Minors of Boolean functions with respect to clique functions and hypergraph homomorphisms

Each clone C on a fixed base set A induces a quasi-order on the set of all operations on A by the following rule: f is a C -minor of g if f can be obtained by substituting operations from C for the variables of g . By making use of a representation of Boolean functions by hypergraphs and hypergraph homomorphisms, it is shown that a clone C on { 0 , 1 } has the property that the corresponding C -minor partial order is universal if and only if C is one of the countably many clones of clique functions or the clone of self-dual monotone functions. Furthermore, the C -minor partial orders are dense when C is a clone of clique functions.