On, locally internal monotonic operations

This paper deals with monotonic binary operations F: [0, 1](2) --> [0, 1] with the property (called locally internal property) that the value at any point (x,y) is always one of its arguments x,y. After stating a theorem that characterizes this kind of operations, some special cases are studied in detail by considering additional properties of the operation: commutativity, existence of a neutral element and associativity. In case of locally internal, associative monotonic operations with neutral element, a characterization theorem gives an improvement of a well-known theorem of Czogala and Drewniak on idempotent, associative and increasing operations with neutral element, as well as an improvement of a characterization theorem for left (and right) continuous, idempotent uninorms. (C) 2002 Elsevier Science B.V. All rights reserved.