Commutative, associative and monotone functions on horizontal sum of chains

The commutative, associative and monotone (CAM) functions on a horizontal sum of bounded chains are studied. Depending on their range, these functions are divided into two main classes and properties of CAM functions from each of these classes are described. Several necessary and sufficient conditions under which a CAM function defined on a horizontal sum of bounded chains can be expressed as a non-trivial (z)-ordinal sum of semigroups are also introduced. Special classes of CAM functions are discussed and several examples are included. Various construction methods for special CAM functions defined on a horizontal sum of bounded chains, such as t-norms, t-conorms, uninorms and nullnorms, are also presented.