Fuzzy posets on sets
The principal notions introduced and studied in this paper are the notions of fuzzy posets and fuzzy dimensions (for a special class of fuzzy posets). A set \(X\) equipped with functions \(L\), \(G\), \(P\) (``less than'', ``greater than'' respectively ``parallel'', or incompatibility functions) defined on \(X\times X\) and having \([0,1]\) as value set, satisfying the condition \(L+G+P=1\), is called a fuzzy poset. Several standard theorems for posets are proved in the setting of fuzzy posets. Furthermore, using a rounding-off procedure, the skeleton of a fuzzy poset is defined, and results on the behaviour of deformations of skeletons of fuzzy posets as it relates to the structure of these fuzzy posets are proved. The fuzzy dimension is defined as the intersection of the least number of fuzzy chains which yield the fuzzy poset. It is clear that it is possible to arrive in this manner at a reasonable theory of fuzzy posets.
scientific article; zbMATH DE number 1563543