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Zbl 0538.08003
Bandelt, Hans-J.
Tolerances on median algebras. (English)
[J] Czech. Math. J. 33(108), 344-347 (1983). ISSN 0011-4642; ISSN 1572-9141/e

A ternary algebra M whose ternary operation (abc) satisfies the identities $(aab)=a,\quad(abc)=(bac)=(bca),$ and $((abc)de)=(a(bde)(cde))$ for all $a,b,c,d,e\quad in\quad M$ 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). \par 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.
[G.A.Fraser]

MSC 2000:: *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

Keywords: ternary algebra; tolerances; median algebras; tree algebras; tolerance extension property

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