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Some kinds of statistical unbounded convergence were studied and investi-gated with respect to the solid topology and order convergence. In this paper, we study the concept of statistical unbounded order convergence in Riesz spaces by a topology -free technique with the order convergence on Riesz spaces. Moreover, we give some relations with other kinds of statistical convergences.

In this paper we prove an Ozguç, Yurdakadim and Taş version of the Korovkin-type approximation by operators in the sense of the power series method. That is, we try to extend the Korovkin approximation theorems, obtained by Ozguç and Taş in 2016, and Taş and Yurdakadim in 2017, for concrete classes of Banach spaces to the class of Riesz spaces. Some applications are presented.

We discuss and compare a number of notions of modulars appeared in literature, among which there is a selection of the well known ones. We highlight the connections between the various definitions and provide several examples, taken from existing literature, recalling known results and completing the picture with some original considerations

The statistical convergence is handled for sequences with the natural density, in general. In a recent paper, the statistical convergence for nets in Riesz spaces has been studied and investigated by developing topology-free techniques in Riesz spaces. In this paper, we introduce the statistically topological convergence for nets on locally solid Riesz spaces with solid topologies. Moreover, we introduce the statistical continuity on locally solid Riesz spaces., Comment: 15

The statistically multiplicative convergence in Riesz algebras was studied and investigated with respect to the solid topology. In the present paper, the statistical convergence with the multiplication in Riesz algebras is introduced by developing topology-free techniques using the order convergence in vector lattices. Moreover, we give some relations with the other kinds of convergences such as the order statistical convergence, the $mo$-convergence, and the order convergence.

2-s2.0-85101180282 A net (x(alpha))(alpha is an element of A) in an f-algebra E is called multiplicative order convergent to x is an element of E if vertical bar x(alpha )- x vertical bar . u ->(o) 0 for all u is an element of E+. This convergence was introduced and studied on f-algebras with the order convergence. In this paper, we study a variation of this convergence for normed Riesz algebras with respect to the norm convergence. A net (x(alpha))(alpha is an element of A) in a normed Riesz algebra E is said to be multiplicative norm convergent to x is an element of E if parallel to vertical bar x(alpha) - x vertical bar . u parallel to -> 0 for each u is an element of E+. We study this concept and investigate its relationship with the other convergences, and also we introduce the mn-topology on normed Riesz algebras.© 2021, Hacettepe University. All rights reserved.

In this paper, we introduce the statistically multiplicative convergent sequences in locally solid Riesz algebras with respect to the algebra multiplication and the solid topology. We study on this concept and we give the notion of $\mathbb{st_m}$-bounded sequence, and also, we prove some relations between this convergence and the other convergences such as the order convergence and the statistical convergence in topological spaces. Also, we give some results related to semiprime $f$-algebras., 15 Pages

International Conference on Mathematical Analysis and Applications in Modeling, ICMAAM 2018 -- 9 January 2018 through 12 January 2018 -- -- 238429 In this paper. we continue to investigate in line of the recent work of Sencimen and Pehlivan, Das and Savas and consider the notion of -statistical order pre Cauchy condition related to a new type of order convergence, namely -statistical order convergence in Riesz spaces and establish some of its basic properties. We mainly investigate their inter-relationship. © 2020, Springer Nature Singapore Pte Ltd. Science and Engineering Research Board, SERB Department of Science and Technology, Ministry of Science and Technology, India, DST Acknowledgements The first author is thankful to TUBA, Tukish Academy of Sciences for arranging a visit during which this work was done. The first author is also thankful to SERB, DST, New Delhi for granting a research project No. SR/S4/MS:813/13 during the tenure of which this work was done.

We extend \L ukasiewicz logic obtaining the infinitary logic $\mathcal{IR}\L$ whose models are algebras $C(X,[0,1])$, where $X$ is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in $\sigma$-complete Riesz spaces with strong unit. The Lindenbaum-Tarski algebra of $\mathcal{IR}\L$ is, up to isomorphism, an algebra of $[0,1]$-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval $[0,1]$.

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(X)\rightarrow C(Y)$$\end{document}C(X)→C(Y) where X, Y are compact Hausdorff spaces. For a simple case, suppose X is metrizable and T is such an order isomorphism. By a theorem of Kaplansky, T induces a homeomorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau :X\rightarrow Y$$\end{document}τ:X→Y. We prove the existence of a homeomorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\times \mathbb {R}\rightarrow Y\times \mathbb {R}$$\end{document}X×R→Y×R that maps the graph of any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C(X)$$\end{document}f∈C(X) onto the graph of Tf. For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let X and Y be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\infty }(X)\rightarrow C^{\infty }(Y)$$\end{document}C∞(X)→C∞(Y). (Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\infty }(X)$$\end{document}C∞(X) is the space of all continuous functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\rightarrow [-\infty ,\infty ]$$\end{document}X→[-∞,∞] that are finite on a dense set.) The third part of the paper considers order isomorphisms T between arbitrary Archimedean Riesz spaces E and F. We prove that such a T extends uniquely to an order isomorphism between their universal completions. (In the absence of linearity this is not obvious.) It follows, that there exist an extremally disconnected compact Hausdorff space X, Riesz isomorphisms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{}$$\end{document}^ of E and F onto order dense Riesz subspaces of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\infty }(X)$$\end{document}C∞(X) and an order isomorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S:C^{\infty }(X)\rightarrow C^{\infty }(X)$$\end{document}S:C∞(X)→C∞(X) such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{Tf}=S\hat{f}$$\end{document}Tf^=Sf^ (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in E$$\end{document}f∈E).

Within the infinitary variety of σ-complete Riesz MV-algebras RMV σ , we introduce the algebraic analogue of a random variable as a homomorphism defined on the free algebra in RMV σ . After a preliminary study of the proposed notion, we use it to define stochastic processes in the framework of non-classical logic (Łukasiewicz logic, more precisely) and we define stochastic independence.

We show that the set of maximal lower bounds of two symmetric matrices with respect to the L\"owner order can be identified to the quotient set $O(p,q)/(O(p)\times O(q))$. Here, $(p,q)$ denotes the inertia of the difference of the two matrices, $O(p)$ is the $p$-th orthogonal group, and $O(p,q)$ is the indefinite orthogonal group arising from a quadratic form with inertia $(p,q)$. We also show that a similar result holds for positive semidefinite maximal lower bounds with maximal rank of two positive semidefinite matrices. We exhibit a correspondence between the maximal lower bounds $C$ of two matrices $A,B$ and certain pairs of subspaces, describing the directions on which the quadratic form associated with $C$ is tangent to the one associated with $A$ or $B$. The present results refines a theorem from Kadison that characterizes the existence of the infimum of two symmetric matrices and a theorem from Moreland, Gudder and Ando on the existence of the positive semidefinite infimum of two positive semidefinite matrices., Comment: 20 pages, 2 figures

Recently Flaminio and Montagna, [FlMo], extended the language of MV-algebras by adding a unary operation, called a state-operator. This notion is introduced here also for effect algebras. Having it, we generalize the Loomis–Sikorski Theorem for monotone σ-complete effect algebras with inter- nal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (RDP) and countable interpolation with an order de- termining system of states is dual to the category of Bauer simplices Ω such that ∂eΩ is an F-space., peer-reviewed

Concavity and supermodularity are in general independent properties. A class of functionals defined on a lattice cone of a Riesz space has the Choquet property when it is the case that its members are concave whenever they are supermodular. We show that for some important Riesz spaces both the class of positively homogeneous functionals and the class of translation invariant functionals have the Choquet property. We extend in this way the results of Choquet [G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953–1954) 131–295] and Konig [H. Konig, The (sub/super) additivity assertion of Choquet, Studia Math. 157 (2003) 171–197].

A Perron-type integral of order 1 and 2 for Riesz-space-valued functions is investigated. Some versions of integration by parts formula for this integral are proved for both orders.

Ph.D. (Mathematics) Our objective in this thesis is to investigate two fundamental questions concerning Riesz operators de ned on a Banach space. Recall that Riesz operators are generalizations of compact operators in the sense that Riesz operators have the same spectral properties as compact operators. However, they do not possess the same algebraic properties as compact operators. Our rst question that we investigate is: When is a Riesz operator a nite rank operator? This question is motivated from the fact that if a compact operator de ned on a Banach space has closed range, then it is a nite rank operator. Also, Ghahramani proved that a compact homomorphism de ned on a C -algebra is a nite rank operator, see . Martin Mathieu, in his paper, generalized the result of Ghahramani by proving that a weakly compact homomorphism de ned on a C -algebra is a nite rank operator...

Bu tez çalışmasında yakın zamanlarda tanımlanan ve metrik uzayların bir genelleştirmesi olan vektör metrik uzaylar kavramı Çevik ve Altun (2009) ile Çevik (2014, 2015) çalışmaları temel alınarak incelenmiştir. Vektör metrik uzay kavramı ile klasik metrik uzaylar arasındaki paralellikler ve farklılıklar ortaya konmuş, konuyla ilgili ön bilgiler bir araya getirilmiş, kanıtlardaki boşluklar doldurularak gerekli bilgiler verilmiş ve bazı sonuçlar elde edilmiştir. In this thesis work, the concept of vector metric spaces which has been defined recently, and is generalization of metric spaces are studied on the basis of studies by Çevik and Altun (2009) and Çevik (2014, 2015). While definitions and theorems about the subject are reviewed, parallelism and differences between metric spaces and vector metric spaces are revealed, and gaps in proofs are filled by providing the necessary knowledge in an attempt to obtain some results.

In this note, we carry out investigations related to the mixed impact of ordering and topological structure of a locally convex solid Riesz space(X,τ)and a scalar valued sequence spaceλ, on the vector valued sequence spaceλ(X)which is formed and topologized with the help ofλandX, and vice versa. Besides,we also characterizeo-matrix transformations fromc(X),ℓ∞(X)to themselves,cs(X)toc(X)and derive necessary conditions for a matrix of linear operators to transformℓ1(X)into a simple ordered vector valued sequence spaceΛ(X).

The general form of continuous linear functionals on an Orlicz sequence space1ϕ(non-separable and non-locally convex in general) is obtained. It is proved that the spacehϕis anM-ideal in1ϕ.

Some convergence theorems have been obtained for the $GH_k$ integral for functions defined in abstract topological spaces and with values in Riesz spaces.

Thesis (Ph.D. (Mathematics))--North-West University, Potchefstroom Campus, 2004. This study revolves around operators defined by conditional expectations and operators generated by random measures. Studies of operators in function spaces defined by conditional expectations first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26]. N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their averaging properties were studied by P.G. Dodds and C.B. Huijsmans and B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert [17] studied their relationship with multiplication operators in C*-modules. It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special cases of kernel operators that were, inter alia, studied by A.R. Schep in [25] were special cases of conditional expectation operators. On the other hand, operators generated by random measures or pseudo-integral operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30], building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late 1970's and early 1980's. In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on Multiplication Conditional Expectation-representable (MCE-representable) operators. We also generalize the result of A. Sourour [27] and show that order continuous linear maps between ideals of almost everywhere finite measurable functions on u-finite measure spaces are MCE-representable. This fact enables us to easily deduce that sums and compositions of MCE-representable operators are again MCE-representable operators. We also show that operators generated by random measures are MCE-representable. The first chapter gathers the definitions and introduces notions and concepts that are used throughout. In particular, we introduce Riesz spaces and operators therein, Riesz and Boolean homomorphisms, conditional expectation operators, kernel and absolute T-kernel operators. In Chapter 2 we look at MCE-operators where we give a definition different from that given by J.J. Grobler and B. de Pagter in [8], but which we show to be equivalent. Chapter 3 involves random measures and operators generated by random measures. We solve the problem (positively) that was posed by A. Sourour in [28] about the relationship of the lattice properties of operators generated by random measures and the lattice properties of their generating random measures. We show that the total variation of a random signed measure representing an order bounded operator T, it being the difference of two random measures, is again a random measure and represents ITI. We also show that the set of all operators generated by a random measure is a band in the Riesz space of all order bounded operators. In Chapter 4 we investigate the relationship between operators generated by random measures and MCE-representable operators. It was shown by A. Sourour in [28, 271 that every order bounded order continuous linear operator acting between ideals of almost everywhere measurable functions is generated by a random measure, provided that the measure spaces involved are standard measure spaces. We prove an analogue of this theorem for the general case where the underlying measure spaces are a-finite. We also, in this general setting, prove that every order continuous linear operator is MCE-representable. This rather surprising result enables us to easily show that sums, products and compositions of MCE-representable operator are again MCE-representable. Key words: Riesz spaces, conditional expectations, multiplication conditional expectation-representable operators, random measures. Doctoral

This study revolves around operators defined by conditional expectations and operators generated by random measures. Studies of operators in function spaces defined by conditional expectations first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26]. N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their averaging properties were studied by P.G. Dodds and C.B. Huijsmans and B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert [17] studied their relationship with multiplication operators in C*-modules. It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special cases of kernel operators that were, inter alia, studied by A.R. Schep in [25] were special cases of conditional expectation operators. On the other hand, operators generated by random measures or pseudo-integral operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30], building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late 1970's and early 1980's. In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on Multiplication Conditional Expectation-representable (MCE-representable) operators. We also generalize the result of A. Sourour [27] and show that order continuous linear maps between ideals of almost everywhere finite measurable functions on u-finite measure spaces are MCE-representable. This fact enables us to easily deduce that sums and compositions of MCE-representable operators are again MCE-representable operators. We also show that operators generated by random measures are MCE-representable. The first chapter gathers the definitions and introduces notions and concepts that are used throughout. In particular, we introduce Riesz spaces and operators therein, Riesz and Boolean homomorphisms, conditional expectation operators, kernel and absolute T-kernel operators. In Chapter 2 we look at MCE-operators where we give a definition different from that given by J.J. Grobler and B. de Pagter in [8], but which we show to be equivalent. Chapter 3 involves random measures and operators generated by random measures. We solve the problem (positively) that was posed by A. Sourour in [28] about the relationship of the lattice properties of operators generated by random measures and the lattice properties of their generating random measures. We show that the total variation of a random signed measure representing an order bounded operator T, it being the difference of two random measures, is again a random measure and represents ITI. We also show that the set of all operators generated by a random measure is a band in the Riesz space of all order bounded operators. In Chapter 4 we investigate the relationship between operators generated by random measures and MCE-representable operators. It was shown by A. Sourour in [28, 271 that every order bounded order continuous linear operator acting between ideals of almost everywhere measurable functions is generated by a random measure, provided that the measure spaces involved are standard measure spaces. We prove an analogue of this theorem for the general case where the underlying measure spaces are a-finite. We also, in this general setting, prove that every order continuous linear operator is MCE-representable. This rather surprising result enables us to easily show that sums, products and compositions of MCE-representable operator are again MCE-representable. Key words: Riesz spaces, conditional expectations, multiplication conditional expectation-representable operators, random measures. Thesis (Ph.D. (Mathematics))--North-West University, Potchefstroom Campus, 2004.

Some versions of Henstock-Kurzweil integral with respect to different derivation bases for functions with values in Dedekind complete Riesz spaces are studied. Fundamental Theorem of Calculus are proved for these integrals and an application to Walsh series is given.

Recently a connection has been found between the improper Kurzweil-Henstock integral on the real line and the integral over a compact space. In this paper these results are extended to a Pettis-type integral for the case of functions with values in Riesz spaces with ``enough" order continuous functionals.

This paper is a step by step account of the Riesz approach to the Lebesgue integral. Besides, motivating the use of ``almost everywhere'' tools, we eliminate unnecessary equivalences, and we give a simple representation of a complete space of integrable functions, usually missing from classical treatises.

Si definisce un integrale del tipo "Burkill-Cesari" per funzioni d'insieme a valori in spazi di Riesz Dedekind completi. Si introduce un concetto di quasi-additività, simile a quello introdotto da Lamberto Cesari in [5]. Si provano alcuni teoremi analoghi a quelli classici, e si confronta l'integrale introdotto con quello di Riemann e con quello monotono di cui in [1]. A definition of "Burkill-Cesari type integral" is given, for set functions, with values in Dedekind complete Riesz space. A concept of quasi-additivity is introduced, similar to the one introduced by Lamberto Cesari in [5]. Some theorems analogous to the classical ones are proved. Moreover, we give a comparison with the "Riemann-integral" and the "monotone integral" defined in [1].

A monotone integral is given for scalar function, with respect to Riesz space values means, and also a necessary and sufficient condition to obtain a Radon-Nikodym density for two means.

In [3] we did not give explicitly the definition of measurability for realvalued functions, with respect to finitely additive measures with values in a Dedekind complete Riesz space. We note that, in [3], all involved functions are intended to be measurable. We now report the definition of measurability, which we gave in [2] (Definition 3.2).

Conditional expectations operators acting on Riesz spaces are shown to commute with a class of principal band projections. Using the above commutation property, conditional expectation operators on Riesz spaces are shown to be averaging operators. Here the theory of f-algebras is used when defining multiplication on the Riesz spaces. This leads to the extension of these conditional expectation operators to their so-called natural domains, i.e., maximal domains for which the operators are both averaging operators and conditional expectations. The natural domain is in many aspects analogous to L 1 .