An analysis of the logic of Riesz spaces with strong unit

Nola, Antonio Di; Lapenta, Serafina; Leustean, Ioana

doi:10.1016/j.apal.2017.10.006

Cited by 8 publications

(9 citation statements)

References 29 publications

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“…We now give the definition of Riesz MV-algebras (RMV-algebras, for short); they are MV-algebras endowed with a scalar multiplication by elements of the real interval [0, 1]. RMV-algebras were introduced in Di Nola and Leuştean [18] and further studied in Di Nola, Lapenta, and Leuştean [14,15]. Definition 2.6 (RMV-algebra) A Riesz MV-algebra (RMV-algebra for short) is an MV-algebra A endowed with an external multiplication f r for every real number r in [0, 1], satisfying the following conditions.…”

Section: Example 23mentioning

confidence: 99%

Sheaf representations and locality of Riesz spaces with order unit

2021

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We present an algebraic study of Riesz spaces (=real vector lattices) with a (strong) order unit. We exploit a categorical equivalence between those structures and a variety of algebras called RMV-algebras. We prove two different sheaf representations for Riesz spaces with order unit: the first represents them as sheaves of linearly ordered Riesz spaces over a spectral space, the second represent them as sheaves of "local" Riesz spaces over a compact Hausdorff space. Motivated by the latter representation we study the class of local RMV-algebras. We study the algebraic properties of local RMV-algebra and provide a characterisation of them as special retracts of the real interval [0,1]. Finally, we prove that the category of local RMV-algebras is equivalent to the category of all Riesz spaces.

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Section: Example 23mentioning

confidence: 99%

Sheaf representations and locality of Riesz spaces with order unit

2021

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“…If we denote by RL the Lindenbaum-Tarski algebra of the logic of Riesz MV-algebras R L and by RL n the Lindenbaum-Tarski algebra in n propositional variables, we can prove that RL n is isomorphic to RM V n . In [10,Theorem 2.15] it is proved that the norm-completion of RL n coincides with C([0, 1] n ), where both algebras are endowed with the unit norm defined above.…”

Section: Lukasiewicz Logic and Riesz Mv-algebrasmentioning

confidence: 99%

“…Thus, IRL n is the σ-completion of RL n -which is the algebra of piecewise linear functions with real coefficients -and RL n has C([0, 1] n ) -which is the algebra of [0, 1]-valued continuous functions defined over [0, 1] n -as its norm completion, see [10,Theorem 2.15]. Since norm completions, Dedekind σ-completions and Dekekind completions of Riesz Spaces are a deeply investigated subject, we shall see in the next subsection how this can be exploited in a logical setting.…”

Section: The Logic Ir Lmentioning

confidence: 99%

“…Indeed, in Definition 3.4 we have defined a notion of logical order convergence and in Proposition 3.6 we have proved that, in the Lindenbaum-Tarski algebra of IR L, the limit is unique and the MV-algebraic operations are continuous with respect to the limit. It was proved in [10] that in RL n , the Lindenbaum-Tarski algebra of the logic R L, Definition 3.4 can be strengthened in such way that it corresponds to the notion of uniform convergence: if we denote by f ψ the function that corresponds (via isomorphism) to [ψ] in RL, we have the following theorem. Such a definition is very similar to Definition 3.4, but it is stronger, as it requires a peculiar sequence of formulas, the point being that formulas of the shape η r correspond to constant functions in RL.…”

Section: Limits Compact Hausdorff Spaces and Completionsmentioning

confidence: 99%

“…In [10] we have discussed the norm-completion of RL n with respect to the unit norm above mentioned. The previous proposition linked closely together norm-completion and σ-completion of RL n , indeed we have the following.…”

Section: Limits Compact Hausdorff Spaces and Completionsmentioning

confidence: 99%

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Infinitary logic and basically disconnected compact Hausdorff spaces

Nola

Lapenta

Leustean

2018

Journal of Logic and Computation

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We extend Lukasiewicz logic obtaining the infinitary logic 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 Dedekind σ-complete Riesz spaces with strong unit. The Lindenbaum-Tarski algebra of 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].[25] led to a categorical equivalence between Riesz MV-algebras and Riesz spaces with strong unit. Consequently, one can develop a logical system that extends Lukasiewicz logic and has Riesz MV-algebras as models [11].The strong connection between functional analysis and Riesz spaces should be reflected by their underlying logical systems and in the present paper we test the expressive power of the "logic of Riesz Spaces" in order to provide a logical system that is closer to what we may think of as the "logic of (some) Hausdorff spaces". Remarkably, by adding a countable disjunction to the logic of Riesz MV-algebras we were able to obtain the desired bridge between logic, topology and functional analysis.Our construction rests on Kakutani's duality between abstract M-spaces and compact Hausdorff spaces [18]. Indeed, from a categorical point of view, the norm-complete Riesz MV-algebras 1 are equivalent to M-spaces, and henceforth dual to compact Hausdorff spaces [11]. Our aim is to express the property of being "norm-complete" in a logical setting. Thus, in [10] we introduced the limit of a sequence of formulas, a syntactic notion whose semantic counterpart is the uniform limit of the corresponding sequence of term functions. This notion is slightly stronger than the usual notion of order convergence and this remark has been the starting point of the present development.In Section 3 we define the system IR L, that stands for Infinitary Riesz Logic, which expands the logic of Riesz spaces with denumerable disjunctions and conjunctions. In this way we define a logical system whose models are isomorphic to unit intervals of σ-complete M-spaces and, consequently, strongly related to basically disconnected compact Hausdorff spaces. The Lindenbaum-Tarski algebra in n variables is concretely characterized as the algebra of all Borel functions f : [0, 1] n → [0, 1]. Moreover, our system enjoys standard completeness: the real interval [0, 1] endowed with a structure of σ-complete Riesz MV-algebra is a standard model.The present approach via infinitary logic is built upon the work of C.R. Karp -of which the monograph [19] is a complete treatise -where we replace the classical axioms of Boolean logic with the axioms of Riesz Lukasiewicz logic.After the needed preliminaries, we define the logical system IR L in Section 3, where we also prove a general completeness theorem. The link between the models of IR L and Kakutani's duality is provided in Section 4. In the last section we prove the Loomis-Sikorski theorem for Riesz MV-algebras, based on the well-known similar result f...

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Rota's Fubini lectures: The first problem

Mundici

2021

Advances in Applied Mathematics

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An analysis of the logic of Riesz spaces with strong unit

Cited by 8 publications

References 29 publications

Sheaf representations and locality of Riesz spaces with order unit

Sheaf representations and locality of Riesz spaces with order unit

Infinitary logic and basically disconnected compact Hausdorff spaces

Rota's Fubini lectures: The first problem

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