### A characterization of quantic quantifiers in orthomodular lattices.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We give a representation of an observable on a fuzzy quantum poset of type II by a pointwise defined real-valued function. This method is inspired by that of Kolesárová [6] and Mesiar [7], and our results extend representations given by the author and Dvurečenskij [4]. Moreover, we show that in this model, the converse representation fails, in general.

A finite orthomodular lattice in which every maximal Boolean subalgebra (block) has the same cardinality $k$ is called $\lambda $-regular, if each atom is a member of just $\lambda $ blocks. We estimate the minimal number of blocks of $\lambda $-regular orthomodular lattices to be lower than of equal to ${\lambda}^{2}$ regardless of $k$.

MV-algebras were introduced by Chang, 1958 as algebraic bases for multi-valued logic. MV stands for “multi-valued" and MV algebras have already occupied an important place in the realm of nonstandard (mathematical) logic applied in several fields including cybernetics. In the present paper, using the Loomis–Sikorski theorem for $\sigma $-MV-algebras, we prove that, with every element $a$ in a $\sigma $-MV algebra $M$, a spectral measure (i. e. an observable) ${\Lambda}_{a}:\mathcal{B}\left([0,1]\right)\to \mathcal{B}\left(M\right)$ can be associated, where $\mathcal{B}\left(M\right)$ denotes the Boolean $\sigma $-algebra...

We prove that the interval topology of an Archimedean atomic lattice effect algebra $E$ is Hausdorff whenever the set of all atoms of $E$ is almost orthogonal. In such a case $E$ is order continuous. If moreover $E$ is complete then order convergence of nets of elements of $E$ is topological and hence it coincides with convergence in the order topology and this topology is compact Hausdorff compatible with a uniformity induced by a separating function family on $E$ corresponding to compact and cocompact elements....

Does there exist an atomic lattice effect algebra with non-atomic subalgebra of sharp elements? An affirmative answer to this question (and slightly more) is given: An example of an atomic MV-effect algebra with a non-atomic Boolean subalgebra of sharp or central elements is presented.

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.