  This paper investigates tail-biting trellis realizations for linear block
codes. Intrinsic trellis properties are used to characterize irreducibility on
given intervals of the time axis. It proves beneficial to always consider the
trellis and its dual simultaneously. A major role is played by trellis
properties that amount to observability and controllability for fragments of
the trellis of various lengths. For fragments of length less than the minimum
span length of the code it is shown that fragment observability and fragment
controllability are equivalent to irreducibility. For reducible trellises, a
constructive reduction procedure is presented. The considerations also lead to
a characterization for when the dual of a trellis allows a product
factorization into elementary ("atomic") trellises.
