Classical causal set theory models spacetime causality by locally finite partial orders. Because partial orders are antisymmetric, they cannot represent distinct elements that are mutually causally related, as would occur in a discrete analog of a chronology-violating region. We develop a locally finite discrete framework that allows closed causal loops and yields a locally finite partial order upon identifying mutually related elements. Each class of mutually related elements is collapsed to one point in the induced poset, and its cardinality is assigned as a finite weight to the corresponding point. Within this setting we define maximum causal time using simple finite paths and prove an equidistance characterization: members of one chronology-violating class have the same discrete proper-time distance from any event in their common past or future, with a finite correction for the closed path inside the class. We also study the special case in which the relation is generated by iterating a single self-map $f:X\to X$; cycles of $f$ then give explicit finite closed causal loops. Finally, we define two finite counting invariants: the proportion of elements in chronology-violating classes and the normalized mean of $|I(x,x)|-1$. We study the mathematical properties of these discrete causal structures; no equations of motion or probabilities for their occurrence are specified. The construction does not assert a continuum-discrete correspondence in the chronology-violating case, and ordinary causets are recovered when all equivalence classes are singletons.

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• Quantum gravity
• Spacetime topology & causal structure