Regular Filter, Associated Filter and Their Properties
Mary Elizabeth Antony1, Sabna K.S2, Mangalambal N.R3

1Mary Elizabeth Antony, Centre for Research in Mathematical Sciences, Calicut University, St.Joseph’s College, Irinjalakuda, Department of Mathematics, Mar Athanasius College, Kothamangalam (Kerala), India
2Sabna K.S, Department of Mathematics, Calicut University/ K.K.T.M. Government College, Pullut (Kerala), India
3Mangalambal N.R. Department of Mathematics, Calicut University, St. Joseph’s College, Irinjalakuda (Kerala), India

Manuscript received on 18 April 2019 | Revised Manuscript received on 25 April 2019 | Manuscript published on 30 April 2019 | PP: 1645-1649 | Volume-8 Issue-4, April 2019 | Retrieval Number: D6761048419/19©BEIESP
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Abstract: The action 𝝈 ∶ 𝑳 × 𝑱 → 𝑱 of a locale 𝑳 on a join semi-lattice 𝑱, with bottom element 𝟎𝑱 , establishes the concept of L-slices (𝝈, 𝑱)which was introduced in [3]. Through the action , the join semilattice displays significant changes in their structure. Our study is based on the set of all elements of the locale which leaves 𝒙 ∈ (𝝈, 𝑱) fixed, under the action 𝝈 .This set is denoted as 𝑭𝒙 . Also,𝑭𝒙 = {𝒂 ∈ 𝑳 |𝝈(𝒂, 𝒙) = 𝒙}is a filter on the locale L .In this study ,we illustrate the properties of the filter𝑭𝒙of the locale generated by the L-slices . We put forth a study of two different types of filters, called the regular filter and the associated filter. The properties of these filters are studied. The collection of regular filters is separated into equivalence classes. Analogous to sequential continuity in topological spaces , we define the continuity of slice morphisms in terms of these filters. Also, the regular and associated filters on the locale L helps us characterise a new type of L-slices called the R-A slice.
Keywords: Locale, Filters, L-slices , R-A slice

Scope of the Article: Applied Mathematics and Mechanics