Lattice of full soft Lie algebra

Subject Areas : Statistics

A. Akbar Estaji ¹ , H. Eghdami ² , T. Haghdadi ³

1 - Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
2 - Faculty of Mathematical Science and Statistics, University of Birjand, Birjand, Iran.
3 - 1) Faculty of Mathematical Science and Statistics, University of Birjand, Birjand, Iran. 2) Faculty of Basic Siences, Birjand University of Technology, Birjand, Iran

Received: 2017-11-30 Accepted : 2018-09-01 Published : 2018-10-23

Keywords: مشبکه, جبر لی, اتم, مشبکه فشرده,

Abstract :

In ‎this ‎paper, ‎we ‎study ‎the ‎relation ‎between ‎the ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎with ‎the ‎lattice theory. ‎We ‎introduce ‎the ‎concepts ‎of ‎the ‎lattice ‎of ‎soft ‎sets, ‎full ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎and next, we ‎verify ‎some ‎properties ‎of ‎them. We ‎prove ‎that ‎the ‎lattice ‎of ‎the ‎soft ‎sets ‎on ‎a fixed parameter set is isomorphic to the power set of a set with respect to set inclusion. In ‎particular, ‎we ‎characterize ‎the ‎atoms ‎in ‎the ‎lattice ‎of ‎the ‎soft ‎sets and the lattice of full soft Lie algebras. ‎After that, ‎we ‎introduce ‎the ‎compact ‎elements in the full soft Lie algebras ‎and ‎we present the necessary and sufficient conditions for the compactness and atomicness ‎of the lattice of full soft Lie algebras. We show that if a full soft Lie algebra is a compact element of the ‎lattice ‎of full soft Lie algebra then the parameter set is finite and the Lie algebra is finitely generated. In the sequel, we study the relationship between prime ‎and ‎maximal ‎ideals in Lie algebras and the prime ‎and ‎maximal ‎elements in the lattice of soft Lie algebras.

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