Skip to main content

On Disjoint Sums in the Lattice of Linear Topologies


Let M be a vector space over a skew-field equipped with the discrete topology, \(\mathcal{L}\)(M) be the lattice of all linear topologies on M ordered by inclusion,and τ*, τ0, τ1\(\mathcal{L}\)(M). We write τ1 = τ* ⊔ τ0 or say that τ1 is a disjoint sum of τ* and τ0 if τ1 = inf{τ0, τ*} and sup{τ0, τ*} is the discrete topology. Given τ1, τ0\(\mathcal{L}\)(M), we say that τ0 is a disjoint summand of τ1 if τ1 = τ* ⊔ τ0 for a certain τ*\(\mathcal{L}\)(M). Some necessary and some sufficient conditions are proved for τ0 to be a disjoint summand of τ1.

This is a preview of subscription content, access via your institution.


  1. 1.

    V. I. Arnautov, S. T. Glavatski, and A. V. Mikhalev, Introduction to the Theory of Topological Rings and Modules, Marcel Dekker, New York (1996).

    Google Scholar 

  2. 2.

    V. I. Arnautov and K. M. Filippov, “On coverings in the lattice of module topologies,” Buletinul Academiei de Stiinte a Republicii Moldova, Matematica, No. 2(30), 7–18 (1999).

    Google Scholar 

  3. 3.

    G. Birkhoff, Lattice Theory, Providence, Rhode Island (1967).

    Google Scholar 

  4. 4.

    S. Warner, Topological Rings, North-Holland, Amsterdam (1993).

    Google Scholar 

Download references

Author information



Additional information


Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 3–18, 2003.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Arnautov, V.I., Filippov, K.M. On Disjoint Sums in the Lattice of Linear Topologies. J Math Sci 128, 3335–3344 (2005).

Download citation


  • Vector Space
  • Discrete Topology
  • Linear Topology