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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)].

##### MSC:
 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
##### Keywords:
probability theory on $$D$$-posets of fuzzy sets