Some Topological Properties Of The Spectrum Of Prime D-Filters Of Distributive Lattices
i-manager’s Journal on Mathematics · 2021
In 1968, the theory of relative annihilators has been introduced in lattices by Mandelker (1970) and he characterized distributive lattices in terms of their relative annihilators. Later, many authors introduced the concept of annihilators in the structures of rings as well as lattices and characterized several algebraic structures in terms of annihilators. Speed (1969) and Cornish (1972 Cornish ( , 1974 Cornish ( , 2009) ) made an extensive study of annihilators in distributive lattices and characterized some algebraic structures like normal lattices and quasi complemented lattices. In 2012, Rao (2012) introduced the concept of normal filters in distributive lattices. Rao investigated some properties of the space of all prime normal filters of distributive lattices in topological sense. In 2013 , Rao et al. (2013) introduced the notion of Boolean filters of distributive lattices and then derived a set of equivalent conditions for a pseudo-complemented distributive lattice to become a Boolean algebra. Recently in 2013, the concepts of e-filters and D-filters (Rao, 2013) are introduced and characterized in MS-algebras. Some topological properties of the class of all prime e-filters of MS-algebras are studied. In this paper, the author introduced the concept of D-prime filters and then a set of equivalent conditions is derived for a prime e-filter to become a D-prime filter in topological terms. In 2016, Rao and Badawy (2016) studied the properties of co-annihilator filters of distributive lattices. In this paper, the authors investigated some topological properties of the space of all prime µ-filters of distributive lattices. A necessary and sufficient condition has been derived for the space of all prime µ-filters of a distributive lattice to be a Hausdorff space (Katrinak, 1966) . In this paper, some properties of prime D-filters are investigated in terms of the Cartesian products of distributive lattices. Prime D-filters of product lattices are characterized. Some properties of minimal prime D-filters of distributive lattices are given. For any collection Y of minimal prime D-filters of a distributive lattice, the closure of Y is characterized. An equivalent condition is derived for the space of all minimal prime D-filters of a distributive lattice to become a compact set. A necessary and sufficient condition is given for the space of all minimal prime D-filters of a distributive lattice to become a closed set. Finally, it is proved that the space of all minimal prime D-filters of a distributive lattice is a Hausdorff space. (1967) and Sankappanavar and Burris (1981) are to be referred for the elementary notions and notations of