  We study the magnetic and thermodynamic properties of a spin-$1/2$ Ising
system containing three layers, each of which is composed exclusively of one
out of two possible types of atoms, \textbf{A} or \textbf{B}. The \textbf{A-A}
and \textbf{B-B} bonds are ferromagnetic while the \textbf{A-B} bonds are
antiferromagnetic. The study is performed through Monte Carlo simulations using
the Wolff algorithm and the data are analyzed with the aid of the
multiple-histogram reweighting technique and finite-size scaling tools. We
verify the occurrence of a compensation phenomenon and obtain the compensation
and critical temperatures of the model as functions of the Hamiltonian
parameters. The influence of each parameter on the overall behavior of the
system is discussed in detail and we present our results in the form of phase
diagrams dividing the parameter space in regions where the compensation
phenomenon is present or absent. Our results may provide invaluable information
for experimentalists seeking to build materials with desired characteristics.
