Abstract
It is well known that for a chain finite orthomodular lattice, all congruences are factor congruences, so any directly irreducible chain finite orthomodular lattice is simple. In this paper it is shown that the notions of directly irreducible and simple coincide in any variety generated by a set of orthomodular lattices that has a uniform finite upper bound on the lengths of their chains. The prototypical example of such a variety is any variety generated by a set ofn dimensional orthocomplemented projective geometries.
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Harding, J. Irreducible orthomodular lattices which are simple. Algebra Universalis 29, 556–563 (1992). https://doi.org/10.1007/BF01190781
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DOI: https://doi.org/10.1007/BF01190781