Tolerances on median algebras. (English) Zbl 0538.08003

Bandelt, Hans-J.

Czech. Math. J. 33(108), 344-347 (1983).

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).
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, congruence relations
08A05	Structure theory of algebraic structures
20N10	Ternary systems (heaps, semiheaps, heapoids, etc.)
06A12	Semilattices
06D05	Structure and representation theory of distributive lattices

Keywords:

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

References:

[1]	H.-J. Bandelt, J. Hedlíková: Median algebras. Discrete Math. 45 (1983), 1-30. · Zbl 0506.06005 · doi:10.1016/0012-365X(83)90173-5
[2]	I. Chajda: On the tolerance extension property. Časopis pěst. mat. 103 (1978), 327-332. · Zbl 0391.06008
[3]	I. Chajda, B. Zelinka: Lattices of tolerances. Časopis pěst. mat. 102 (1977), 10-24. · Zbl 0354.08011
[4]	I. Chajda, B. Zelinka: Minimal compatible tolerances on lattices. Czech. Math. J. 27 (1977), 452-459. · Zbl 0379.06002
[5]	J. Niederle: Relative bicomplements and tolerance extension property in distributive lattices. Časopis pěst. mat. 103 (1978), 250-254. · Zbl 0393.06005
[6]	J. Nieminen: Tolerance relations on simple ternary algebras. Archivum Math. (Brno) 13 (1977), 105-110. · Zbl 0368.08001
[7]	J. Nieminen: The ideal structure of simple ternary algebras. Colloq. Math. 40 (1978), 23-29. · Zbl 0415.06002
[8]	M. Sholander: Trees, lattices, order, and betweenness. Proc. Amer. Math. Soc. 3 (1952), 369-381.
[9]	M. Sholander: Medians, lattices, and trees. Proc. Amer. Math. Soc. 5 (1954), 808-812. · Zbl 0056.26201 · doi:10.2307/2031872
[10]	B. Zelinka: Tolerances and congruences on tree algebras. Czech. Math. J. 25 (1975), 634 to 637. · Zbl 0327.08002