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

Uniqueness and non-uniqueness of chains on half lines