Fuzzy semistar operations of finite character on integral domains

This paper is devoted to the construction of examples of fuzzy semistar operations on integral domains. A special case of interest is the fuzzy semistar operation of finite character. We show that the set of fuzzy semistar operations of finite character on an integral domain is a complete lattice. We also prove that the fuzzy semistar operation on an integral domain defined by a finite collection of overrings and union preserving fuzzy semistar operations of finite character on the overrings is of finite character. An example is constructed to prove that if the collection of overrings is infinite, the corresponding fuzzy semistar operation may not be of finite character. We describe the set of fuzzy semistar operation of finite character on valuation domains.