Mykytsei O. Lattice-valued predicates on continuous semilattices.

The thesis is devoted to lattice-valued monotonic predicates on continuous semilattices, which are natural generalizations of non-additive real-valued and lattice-valued measures. Generalization of the notion of monotonic predicate to functions with values in completely distributive lattices is justified. It is proved that the sets of L-valued monotonic predicates are free objects over the respective continuous semiattices. The notion of compatibility P:S× S'→{0,1} between continuous semilattices S and S' is introduced. Several subclasses of the set C_w (S,S') of such compatibilities are introduced, and it is shown that many specific kinds of mathematical objects can be regarded as compatibilities. A modification of classical Lawson duality is proposed. For complete stably continuous semilattices S and S' with zeros a complete continuous semilattice S⊠S' with zero is introduced so that it contains both S and S'. For normalized monotonic predicates c⊠c^'∈M_[L] (S⊠S^') the symmetrical tensor product c⊠c^':S⊠S'→L^op is defined. This product is an L -valued normalized monotonic predicate on the complete semilattice S⊠S'. It is shown that the constructions of lattices of normalized L -capacities and of normalized L ̃-capacities on continuous semigroups with zeros are functorial and linked via two classic dualities and a functor isomorphism. Crisp and L-fuzzy ambiguous representations between continuous semilattices are defined. Subclasses of pseudoinvertible ambiguous and pseudoinvertible L-ambiguous representations are defined and investigated. An embedding of the category L-fuzzy ambiguous representations into the category of continuous L-semimodules is constructed. An embedding of the category of continuous semilattices and pseudoinvertible L-ambiguous representations into category of continuous L-semimodules as a full subcategory is constructed. An interpretation of L-ambiguous representations and respective L -linear operators as predicate transformers is proposed. Monotone predicates are applied to the analysis of two player rough games. Monotonic payoff predicates w ̅_i∈M_([(L]) ̃ ) S_i і w ̅_(-i)∈M_([L]) S_(-i)are obtained to describe lower estimates of the game result for rational players. Recurrent formulae are found, which makes it possible to calculate these predicates backwards from the last possible move of the game. It is proved that for a rough extensive-form real-valued zero-sum finite games, the least upper bound of the minimal gain of the first player is equal to the greatest lower bound of the maximal loss of the second player.