  Entanglement entropy is now widely accepted as having deep connections with
quantum gravity. It is therefore desirable to understand it in the context of
causal sets, especially since they provide in a natural manner the UV cutoff
needed to render entanglement entropy finite. Formulating a notion of
entanglement entropy in a causal set is not straightforward because the type of
canonical hypersurface-data on which its definition typically relies is not
available. Instead, we appeal to the more global expression given in
arXiv:1205.2953 which, for a gaussian scalar field, expresses the entropy of a
spacetime region in terms of the field's correlation function within that
region (its "Wightman function" $W(x,x')$). Carrying this formula over to the
causal set, one obtains an entropy which is both finite and of a Lorentz
invariant nature. We evaluate this global entropy-expression numerically for
certain regions (primarily order-intervals or "causal diamonds") within causal
sets of 1+1 dimensions. For the causal-set counterpart of the entanglement
entropy, we obtain, in the first instance, a result that follows a (spacetime)
volume law instead of the expected (spatial) area law. We find, however, that
one obtains an area law if one truncates the commutator function ("Pauli-Jordan
operator") and the Wightman function by projecting out the eigenmodes of the
Pauli-Jordan operator whose eigenvalues are too close to zero according to a
geometrical criterion which we describe more fully below. In connection with
these results and the questions they raise, we also study the "entropy of
coarse-graining" generated by thinning out the causal set, and we compare it
with what one obtains by similarly thinning out a chain of harmonic
oscillators, finding the same, "universal" behaviour in both cases.
