Zariski Topology on L-Slices
Mary Elizabeth Antony1, Sabna.K.S2, Manglambal N.R3
1Mary Elizabeth Antony, Research Scholar, Department of Mathematical Science, ST Joseph’s College, Irinjalakuda. Kerala, India.
2Sabna K.S, Research Scholar, Department of Mathematical Science, ST Joseph’s College, Irinjalakuda. Kerala, India.
3Manglambal N.R Research Scholar, Department of Mathematical Science, ST Joseph’s College, Irinjalakuda. Kerala, India.
Manuscript received on 01 February 2019 | Revised Manuscript received on 07 February 2019 | Manuscript Published on 13 February 2019 | PP: 274-276 | Volume-8 Issue- 4S February 2019 | Retrieval Number: DS2874028419/2019©BEIESP
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© The Authors. Blue Eyes Intelligence Engineering and Sciences Publication (BEIESP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: Ahe action of a locale 𝑳 on a join semilattice 𝑱 gives us the newly introduced notion of 𝑳-slices (𝝈, 𝑱) .We have tried to extend the idea of Zariski topology on modules to 𝑳-slices. Given a locale 𝑳 and a L-slice (𝝈, 𝑱) , for 𝒎 ∈ (𝝈, 𝑱)and 𝒓 ∈ 𝑳 ,we have constructed (𝝈, 𝑱) ideals 𝒓 ↳ 𝒎 = 𝒏 ∈ 𝝈, 𝑱 𝝈 𝒓, 𝒏 ≤ 𝒎}. Their properties and characteristics are studied. Similarly, for a given L-slice (𝝈, 𝑱) and 𝒏, 𝒎 ∈ 𝝈, 𝑱 , we examine the properties of 𝑳 -ideals [𝒓 ↳ 𝒏] = 𝒓 ∈ 𝑳 𝝈 𝒓, 𝒏 ≤ 𝒎} . We introduce the notion of 𝑳-prime elements on (𝝈, 𝑱 ) and their properties are discussed. The collection of L-prime elements is defined as Spec((𝝈, 𝑱) and we examine the possibility of existence of Zariski topology on it.
Keywords: Join Semilattice, L-Slices, Zariski Topology, L-Ideals.
Scope of the Article: Applied Mathematics and Mechanics