. It is also shown that each l-group contains a largest archimedean convex l-subgroup which admits a unique scalar multiplication.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

Fundamental relations and identities of fuzzy hyperalgebras

2021 ◽  
pp. 1-10
Author(s):  
Narjes Firouzkouhi ◽  
Abbas Amini ◽  
Chun Cheng ◽  
Mehdi Soleymani ◽  
Bijan Davvaz
Keyword(s):  
Universal Algebra ◽  
Equivalence Relation ◽  
Polynomial Function ◽  
Equivalence Relations ◽  
Fundamental Relation ◽  
Term Function ◽  
Biological Phenomena ◽  
Definition Of

Inspired by fuzzy hyperalgebras and fuzzy polynomial function (term function), some homomorphism properties of fundamental relation on fuzzy hyperalgebras are conveyed. The obtained relations of fuzzy hyperalgebra are utilized for certain applications, i.e., biological phenomena and genetics along with some elucidatory examples presenting various aspects of fuzzy hyperalgebras. Then, by considering the definition of identities (weak and strong) as a class of fuzzy polynomial function, the smallest equivalence relation (fundamental relation) is obtained which is an important tool for fuzzy hyperalgebraic systems. Through the characterization of these equivalence relations of a fuzzy hyperalgebra, we assign the smallest equivalence relation α i 1 i 2 ∗ on a fuzzy hyperalgebra via identities where the factor hyperalgebra is a universal algebra. We extend and improve the identities on fuzzy hyperalgebras and characterize the smallest equivalence relation α J ∗ on the set of strong identities.

scholarly journals A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽  
2021 ◽  
Author(s):  
Péter Vrana
Keyword(s):  
Compact Hausdorff Space ◽  
Polynomial Growth ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Equivalent Condition ◽  
Real Numbers ◽  
Asymptotic Spectrum ◽  
Ring Of Continuous Functions ◽  
Archimedean Property ◽  
Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

Microbrook, Mesobrook, Macrobrook

2019 ◽  
Vol 2 (4) ◽  
pp. 245-253 ◽  
Author(s):  
Sebastian Jilke ◽  
Asmus Leth Olsen ◽  
William Resh ◽  
Saba Siddiki
Keyword(s):  
Problem Solving ◽  
Public Administration ◽  
Theory And Practice ◽  
Levels Of Analysis ◽  
Explicit Characterization ◽  
Methodological Perspective ◽  
Level Of Analysis ◽  
Effective Problem

Abstract This article assesses the field of public administration from a conceptual and methodological perspective. We urge public administration scholars to resolve the ambiguities that mire our scholarship due to the inadequate treatment of levels of analysis in our research. Overall, we encourage methodological accountability through a more explicit characterization of one’s research by the level of analysis to which it relates. We argue that this particular form of accountability is critical for effective problem solving for advancing theory and practice.

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