 Advances in Pure Mathematics, 2012, 2, 41-44 http://dx.doi.org/10.4236/apm.2012.21010 Published Online January 2012 (http://www.SciRP.org/journal/apm) L-Topological Spaces Based on Residuated Lattices Zhudeng Wang1, Xuejun Liu2 1Department of Mathematics, Yancheng Teachers University, Yancheng, China 2School of Computer and Information Technology, Zhejiang Wanli University, Ningbo, China Email: zhudengwang2004@yahoo.com.cn, lm88134005@126.com Received September 14, 2011; revised October 10, 2011; accepted October 20, 2011 ABSTRACT In this paper, we introduce the notion of L-topological spaces based on a complete bounded integral residuated lattice and discuss some properties of interior and left (right) closure operators. Keywords: Residuated Lattice; L-Topological Space; Interior Operator; Left (Right) Closure Operator 1. Introduction xyzy xz . Residuation is a fundamental concept of ordered struc-tures and the residuated lattices, obtained by adding a residuated monoid operation to lattices, have been applied in several branches of mathematics, including L-groups, ideal lattices of rings and multivalued logic. Commuta-tive residuated lattices have been studied by Krull, Dil-worth and Ward. These structures were generalized to the non-commutative situation by Blount and Tsinakis . Definition 1.1. [1-4]. A residuated lattice is an algebra of type (2, 2, 2, 2, 2, 0, 0) sat-isfying the following conditions: ,,,, ,,0,1LL(L1) is a lattice, ,,L,,1L(L2) is a monoid, i.e., is associative and 11xxx for any xL, (L3) xyz if and only if xyz if and only if yxz,, for any xyz L. Generally speaking, 1 is not the top element of L. A residuated lattice with a constant 0 is called a pointed residuated lattice or full Lambek algebra (FL-algebra, for short). If 1x for all xL, then L is called integral residuated lattice. An FL-algebra L which satisfies the condition 01x for all xL is called FLw-algebra or bounded integral residuated lattice (see ). Clearly, if L is an FLw-algebra, then ,,,0,1Lab11 is a bounded lattice. A bounded integral residuated lattice is called com-mutative (see ) if the operation is commutative. We adopt the usual convention of representing the monoid operation by juxtaposition, writing ab for . The following theorem collects some properties of bounded integral residuated lattices (see [1-4,6]. Theorem 1.1. Let L be a bounded integral residuated lattice. Then the following properties hold. 1) , 1xxxxxxx . 2) ,3) xxyxy,xyx x y ,xyxy. yxxy ,xyyz xz  4) yzxy xz,. 5) If xythen x,,zyzzxzy,xzy z, zyzz xzy .zxzy and x6) xy if and only if if and only if 1xy1xy. ,xyzy xzxyzxyz . 7)  ,xyzxzyz 8) .xyzxzyz    ,xyzxyx z 9) xyzxyx z . If bounded integral residuated lattice L is complete, then ,xzyLyxzxzyLxyz ,, ,jjababLjJ Thus, it follows from some results in  that Theorem 1.2. Let L be a complete bounded integral residuated lattice and . Then the following properties hold. 1) jJjjJ jab ab,jJ jjJ jab ab and i.e., the operation  is infinitely -distributive. jJ jjJjab ab and 2) .jJ jjJjab ab 3) jJj jJjab ab and jJj jJjab ab, i.e., the two residuation operations and are all right infinitely -dis- tributive (see ). jJ jjJjab ab and 4) jJ jjJjab ab. Copyright © 2012 SciRes. APM Z. D. WANG ET AL. 42 5)   jJjab ,XLjJ jab  . and jJjabjJjab Let us define on L two negations, L and R: 0xx,L0xx, and . RFor any jxxjJbL, it follows from Theo-rems 1.1 and 1.2 that ,LRxx  ,RLxx  ,LLxyxy ,RRxyxy ,RLxyy x ,LRL Lxx  ,RLR Rxx RR ,xyy x  ,LLxyy x ,LL jJjjJjxx  ,RRjJjjJjxx  ,LLjJjjJ jxx  R.RjJjjJ jxxRL A bounded residuated lattice L is called an involutive residuated lattice (see ) if LRxxx  for any xL,,L. In a complete involutive residuated lattice L, RR Lxyy x,.RRxxyy  LLjJj jJjjJjjJjxxx x .XL,XjLjJLR  In the sequel, unless otherwise stated, L always repre-sents any given complete bounded integral residuated lattice with maximal element 1 and minimal element 0. The family of all L-fuzzy set in X will be denoted by For any family of L-fuzzy sets, we will write ,,jJj and jJj,,R to de-note the L-fuzzy sets in X given by  LLRxxxx  ,.jjJjjJjjJjjJxxxx Besides this, we define as follows: 1,0 XXXL1() 1XxxX00X and xxX . 2. L-Topological Spaces A completely distributive lattice L is called a F-lattice, if L has an order-reversing involution ':. When L is a F-lattice, Liu and Luo  studied the concept of L- topology. Below, we consider the notion of L-topological space based on a complete bounded integral residuated lattice. LL.XLDefinition 2.1. Let If  satisfies the fol-lowing three conditions: (LFT1) 0,1XX , (LFT2) ,, , (LFT3) jjJj then  is called an L-topology on X and When ][0,1L, called an L-topological space  an F-topological space.  is called an open subset in . XLEvery element in L,XL L- topological space. Let LR and R . The elements of L and R are called, respectively, left closed subsets and right closed subsets in . XL be an L-topology on X and Definition 2.2. Let  L-fuzzy subset of X. The interior, left closure and right closure of  w.r.t  are, respectively, defined by int ,  ,LLcl  .RRcl  int, Lcl and Rcl are, respectively, called interior, left closure and right closure operators. int, For the sake of convenience, we denote ,clLRcl o and  by , L, and R respec-tively. In view of Definitions 2.1 and 2.2, for any ,XL,o 1,,,LLLLL L    2,,,RRRRR R   where 1,,L 2,,Ro i.e.,  is just the largest open subset contained in , L and L are, respectively, the smallest left closed and right closed subsets containing . For any ,XL,,.Lo LLLL LLL     .RoR Similarly, R,XLLo L Theorem 2.1. If L is an involutive residuated lattice and then 1) L Ro R and R oLR RL; 2) RL  ,,ooLLRR; 3) RL  Copyright © 2012 SciRes. APM Z. D. WANG ET AL. 43Copyright © 2012 SciRes. APM oLRL.oRLRand    Proof. When L is an involutive residuated lattice, .RLLRXLXL,LL 1) If and .RL then RL  Lo L . Thus, L R Similarly, RoRoRL. 2) It follows from 1) that o ,RLL oLR  .LR oR oLLoRRoLR  ,L LRR ,RRLL,LR 3) By 2), we see that R L LRLL .RL oRLRL RR  0 0.X XLR. ,11,0oXXX,,oXL Theorem 2.2. Let . Then the following properties hold: 1) 2) LR , 3) If  then ,oo, LL .RR 4) ,ooo LLL and .RRR 5) .oooLL 6) If ,,Lxyx.yxyL  then LLLRR 7) If ,,Rxyx.yxyL then RRR Proof. By Definition 2.2, it is easy to see that 1)-3) hold. 4) By 2) and 3), we have that .ooo On the other hand, o and oo. Thus, it follows from Definition 2.1 that and so .oooooo We can prove in an analogous way that LLL and .RRR 5) Clearly, .ooo,oo Noting that .ooo oooo we see that  .ooo Thus, 11, 6) There exist  such that 1,LL1LL . If ,,LLLxyxyxyL 11.LLL then 11LLL   Thus, .LLL Clearly, .LLL Therefore, .LLL:XX 7) Similar to (6). Theorem 2.3. Let fLLXL be a mapping. Then the following two statements hold. 1) If the operator f on satisfying the follwing conditions: (C1) 11,XXf (C2) ,XfL (C3) ,,Xfff L then ,XfL is an L-topology on X. Moreover, if the operator f also fulfills (C4) ,Xfff L, then with the L-topology  of,XL. for every i.e., f is the interior operator w.r.t  2) If the operator f on satisfying the follwing conditions: XL(C1') 00,XXf (C2') ,XfL (C3') ,,Xfff L then a) when ,,LLLxyxyxyL  1,LLXfL  is an L-topology on X, moreover, if the operator f also fulfills (C4) ,Xfff L : and LXXLL is a bijection, then with the L-topology 1, ,XLfL i.e. , f is the left closure operator w.r.t 1; b) when ,,RRRxyxyxyL  2,RR XfL : is also an L-topology on X, moreover if (C4) holds and RXXLL is a bijection, then with the L-topology 2, ,XRfL i.e., f is the right closure op-erator w.r.t 2. Proof. 1) Refer to the proof of Theorem 2.2.2 1 in . 2) Clearly, 10,1.XX If 12 1,, then  121 2121212,LLLLLLLLffff    121 i.e., 1,jjJ. If then Z. D. WANG ET AL. Copyright © 2012 SciRes. APM 44 .LREFERENCES LLjJjjJ jLLjJjffjJ jjJ jf .LLjJ j    K. Blount and C. Tsinakis, “The Structure of Residuated Lattices,” International Journal of Algebra and Compu-tation, Vol. 13, No. 4, 2003, pp. 437-461. Combing with (C2'), we have that  N. Galatos, P. Jipsen, T. Kowalski and H. One, “Residu-ated Lattices: An Algebraic Glimpse at Substructural Lo-gics,” Elsevier, Amsterdam, 2007. jJ jf1jJ  Thus, j 1 and so  is an L-topology on X. For any ,XL L. Z. Liu and K. T. Li, “Boolean Filters and Positive Im-plicative Filters of Residuated Lattices,” Information Sci-ences, Vol. 177, No. 24, 2007, pp. 5725-5738. doi:10.1016/j.ins.2007.07.014 111,,,,LLLLLLLLfff      .  Z. D. Wang and J. X. Fang, “On v-Filters and Normal v-Filters of a Residuated Lattice with a Weak vt-Opera- tor,” Information Sciences, Vol. 178, No. 17, 2008, pp. 3465-3473. doi:10.1016/j.ins.2008.04.003  U. Hohle, “Commutative, Residuated L-Monoids,” In: U. Hohle and E. P. Klement, Eds., Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer Academic Publishers, Boston, Dordrecht, 1995, pp. 53-106. i.e., LLff Moreover, if (C4) holds and :LXLXL is a bijection, then  1,.XLLLfLf    ,  A. M. Radzikowska and E. E. Kerre, “Fuzzy Rough Sets Based on Residuated Lattices,” In: J. F. Peter et al., Eds., Transactions on Rough Sets II, LNCS 3135, 2004, pp. 278-296. fTherefore, L i.e., f is the left closure op-erator w.r.t  Z. D. Wang and Y. D. Yu, “Pseudo-t-Norms and Implica-tion Operators on a Complete Brouwerian Lattice,” Fuzzy Sets and Systems, Vol. 132, No. 1, 2002, pp. 113-124. doi:10.1016/S0165-0114(01)00210-X 1. We can prove in an analogous way that 2 is an L- topology on X and the correspond ing f is the right closure operator w.r.t  Z. D. Wang and J. X. Fang, “Residual Operations of Left and Right Uninorms on a Complete Lattice,” Fuzzy Sets and Systems, Vol. 160, No. 1, 2009, pp. 22-31. doi:10.1016/j.fss.2008.03.001 2. 3. Acknowledgements This work is supported by Science Foundation of Yancheng Teachers University (11YSYJB0201).  Y. M. Liu and M. K. Luo, “Fuzzy Topology,” World Scientific Publishing, Singapore, 1997.