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Keywords:
De Morgan residuated lattice; $\alpha $-ideal; prime $\alpha $-ideal; quasi-complemented De Morgan residuated lattice; hull-kernel topology
De Morgan residuated lattice; $\alpha $-ideal; prime $\alpha $-ideal; quasi-complemented De Morgan residuated lattice; hull-kernel topology
Summary:
The main purpose of this paper is to study the lattice structure of $\alpha$-ideals and investigate the hull-kernel topology on prime $\alpha$-ideals in a De Morgan residuated lattice. First, the lattice structure of all $\alpha$-ideals in a De Morgan residuated lattice is studied and it is proved that the set of all $\alpha$-ideals forms a complete Heyting algebra. Furthermore, by means of annihilator operators, the concept of quasi-complemented De Morgan residuated lattices is proposed and some equivalent conditions are derived for them. Finally, the space of all prime $\alpha$-ideals is studied and it is stated that the space is a spectral space. In addition, some conditions are given for the space to be a Hausdorff space. Some topological characterizations of quasi-complemented De Morgan residuated lattices are demonstrated.
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