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Tolerances on median algebras. (English) Zbl 0538.08003

A ternary algebra M whose ternary operation (abc) satisfies the identities $\left(aab\right)=a,\phantom{\rule{1.em}{0ex}}\left(abc\right)=\left(bac\right)=\left(bca\right),$ and $\left(\left(abc\right)de\right)=\left(a\left(bde\right)\left(cde\right)\right)$ for all $a,b,c,d,e\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}M$ is called a median algebra. Several papers have studied tolerances, reflexive and symmetric compatible relations on algebras. These relations are well understood for distributive lattices, median algebras, and tree algebras (median algebras in which any (abd), (acd), (bcd) are not distinct).

This paper gives the main facts, provides simple proofs, and extends some previous results concerning tolerances. The following are typical results. If $\xi$ is a reflexive, symmetric relation on a tree algebra, then $\xi$ is a tolerance if and only if all blocks of $\xi$ are convex. A median algebra has the tolerance extension property if and only if it is a tree algebra. The lattice of all tolerances on a median algebra is distributive.

Reviewer: G.A.Fraser
##### MSC:
 08A30 Subalgebras of general algebraic systems 08A05 Structure theory of general algebraic systems 20N10 Ternary systems (group theory) 06A12 Semilattices 06D05 Structure and representation theory of distributive lattices