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Modular atomic effect algebras and the existence of subadditive states

Zdena Riečanová — 2004


Lattice effect algebras generalize orthomodular lattices and M V -algebras. We describe all complete modular atomic effect algebras. This allows us to prove the existence of ordercontinuous subadditive states (probabilities) on them. For the separable noncomplete ones we show that the existence of a faithful probability is equivalent to the condition that their MacNeille complete modular effect algebra.

The existence of states on every Archimedean atomic lattice effect algebra with at most five blocks

Zdena Riečanová — 2008


Effect algebras are very natural logical structures as carriers of probabilities and states. They were introduced for modeling of sets of propositions, properties, questions, or events with fuzziness, uncertainty or unsharpness. Nevertheless, there are effect algebras without any state, and questions about the existence (for non-modular) are still unanswered. We show that every Archimedean atomic lattice effect algebra with at most five blocks (maximal MV-subalgebras) has at least one state, which...

Archimedean atomic lattice effect algebras in which all sharp elements are central

Zdena Riečanová — 2006


We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it.

Lattice effect algebras densely embeddable into complete ones

Zdena Riečanová — 2011


An effect algebraic partial binary operation ø p l u s defined on the underlying set E uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion E ^ of E there exists an effect algebraic partial binary operation ^ then ^ need not be an extension of . Moreover, for an Archimedean atomic lattice effect algebra E we give a necessary and sufficient condition for that ^ existing on E ^ is an extension of defined on E . Further we show that such ^ extending exists at most...

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