Scattered through the literature various structures occur that generalize median graphs and median algebras: quasi‐median graphs, retracts of Hamming graphs, graphs of finite windex, isotropic media, subdirect products of simplex algebras, certain ternary algebras, and so on. We connect them all up, provide some new generalizations of quasi‐median graphs, some new sets of axioms for quasi‐median algebras, and some shorter proofs as well.
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Bandelt et al. (1994) studied this question.