  We establish a one-to-one correspondence between one-sided and two-sided
regular systems of conditional probabilities on the half-line that preserves
the associated chains and Gibbs measures. As an application, we determine
uniqueness and non-uniqueness regimes in one-sided versions of ferromagnetic
Ising models with long range interactions. Our study shows that the interplay
between chain and Gibbsian theories yields more information than that contained
within the known theory of each separate framework. In particular: (i) A
Gibbsian construction due to Dyson yields a new family of chains with phase
transitions; (ii) these transitions show that a square summability uniqueness
condition of chains is false in the general non-shift-invariant setting, and
(iii) an uniqueness criterion for chains shows that a Gibbsian conjecture due
to Kac and Thompson is false in this half-line setting.
