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\(D\)-posets of fuzzy sets. (English) Zbl 0797.04011
A partially ordered set \(F\) is \(D\)-poset, if a partial binary operation - - is given defining \(a-b\) whenever \(b\leq a\) in such a way that some axioms are satisfied: (i) \(a- b\leq a\), (ii) \(a- (a- b)= b\), (iii) \(a\leq b\leq c\) implies \(c- b\leq c- a\) and \((c- a)- (c-b)= b- a\). In the paper, the author considers the special case of a \(D\)-poset, where elements of \(F\) are real functions \(f: X\to \langle 0,1\rangle\). In the paper the probability theory on \(D\)-posets of fuzzy sets is exposed. Of course, the general analogue is also known and it has been realized in the paper by F. Chovanec and the author [Math. Slovaca 44, No. 1, 21-34 (1994; Zbl 0789.03048)].

03E72 Theory of fuzzy sets, etc.
06A06 Partial orders, general
60B99 Probability theory on algebraic and topological structures
03G12 Quantum logic
28E10 Fuzzy measure theory