Since almost fifty years, Shannon entropy measure has been used as a very powerful tool when quantifying and processing the amount of randomness contained in probability distributions. In this paper we propose a lattice-valued entropy measure H ascribing to each lattice-valued possibilistic distribution π the value H(π) defined as the expected value (in the sense of lattice-valued Sugeno integral with infimum in the role of t-norm) of certain nonincreasing function of the values ascribed to the elements of the basic space by the possibilistic distribution π in question. The main result reads that, for completely distributive complete lattices, the entropy value ascribed to possibilistically independent product of a finite number of lattice-valued possibilistic distributions is defined by the supremum of the entropy values ascribed to particular distributions.

We investigate a partial non-numerical possibilistic measure taking its values in a complete lattice and defined on a nested system of subsets of the universe under consideration. Our aim is to extend this measure conservatively to the power-set of all systems of this universe using the same idea as that when introducing outer measures. Hence, we ascribe to each subset of the universe its minimal (in the sense of set inclusion) covering by a set from the nested domain and define its possibility degree as identical with the value ascribed to this covering. Also analyzed is the case when they are two lattice-valued possibilistic measures on the same universe, each with its own nested domain and our aim is to define one possibilistic measure on the power-set in question in a way sophistically taking profit of the information offered by both the particular partial lattice-valued measures.

Fuzzy measures and possibilistic measures taking their values in partially ordered sets, in particular, in semilattices and lattices, are introduced. Suggested and investigated in more detail are Cartesian products and marginals of poset-valued fuzzy and possibilistic measures. The conditions under which the resulting set functions possess the properties of poset-valued fuzzy and possibilistic measures are analyzed and some relevant assertions are stated and proved.

It is a well-known fact that the Dempster combination rule for combination of uncertainty degrees coming from two or more sources is legitimate only if the combined empirical data, charged with uncertainty and taken as random variables, are statistically (stochastically) independent. We shall prove, however, that for a particular but large enough class of probability measures, an analogy of Dempster combination rule, preserving its extensional character but using some nonstandard and boolean-like structures over the unit interval of real numbers, can be obtained without the assumption of statistical independence of input empirical data charged with uncertainty.

A possibility to define a binary operation over the space of pairs of belief functions, inverse or dual to the well-known Dempster combination rule in the same sense in which substraction is dual with respect to the addition operation in the space of real numbers, can be taken as an important problem for the purely algebraic as well as from the application point of view. Or, it offers a way how to eliminate the modification of a belief function obtained when combining this original belief function with other pieces of information, later proved not to be reliable. In the space of classical belief functions definable by set-valued (generalized) random variables defined on a probability space, the invertibility problem for belief functions, resulting from the above mentioned problem of "dual" combination rule, can be proved to be unsolvable up to trivial cases. However, when generalizing the notion of belief functions in such a way that probability space is replaced by more general measurable space with signed measure, inverse belief functions can be defined for a large class of belief functions generalized in the corresponding way. "Dual" combination rule is then defined by the application of the Dempster rule to the inverse belief functions.

Possibility measures have been conceived by Zadeh [1], and developed by, e. g., Dubois, Prade and others, as very simple, if compared with, e.g., probability theory, but still nontrivial and reasonable uncertainty measures. The relatively poor descriptional and operational abilities of possibility measures seem to be closely related to the standard linear ordering relation and the corresponding supremum and infimum operations defined over the unit interval of real numbers. Having discussed, very briefly, the possibilities how to overcome these limitations, we propose and investigate possibility measures taking their values in the well-known Cantor subset of the unit interval but defined with respect to a nonstandard operation of supremum resulting from a nonstandard partial ordering of real numbers from the Cantor set. Such a nonstandard possibility measure is proved to be a sufficient tool to define an infinite sequence of probability measures defined over countable sets, consequently, the distribution of each continuous real-valued random variable defined over a general abstract probability space can be defined by an appropriate nonstandard possibility measure.