scholarly journals Noncommutative Poincaré recurrence theorem

2001 ◽  
Vol 89 (1) ◽  
pp. 1-6
Author(s):  
Andrzej Łuczak
Keyword(s):  
Poincaré Recurrence Theorem ◽  
Recurrence Theorem ◽  
Poincare Recurrence ◽  
Poincaré Recurrence

Strong Poincaré recurrence theorem in MV-algebras

2010 ◽  
Vol 60 (5) ◽  
Author(s):  
Beloslav Riečan
Keyword(s):  
Probability Space ◽  
Mv Algebra ◽  
Measure Preserving Transformation ◽  
Poincaré Recurrence Theorem ◽  
Recurrence Theorem ◽  
Poincare Recurrence ◽  
Poincaré Recurrence ◽  
Almost All ◽  
Poincare Theorem ◽  
Mv Algebras

AbstractThe classical Poincaré strong recurrence theorem states that for any probability space (Ω, ℒ, P), any P-measure preserving transformation T, and any A ∈ ℒ, almost all points of A return to A infinitely many times. In the present paper the Poincaré theorem is proved when the σ-algebra ℒ is substituted by an MV-algebra of a special type. Another approach is used in [RIEČAN, B.: Poincaré recurrence theorem in MV-algebras. In: Proc. IFSA-EUSFLAT 2009 (To appear)], where the weak variant of the theorem is proved, of course, for arbitrary MV-algebras. Such generalizations were already done in the literature, e.g. for quantum logic, see [DVUREČENSKIJ, A.: On some properties of transformations of a logic, Math. Slovaca 26 (1976), 131–137.

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