  An optical cluster finder inevitably suffers from projection effects, where
it misidentifies a superposition of galaxies in multiple halos along the
line-of-sight as a single cluster. Using mock cluster catalogs built from
cosmological N-body simulations, we quantify the impact of these projection
effects with a particular focus on the observables of interest for cluster
cosmology, namely the cluster lensing and the cluster clustering signals. We
find that "observed" clusters, i.e. clusters identified by our cluster finder
algorithm, exhibit lensing and clustering signals that deviate from
expectations based on a statistically isotropic halo model -- while both
signals agree with halo model expectations on small scales, they show
unexpected boosts on large scales, by up to a factor of 1.2 or 1.4
respectively. We identify the origin of these boosts as the inherent selection
bias of optical cluster finders for clusters embedded within filaments aligned
with the line-of-sight, and show that a minority ($\sim 30\%$) of such clusters
within the entire sample is responsible for this observed boost. We discuss the
implications of our results on previous studies of optical cluster, as well as
prospects for identifying and mitigating projection effects in future cluster
cosmology analyses.
