SOME PROPERTIES OF H-IRREDUCIBLE LATTICES
Joanna Grygiel · 2004
We describe the method of decomposing lattices via tolerance relations and introduce the notion of H-irreducibility. Then we characterize some classes of H-irreducible lattices, which appears to be useful for description of blocks of the skeleton tolerance of flnite modular and distributive lattices. In the case of big and complex flnite lattices, which often arise in data processing, it is sometimes useful to split the lattice up into simpler parts. One of methods of decomposing lattices is provided by tolerance relations, which can be regarded as a natural generalization of equivalence relations. The method is especially valuable in the case of modular lattices as the smallest glued tolerance (so called skeleton tolerance) decomposes a flnite modular lattice into blocks being its maximal atomistic intervals (and the maximal boolean intervals if the lattice is distributive). These intervals can be glued up together into the former lattice by use of construction described by Herrmann (see [8]). As we are going to show here, atomistic intervals are H-irreducible, i.e. their skeleton tolerance does not decompose them. We consider other examples of H-irreducible lattices and prove that in the non-modular case the blocks of the skeleton tolerance need not be H-irreducible.