We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set-representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non-Boolean situations. On the other hand, a difference-closed system can be constructed such that even two-valued states do not allowfor extensions. Finally,we consider these questions in a σ-complete setup and find a large class of such systems with rather interesting state properties.
States on systems of sets that are closed under symmetric difference / DE SIMONE, Anna; Navara, Mirko; Pták, Pavel. - In: MATHEMATISCHE NACHRICHTEN. - ISSN 0025-584X. - 288:17-18(2015), pp. 1995-2000. [10.1002/mana.201500029]
States on systems of sets that are closed under symmetric difference
DE SIMONE, ANNA;
2015
Abstract
We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set-representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non-Boolean situations. On the other hand, a difference-closed system can be constructed such that even two-valued states do not allowfor extensions. Finally,we consider these questions in a σ-complete setup and find a large class of such systems with rather interesting state properties.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.