A conceptual breakthrough of Jacob and Monod was to recognize the distinction between regulatory substances and sites where they act to change gene expression.  A former soldier, Jacob used the analogy of a bomber that would release its lethal cargo upon receipt of a special radio transmission or signal.  A working system requires both a ground transmitter and a receiver in the airplane.  Now, suppose that the usual transmitter is broken.  This system can be made to work by introduction of a second, functional transmitter.  In contrast, he said, consider a bomber with a defective receiver.  The behavior of this bomber cannot be changed by introduction of a second, functional aeroplane.
To analyze regulatory mutants of the lac operon, Jacob developed a system by which a second copy of the lac genes (lacI with its promoter, and lacZYA with promoter and operator) could be introduced into a single cell.  A culture of such bacteria, which are diploid for the lac genes but otherwise normal, is then tested for the regulatory phenotype.  In particular, it is determined whether LacZ and LacY are made even in the absence of IPTG (due to the lactose repressor produced by the mutant gene being non-functional).  This experiment, in which genes or gene clusters are tested pairwise, is called a complementation test.
This test is illustrated in the figure (lacA is omitted for simplicity).  First, certain haploid states are shown (i.e. the cell carries only a single copy of the lac genes).  Panel (a) shows repression, (b) shows induction by IPTG, and (c) and (d) show the effect of a mutation to the lacI gene or to the operator, respectively.  In panel (e) the complementation test for repressor is shown.  If one copy of the lac genes carries a mutation in lacI, but the second copy is wild type for lacI, the resulting phenotype is normal—but lacZ is expressed when exposed to inducer IPTG.  Mutations affecting repressor are said to be recessive to wild type (and that wild type is dominant), and this is explained by the fact that repressor is a small protein which can diffuse in the cell.  The copy of the lac operon adjacent to the defective lacI gene is effectively shut off by protein produced from the second copy of lacI.
If the same experiment is carried out using an operator mutation, a different result is obtained (panel (f)).  The phenotype of a cell carrying one mutant and one wild type operator site is that LacZ and LacY are produced even in the absence of the inducer IPTG; because the damaged operator site, does not permit binding of the repressor to inhibit transcription of the structural genes.  The operator mutation is dominant.  When the operator site where repressor must bind is damaged by mutation, the presence of a second functional site in the same cell makes no difference to expression of genes controlled by the mutant site.
A more sophisticated version of this experiment uses marked operons to distinguish between the two copies of the lac genes and show that the unregulated structural gene(s) is(are) the one(s) next to the mutant operator (panel (g).  For example, suppose that one copy is marked by a mutation inactivating lacZ so that it can only produce the LacY protein, while the second copy carries a mutation affecting lacY and can only produce LacZ.  In this version, only the copy of the lac operon that is adjacent to the mutant operator is expressed without IPTG.  We say that the operator mutation is cis-dominant, it is dominant to wild type but affects only the copy of the operon which is immediately adjacent to it.
This explanation is misleading in an important sense, because it proceeds from a description of the experiment and then explains the results in terms of a model.  But in fact, it is often true that the model comes first, and an experiment is fashioned specifically to test the model.  Jacob and Monod first imagined that there must be a site in DNA with the properties of the operator, and then designed their complementation tests to show this.
The dominance of operator mutants also suggests a procedure to select them specifically.  If regulatory mutants are selected from a culture of wild type using phenyl-Gal, as described above, operator mutations are rare compared to repressor mutants because the target-size is so small.  But if instead we start with a strain which carries two copies of the whole lac region (that is diploid for lac), the repressor mutations (which still occur) are not recovered because complementation by the second, wild type lacI gene confers a wild type phenotype.  In contrast, mutation of one copy of the operator confers a mutant phenotype because it is dominant to the second, wild type copy.