Annihilators in Normal Autometrized Algebras
Ivan Chajda, Jiří Rachůnek · Czechoslovak Mathematical Journal · 2001
The concepts of an annihilator and a relative annihilator in an autometrized l-algebra are introduced. It is shown that every relative annihilator in a normal autometrized l-algebra $$\mathcal{A}$$ is an ideal of $$\mathcal{A}$$ and every principal ideal of $$\mathcal{A}$$ is an annihilator of $$\mathcal{A}$$ . The set of all annihilators of $$\mathcal{A}$$ forms a complete lattice. The concept of an I-polar is introduced for every ideal I of $$\mathcal{A}$$ . The set of all I-polars is a complete lattice which becomes a two-element chain provided I is prime. The I-polars are characterized as pseudocomplements in the lattice of all ideals of $$\mathcal{A}$$ containing I.