Fuzzy boundary and characteristic properties of order-homomorphisms (Q756180)

A fuzzy topological space (L(M),\(\delta\)) is studied where L is a completely distributive lattice with an order-reversing involution, M the collection of all molecules in L and \(\delta\) a topology on L. The concept \(\partial A\) of the boundary of an element A in a fuzzy topological space (L,\(\delta\)) is introduced as are the concepts of boundary-preserving, interior-preserving and closure-preserving order- homomorphisms. All these concepts and the relations among them are studied. Some of the results are as follows: \(\partial (A\vee B)\leq \partial A\vee \partial B\), \(\partial (A\wedge B)\leq \partial A\vee \partial B\). Equivalents of continuous (open, closed) order-homomorphisms \(f: (L_ 1,\delta_ 1)\to (L_ 2,\delta_ 2)\) are given. Let \(f: (L_ 1,\delta_ 1)\to (L_ 2,\delta_ 2)\) be an order-homomorphism and a bijection; there are 27 equivalences for f to be a homeomorphism.