L-valued Random Variables And The Operation (?) On Heyting Algebras

Abstract: Researches of this thesis including two respects (both relate to lattice theory and topology). On one hand, we have investigated L-valued random variables (L is a closed-set lattice or moleculely generated lattice). Receiving already has inspiration of thought , method and skill studying L-topological space in documents, we have defined a crisp topology TFLC on FLC(Rn) = {A ∈ LRn | for each α in Copr(L), {x ∈ Rn | A(x) ≥α} is non-empty compact set of Rn} and several kinds of metric on L-subset groups, some properties have discussed of them. On this basis, we have defined several L-valued random variables and discussed preliminary nature of them. At the same time, we have studied belief measure induced by L-valued random variables and its nature. On the other hand, a kind of operation (?) is defined on Heyting algebras with minimal elements, and properties of (?) and nucleus, including relationships between (?) and nucleus, are investigated.The content of this text divides into four chapters altogether:The chapter one is prepare knowledge, which has introduced the topology on superspace and the concept of measurable multivalued mapping and random sets. On random F-set's basis, we have given the concept of random L-set and its nature.The chapter two is main part of this paper, L-valued random variables are studied (L is a closed set lattice or moleculely generated lattice). At first, we have introduced the concepts of FcL(Rn), FcocL(Rn), FcLb(Rn), FcocLb(Rn), and have proved that (K₀, h) is a complete metric space. Next we have defined a crisp topology TcFL on FcL(Rn) and three kind of metric on L-subset groups, and we have proved that (FcL(Rn),TFcL) is a separable space and under the mapping Ï€ = {Ï€_α}α ∈J ( Ï€_α : FcL(Rn) → K₀, J is Copr(L)), we have gotten that if β ∈ σ{TFcL), Ï€(β) ∈ σ(Tph); if A ∈σ (Tph), Ï€^-1(A) ∈ σ(TFcL) (ph is a metric on the product space K₀|J|, Tph is the product topology induced by the metric ph). Finally, we have defined several kinds of L-valued random variables on this basis and discussed preliminary nature of them.