Ismael Akray & Haval M. Mohammed Salih

Department of Mathematics, Faculty of Science, Soran University/Erbil/Iraq

Department of Mathematics, Faculty of Science, Soran University/Erbil/Iraq

DOI: https://doi.org/10.17656/jzs.10576

In this paper, we define Rad-supplemented lattice and its dual soc-supplemented lattice. Furthermore, we show that in compactly generated lattices the first one is equivalent to supplemented lattice. Also, we study some properties of soc-supplemented lattices. Finally, we define cofinitely soc-supplemented lattice (briefly css-lattice) and proved that an arbitrary join of css-lattices is a css-lattice.

Supplemented lattice; radical and socle of lattice; cofinite element

[1] Ismael Akray, Adil Kadir Jabbar and Reza Sazeedeh, "On Soc- -s-modules". Journal of Koya Uiversity, Vol. 24, No. 6, pp 73 – 90. (2012).

[2] Ismael Akray, "Cofinitely soc-supplemented modules". Journal of Garmian University, Vol. 2, No. 2, pp 23 – 32. (2015).

[3] G. Birkhoff, "Lattice theory". American Mathematical society, (1948).

[4] G. Calugareanu, "Lattice Concepts of Module Theory". Kluwer Texts in the Mathematical Sciences (2000).

[5]B. A. Davey and H. A. Priestley, "Introduction to lattices and order". Cambridge University Press (2002).

[6]M. L. Galvao and P. F. Smith, "Chain conditions in modular lattices", Coll. Math., Vol. 76, No. 1, pp 85-98. (1998).

[7]D. Keskin, "An approach to extending and lifting modules by modular lattices", Indian J. Pure Appl. Math., Vol. 33, No. 1, pp 81-86. (2002).

[8]B. Stenstrom, "Radicals and socles of lattices", Arch. Math., Vol. 20, pp 258-261. (1969).

Department of Mathematics, Faculty of Science, Soran University/Erbil/Iraq

Department of Mathematics, Faculty of Science, Soran University/Erbil/Iraq

DOI: https://doi.org/10.17656/jzs.10576

**Abstract**In this paper, we define Rad-supplemented lattice and its dual soc-supplemented lattice. Furthermore, we show that in compactly generated lattices the first one is equivalent to supplemented lattice. Also, we study some properties of soc-supplemented lattices. Finally, we define cofinitely soc-supplemented lattice (briefly css-lattice) and proved that an arbitrary join of css-lattices is a css-lattice.

**Key Words:**Supplemented lattice; radical and socle of lattice; cofinite element

**References**

[1] Ismael Akray, Adil Kadir Jabbar and Reza Sazeedeh, "On Soc- -s-modules". Journal of Koya Uiversity, Vol. 24, No. 6, pp 73 – 90. (2012).

[2] Ismael Akray, "Cofinitely soc-supplemented modules". Journal of Garmian University, Vol. 2, No. 2, pp 23 – 32. (2015).

[3] G. Birkhoff, "Lattice theory". American Mathematical society, (1948).

[4] G. Calugareanu, "Lattice Concepts of Module Theory". Kluwer Texts in the Mathematical Sciences (2000).

[5]B. A. Davey and H. A. Priestley, "Introduction to lattices and order". Cambridge University Press (2002).

[6]M. L. Galvao and P. F. Smith, "Chain conditions in modular lattices", Coll. Math., Vol. 76, No. 1, pp 85-98. (1998).

[7]D. Keskin, "An approach to extending and lifting modules by modular lattices", Indian J. Pure Appl. Math., Vol. 33, No. 1, pp 81-86. (2002).

[8]B. Stenstrom, "Radicals and socles of lattices", Arch. Math., Vol. 20, pp 258-261. (1969).