Subfunction relations defined by the clones containing all unary operations

Abstract

For a class C of operations on a nonempty base set A, an operation f is called a C-subfunction of an operation g, if f = g(h_1, ..., h_n), where all the inner functions h_i are members of C. Two operations are C-equivalent if they are C-subfunctions of each other. The C-subfunction relation is a quasiorder if and only if the defining class C is a clone. The C-subfunction relations defined by clones that contain all unary operations on a finite base set are examined. For each such clone it is determined whether the corresponding partial order satisfies the descending chain condition and whether it contains infinite antichains.