Continuity in quantitative domains

Scott convergence and fuzzy Scott topology on L-posets

“…Fuzzy order was first introduced by Zadeh [25], from then on, different kinds of fuzzy order have been introduced and studied by different authors (the reader is referred to [4,12,15,16,[26][27][28]). In this paper, we adopt the definition of fuzzy order in [15,16].…”

“…Definition 2.7 ( [16,19,21,22]). Let .X; e/ be an L-poset, A 2 L X , x 0 2 X. x 0 is called a fuzzy join (resp., fuzzy meet) of A denoted by x 0 D tA (resp.,…”

“…Wagner's -categories [12], Rutten's generalized metric spaces [13] and Flagg's continuity spaces [14] are examples of quantitative domain theory. Fan and Zhang [15,16] studied quantitative domain via fuzzy set theory, where fuzzy partial order was clearly proposed. After analysis, it is easily seen that -categories could be regarded as a fuzzy preordered set in [15,16], and a fuzzy partial ordered set (an L-poset) is equivalent to an L-ordered set introduced by Bělohlávek [17,18].…”

“…Fan and Zhang [15,16] studied quantitative domain via fuzzy set theory, where fuzzy partial order was clearly proposed. After analysis, it is easily seen that -categories could be regarded as a fuzzy preordered set in [15,16], and a fuzzy partial ordered set (an L-poset) is equivalent to an L-ordered set introduced by Bělohlávek [17,18]. Later on, Lai and Zhang [19,20] studied complete and directed-complete -categories, and their continuity was also discussed.…”

“…Later on, Lai and Zhang [19,20] studied complete and directed-complete -categories, and their continuity was also discussed. Following [15,16,19], Yao [21,22] studied the continuity of fuzzy dcpos, and further extended the Scott convergence and Scott topology on classical dcpos to fuzzy setting. But the results in [22] do not adapt to fuzzy partially ordered sets as well as in the classical case, even the continuity needs to be modified.…”