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We present the algebraic structures behind the approaches used to work with pairwise comparison matrices and, in general, the representation of preferences. We obtain a general definition of consistency and a universal decomposition in the space of PCMs, which allow us to define a consistency index. Also Arrow's theorem, which is presented in a general form, is relevant. All the presented results can be seen in the main formulations of PCMs, i.e., multiplicative, additive and fuzzy approach, by the fact that each of them is a particular interpretation of the more general algebraic structure needed to deal with these theories. [ABSTRACT FROM AUTHOR]

For any real number p ∈ [ 1 , + ∞) {p\in[1,+\infty)} , we characterise the operations ℝ I → ℝ {\mathbb{R}^{I}\to\mathbb{R}} that preserve p-integrability, i.e., the operations under which, for every measure μ, the set ℒ p ⁢ (μ) {\mathcal{L}^{p}(\mu)} is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind σ-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that ℝ {\mathbb{R}} generates this variety. From this, we exhibit a concrete model of the free Dedekind σ-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve p-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind σ-complete Riesz spaces with weak unit, ℝ {\mathbb{R}} is proved to generate this variety, and a concrete model of the free Dedekind σ-complete Riesz spaces with weak unit is exhibited. [ABSTRACT FROM AUTHOR]

A Riesz structure on a lattice ordered abelian group G is a real vector space structure where the product of a positive element of G and a positive real is positive. In this paper we show that for every cardinal k there is a totally ordered abelian group with at least k Riesz structures, all of them isomorphic. Moreover two Riesz structures on the same totally ordered group are partially isomorphic in the sense of model theory. Further, as a main result, we build two nonisomorphic Riesz structures on the same l-group with strong unit. This gives a solution to a problem posed by Conrad in 1975. Finally we apply the main result to MV-algebras and Riesz MV-algebras. [ABSTRACT FROM AUTHOR]

In this paper we study the concept of generalized geometric difference of six-dimensional rough almost lacunary statistical Cesáro of χ 3 {\chi^{3}} over probabilistic space P is defined by Musielak–Orlicz function. Since the study of convergence in Probabilistic space P is fundamental to probabilistic functional analysis, we feel that the concept of generalized geometric difference of six-dimensional rough almost lacunary statistical Cesáro of χ 3 {\chi^{3}} over probabilistic space P defined by Musielak in a probabilistic space P would provide a more general framework for the subject. [ABSTRACT FROM AUTHOR]

In this paper we study the concept of generalized geometric difference of six-dimensional rough almost lacunary statistical Cesáro of χ 3 {\chi^{3}} over probabilistic space P is defined by Musielak–Orlicz function. Since the study of convergence in Probabilistic space P is fundamental to probabilistic functional analysis, we feel that the concept of generalized geometric difference of six-dimensional rough almost lacunary statistical Cesáro of χ 3 {\chi^{3}} over probabilistic space P defined by Musielak in a probabilistic space P would provide a more general framework for the subject. [ABSTRACT FROM AUTHOR]

We use a landmark result in the theory of Riesz spaces – Freudenthal’s 1936 spectral theorem – to canonically represent any Archimedean lattice-ordered group G with a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriate G-indexed zero-dimensional compactification w G ⁢ Z G {w_{G}Z_{G}} of its space Z G {Z_{G}} of minimal prime ideals.The two further ingredients needed to establish this representation are the Yosida representation of G on its space X G {X_{G}} of maximal ideals, and the well-known continuous surjection of Z G {Z_{G}} onto X G {X_{G}}.We then establish our main result by showing that the inclusion-minimal extension of this representation of G that separates the points of Z G {Z_{G}} – namely, the sublattice subgroup of C ⁡ ( Z G ) {\operatorname{C}(Z_{G})} generated by the image of G along with all characteristic functions of clopen (closed and open) subsets of Z G {Z_{G}} which are determined by elements of G – is precisely the classical projectable hull of G.Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit. [ABSTRACT FROM AUTHOR]