Descending chains and antichains of the unary, linear, and monotone subfunction relations

Order, 23 (2006), 129-142
http://dx.doi.org/10.1007/s11083-006-9036-y

The C-subfunction relations on the set of operations on a finite base set A defined by function classes C are examined. For certain clones C on A, it is determined whether the partial orders induced by the respective C-subfunction relations have infinite descending chains or infinite antichains. More specifically, we investigate the subfunction relations defined by the clone of all functions on A, the clones of essentially at most unary operations, the clones of linear functions on a finite field, and the clones of monotone functions with respect to the various partial orders on A.