### Ordering of observables and characterization of conditional expectation

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In this paper, the authors introduce the notion of conditional expectation of an observable $x$ on a logic with respect to a sublogic, in a state $m$, relative to an element $a$ of the logic. This conditional expectation is an analogue of the expectation of an integrable function on a probability space.

In this paper we construct conditional states on semi-simple MV-algebras. We show that these conditional states are not given uniquely. By using them we construct the joint probability distributions and discuss the properties of these distributions. We show that the independence is not symmetric.

New approach to characterization of orthomodular lattices by means of special types of bivariable functions $G$ is suggested. Under special marginal conditions a bivariable function $G$ can operate as, for example, infimum measure, supremum measure or symmetric difference measure for two elements of an orthomodular lattice.

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