The paper carries out a systematic investigation into the axiomatization problem of the asymmetric chi calculus. As a crucial step in attacking the problem, an open style bisimilarity is defined for each of the eighteen L-bisimilarities and the two are proved to be equal. On top of the open bisimilarities, explicit definitions of the eighteen L-congruences are given, which suggest immediately possible axioms for the congruence relations. In addition to the axioms for strong bisimilarity, the paper proposes altogether twenty one additional axioms, three of which being the well-known tau laws and the other eighteen being new. These axioms help to lift a complete system for the strong bisimilarity to complete systems for the eighteen L-congruences.