A Note on Distributive Triples

Marcin Łazarz

doi:10.18778/0138-0680.48.3.04

Authors

Marcin Łazarz University of Wrocław, Poland, Department of Logic and Methodology of Sciences

DOI:

https://doi.org/10.18778/0138-0680.48.3.04

Keywords:

Distributive triple, dually distributive triple, covering diamond

Abstract

Even if a lattice L is not distributive, it is still possible that for particular elements x, y, z ∈ L it holds (x∨y) ∧z = (x∧z) ∨ (y ∧z). If this is the case, we say that the triple (x, y, z) is distributive. In this note we provide some sufficient conditions for the distributivity of a given triple.

References

G. Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, Vol. XXV, Providence, Rhode Island (1973).
Google Scholar

E. Fried, G. Grätzer, H. Lakser, Projective geometries as cover-preserving sublattices, Algebra Universalis, Vol. 27 (1990), pp. 270–278.
Google Scholar

G. Grätzer, General lattice theory, Birkhäuser, Basel, Stuttgart (1978).
Google Scholar

G. Grätzer, Lattice Theory: Foundation, Birkhäuser, Basel (2011).
Google Scholar

C. Herrmann, A. P. Huhn, Lattices of normal subgroups which are generated by frames, [in:] Proceedings of the Lattice Theory Colloquium, Szeged 1974, Colloq. Math. Soc. János Bolyai, 14, pp. 97–136, North-Holland, Amsterdam (1976).
Google Scholar

B. Jónsson, Equational Classes of Lattices, Mathematica Scandinavica, Vol. 22 (1968), pp. 187–196.
Google Scholar

M. Stern, Semimodular Lattices. Theory and Applications, Cambridge University Press (1999).
Google Scholar