Random sets, q-distributions and quantum groups

Probabilistic Fuzzy Sets and a Related Operator Algebra

“…It can then be shown that taking the union of A with itself n times gives rise to a random set with an average (Arik et al, 1997). The random set obtained in this case is a very special example.…”

“…With the introduction of q-oscillators (Arik and Coon, 1976;Biedenharn, 1989;Bonatsos et al, 1991;Chakrabarti and Jagannathan, 1991;Chaichian, 1990;Chaichian et al, 1991;Fairlie and Zachos, 1991;Hayashi, 1990;Jannussis et al, 1991;Macfarlane, 1989) and quantum groups (Drinfeld, 1986;Fadeev et al, 1989;Jimbo, 1986;Woronowicz, 1987), algebras which do not fall into these categories, but in the limit q → 1 of the deformation parameter reduce to Lie algebra or Lie super-algebra were considered. This paper may be considered as a first step in establishing a relationship between fuzzy sets and quantum physics.…”

“…This algebra, up to a normalization of a, is the q-oscillator algebra and its nontrivial representation (Arik and Coon, 1976) is given by…”

“…In most physical applications 1 − q ∝ h where h is the Planck's constant. Independent of these models but motivated by nonextensive statistical mechanics (Erzan, 1997;Tsallis, 1994) and using the framework of random sets an identification 1 − q = 1 M was made (Arik et al, 1997), where M is the number of elements in the universal set. It was shown that random sets obtained by choosing a random element from the universal set are created by a q-oscillator creation operator and a union of n such random sets has SU q (n) quantum group symmetry (Arik et al, 1997).…”

“…Independent of these models but motivated by nonextensive statistical mechanics (Erzan, 1997;Tsallis, 1994) and using the framework of random sets an identification 1 − q = 1 M was made (Arik et al, 1997), where M is the number of elements in the universal set. It was shown that random sets obtained by choosing a random element from the universal set are created by a q-oscillator creation operator and a union of n such random sets has SU q (n) quantum group symmetry (Arik et al, 1997). When such random sets are transformed into fuzzy sets by taking the one point trace as explained below, the parameter q is fixed and it is related to the membership function µ of an element of the fuzzy set by µ = 1 − q = 1 M .…”