Black holes and singularities in causal set gravity

A precise definition of a black hole has long been absent in causal set theory. I first show that the local finiteness of the theory cannot be interpreted as a complete discretization condition, and the theory still admits continua, which I call singular anti-chains. After providing a couple of suggestive calculations from general relativity to support the argument that a singular anti-chain behaves like a causal singularity in a space-time, I propose a definition of a causal set black hole based on a singular anti-chain. The paper concludes with brief discussions of questions that causal set black holes and singularities may give rise to.