Borzooei, R., Hosseini, F., Zahiri, O. (2018). Convex $L$-lattice subgroups in $L$-ordered groups. Categories and General Algebraic Structures with Applications, 9(1), 139-161.

Rajabali Borzooei; Fateme Hosseini; Omid Zahiri. "Convex $L$-lattice subgroups in $L$-ordered groups". Categories and General Algebraic Structures with Applications, 9, 1, 2018, 139-161.

Borzooei, R., Hosseini, F., Zahiri, O. (2018). 'Convex $L$-lattice subgroups in $L$-ordered groups', Categories and General Algebraic Structures with Applications, 9(1), pp. 139-161.

Borzooei, R., Hosseini, F., Zahiri, O. Convex $L$-lattice subgroups in $L$-ordered groups. Categories and General Algebraic Structures with Applications, 2018; 9(1): 139-161.

Convex $L$-lattice subgroups in $L$-ordered groups

^{1}Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran.

^{2}University of Applied Science and Technology, Tehran, Iran

Abstract

In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result about the set of all convex subgroups of a lattice ordered group and prove that $C(G)$, the set of all convex $L$-lattice subgroups of an $L$-ordered group $G$, is an $L$-complete lattice on height one. Then we use these objects to construct the quotient $L$-ordered groups and state some related results.

[1] Anderson, M. and Feil, T., “Lattice-ordered Groups”, D. Reidel Publishing Co., Dordrecht, 1988. [2] Bˇelohlávek, R., “Fuzzy Relational Systems: Foundations and Principles”, Kluwer Academic Publishers, 2002. [3] Blyth, T.S., “Lattices and Ordered Algebraic Structures”, Springer-Verlag, 2005. [4] Borzooei, R.A., Dvureˇcenskij, A., and Zahiri, O., L-ordered and L-lattice ordered groups, Inform. Sci. 314(1) (2015), 118-134. [5] Borzooei, R. A., Hosseini, F., and Zahiri, O., (Totally) L-ordered groups, J. Intell. Fuzzy Systems 30(3) (2016), 1489-1498. [6] Darnel, M.R., “Theory of Lattice-Ordered Groups”, CRC Press, 1994. [7] Fan, L., A new approach to quantitative domain theory, Electron. Notes Theor. Comput. Sci. 45 (2001), 77-87. [8] Glass, A.M.W., “Partially Ordered Groups”, Word Scientific, 1999. [9] Goguen, J.A., L-fuzzy sets, J. Math. Anal. Appl. 18 (1967), 145-174. [10] Johnstone, P.T., “Stone Spaces”, Cambridge University Press, 1982. [11] Lai, H. and Zhang, D., Complete and directed complete-categories, Theoret. Comput. Sci. 88 (2007), 1-25. [12] Martinek, P., Completely lattice L-ordered sets with and without L-equality, Fuzzy Sets and Systems 166 (2011), 44-55. [13] Ovchinnikov, S.V., Structure of fuzzy binary relations, Fuzzy Sets and Systems 6 (1981), 169-195. [14] Venugopalan, P., Fuzzy ordered sets, Fuzzy Sets and Systems 46 (1992), 221-226. [15] Yao, W., Quantitative domains via fuzzy sets, part I: Continuity of fuzzy directed complete posets, Fuzzy Sets and Systems 161 (2010), 973-987. [16] Yao, W., An approach to fuzzy frames via fuzzy posets, Fuzzy Sets and Systems 166 (2011), 75-89. [17] Yao, W., A survey of fuzzifications of frames, The Papert-Papert-Isbell adjunction and sobriety, Fuzzy Sets and Systems 190 (2012), 63-81. [18] Yao, W. and Lu, L.X., Fuzzy Galois connections on fuzzy posets, Math.Log. Q. 55 (2009), 105-112. [19] Zhang, Q.Y. and Fan, L., Continuity in quantitative domains, Fuzzy Sets and Systems 154 (2005), 118-131. [20] Zhang, D. and Liu, Y.M., L-fuzzy version of stonean representation theory for distributive lattices, Fuzzy Sets and Systems 76 (1995), 259-270. [21] Zhanga, Q., Xie,W., and Fan, L., Fuzzy complete lattices, Fuzzy Sets and Systems 160 (2009), 2275-2291.