Probability Distributions on Temporal Semigroups Probability structures in which [0, ∞) plays the role of time are very important in applications such as reliability, lifetime random variables, renewal processes, and stationarity. This dissertation studies some analogous probability structures on a class of semigroups called temporal semigroups, which capture some of the essential features of [0, ∞) as a model of time. A temporal semigroup has a minimum element, a semigroup operation, a compatible partial order, and a left-invariant measure. Gamma random variables are defined on temporal semigroups and their distribution is shown to have a nice representation. These distributions are used to examine analogs of residual life and total life random variables. Stationary point processes are examined. A characterization using the Laplace functional is determined and used to develop a means for generating new stationary point processes. A simple connection between stationary point processes and the exponential distribution is developed. Temporal semigroups provide additional insight into aging properties. Increasing failure rate, increasing failure rate average, and new better than used aging properties are examined on temporal semigroups isomorphic to ([0,∞), +). Relationships between entropy and the algebraic structure of the free semigroup are studied. Exponential distributions are shown to maximize entropy in a certain sense.