A characterization of tolerance-distributive tree semilattices
Ivan Chajda, Bohdan Zelinka · Czechoslovak Mathematical Journal · 1987
A tolerance on an algebra SU == is a reflexive and symmetric binary relation Ton Ä which has the Substitution Property with respect to J^, i.e., (a^, b^) G e T,..., (a", b") e TimpHes (/(a^, ..., a^), /{b^, ..., b^)) e Tfor each n-ary operation /G #" and any elements a^, ..., a", b^,..., b" of Ä.The set of all tolerances on 51 forms an algebraic lattice LT[^) with respect to the set inclusion (see [4], [5]).Basic properties of this lattice were investigated in [4] and, especially for semilattices, in [5] and [6].An algebra Ж is called congruence-distributive, if the congruence lattice Con{'^) is distributive.It is well-known that lattices and semilattices are congruencedistributive.Although tolerances are a generalization of congruences, the situation with them is quite different.We shall call an algebra Ш tolerance-distributive (or tolerance-modular), if ЬТ{Щ is distributive (or modular, respectively).A class .Jf of algebras is tolerance-distributive (or tolerance-modular), if each Же Jf has this property.It was proved in [2] and [3] that the variety ^ of all distributive lattices is the only non-trivial tolerance-distributive lattice variety.A variety of semilattices is tolerance-modular if and only if it is trivial, see [2].The variety ^ of distributive lattices is the only non-trivial tolerance-modular variety [l].H.-J. Bandelt [l] has investigated a weaker condition: a lattice Lwith the least element О is 0-modular, if it does not contain a minimal non-modular sublattice containing the least element O.He has proved that every lattice Lis tolerance-0-modular (i.e., LT{L) is 0-modular).Our first results for tolerance lattices of semilattices were presented in [5]:(i) The class of all semilattices is not tolerance-0-modular.(ii) The class of all tree semilattices is tolerance-0-modular.Recall that a semilattice 5 is a tree semilattice, if each interval of S (in the induced ordering) is a chain.The result (ii) has motivated our effort to characterize tolerancemodular or tolerance-distributive semilattices among tree semilattices.