Advances in Pure Mathematics, 2012, 2, 4144 http://dx.doi.org/10.4236/apm.2012.21010 Published Online January 2012 (http://www.SciRP.org/journal/apm) LTopological 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 Ltopological spaces based on a complete bounded integral residuated lattice and discuss some properties of interior and left (right) closure operators. Keywords: Residuated Lattice; LTopological Space; Interior Operator; Left (Right) Closure Operator 1. Introduction yzy 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 Lgroups, 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 noncommutative situation by Blount and Tsinakis [1]. Definition 1.1. [14]. 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 11 xx for any L, (L3) yz if and only if yz if and only if xz,, for any yz 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 (FLalgebra, for short). If 1 for all L, then L is called integral residuated lattice. An FLalgebra L which satisfies the condition 01 for all L is called FLwalgebra or bounded integral residuated lattice (see [2]). Clearly, if L is an FLwalgebra, then ,,,0,1L ab 11 is a bounded lattice. A bounded integral residuated lattice is called com mutative (see [5]) 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 [14,6]. Theorem 1.1. Let L be a bounded integral residuated lattice. Then the following properties hold. 1) , 1xxxx xx . 2) ,3) xyxy , yx x y , yxy . xxy , yyz xz 4) zxy xz ,. 5) If y then ,,zyzzxzy, zy z , zyzz xzy .zxzy and 6) y if and only if if and only if 1xy 1xy . , yzy xzxyzxyz . 7) , yzxzyz 8) . yzxzyz , yzxyx z 9) yzxyx z . If bounded integral residuated lattice L is complete, then , zyLyxzxzyLxyz ,, , jj ababLjJ Thus, it follows from some results in [7] that Theorem 1.2. Let L be a complete bounded integral residuated lattice and . Then the following properties hold. 1) JjjJ j ab ab , jJ jjJ j ab ab and i.e., the operation is infinitely distributive. jJ jjJj ab ab and 2) . jJ jjJj ab ab 3) Jj jJj ab ab and Jj jJj ab ab , i.e., the two residuation operations and are all right infinitely dis tributive (see [8]). jJ jjJj ab ab and 4) jJ jjJj ab ab . C opyright © 2012 SciRes. APM
Z. D. WANG ET AL. 42 5) Jj ab , X L jJ j ab . and Jj ab jJj ab Let us define on L two negations, and : 0xx , L0xx , and . R For any j xjJbL , it follows from Theo rems 1.1 and 1.2 that , LR x , RL x , LL yxy , RR yxy , RL yy x , LRL L x , RLR R x RR , yy x , LL yy x , LL JjjJj x , RR JjjJj x , LL JjjJ j x R. R JjjJ j x RL A bounded residuated lattice L is called an involutive residuated lattice (see [3]) if LR xx for any L ,, L . In a complete involutive residuated lattice L, RR L yy x ,. RR xxyy LL Jj jJjjJjjJj xx x . X L ,X jLjJ 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 Lfuzzy set in X will be denoted by For any family of Lfuzzy sets, we will write ,, Jj and Jj ,, R to de note the Lfuzzy sets in X given by LL R xxx ,. jjJjjJjjJjjJ xxx Besides this, we define as follows: 1,0 X XXL 1() 1 X xX 00 X and xX . 2. LTopological Spaces A completely distributive lattice L is called a Flattice, if L has an orderreversing involution ':. When L is a Flattice, Liu and Luo [9] studied the concept of L topology. Below, we consider the notion of Ltopological space based on a complete bounded integral residuated lattice. LL . X L Definition 2.1. Let If satisfies the fol lowing three conditions: (LFT1) 0,1 XX , (LFT2) ,, , (LFT3) jjJj then is called an Ltopology on X and When ][0,1L , called an Ltopological space an Ftopological space. is called an open subset in . X L Every element in L , X L L topological space. Let L R and R . The elements of and are called, respectively, left closed subsets and right closed subsets in . X L be an Ltopology on X and Definition 2.2. Let Lfuzzy subset of X. The interior, left closure and right closure of w.r.t are, respectively, defined by int , , LL cl . RR cl int, cl and cl are, respectively, called interior, left closure and right closure operators. int , For the sake of convenience, we denote ,clL R cl o and by , , and respec tively. In view of Definitions 2.1 and 2.2, for any , X L , o 1 , ,, LL L LL L 2 , ,, RR R RR R where 1,, L 2,, R o i.e., is just the largest open subset contained in , and are, respectively, the smallest left closed and right closed subsets containing . For any , X L , ,. Lo LL LL LL L . RoR Similarly, , X L Lo L Theorem 2.1. If L is an involutive residuated lattice and then 1) Ro R and oLR RL ; 2) L ,, oo LLRR ; 3) L Copyright © 2012 SciRes. APM
Z. D. WANG ET AL. 43 Copyright © 2012 SciRes. APM o LR L . o RL R and Proof. When L is an involutive residuated lattice, . LLR X L X L , LL 1) If and . RL then RL Lo L . Thus, R Similarly, Ro oRL . 2) It follows from 1) that o , RL oLR . LR o o LL o RR o LR , L L R R , RR L L , LR 3) By 2), we see that R L LR L . RL o RL RL R 0 0. X X LR . , 11,0 o XXX ,, o X L Theorem 2.2. Let . Then the following properties hold: 1) 2) R , 3) If then , oo , L . R 4) , o oo L L and . R R 5) . ooo LL 6) If ,, L yx . yxyL then L L RR 7) If ,, R yx . yxyL then R R Proof. By Definition 2.2, it is easy to see that 1)3) hold. 4) By 2) and 3), we have that . o oo On the other hand, o and oo . Thus, it follows from Definition 2.1 that and so . o oo o oo We can prove in an analogous way that L L and . R R 5) Clearly, . ooo , oo Noting that . o oo oooo we see that . ooo Thus, 11 , 6) There exist such that 1, L L 1 L L . If ,, LLL yxyxyL 11 . LLL then 11 LL Thus, . L L Clearly, . L L Therefore, . L L :XX 7) Similar to (6). Theorem 2.3. Let LL X L be a mapping. Then the following two statements hold. 1) If the operator f on satisfying the follwing conditions: (C1) 11, XX f (C2) , X L (C3) ,, X ff L then ,X L is an Ltopology on X. Moreover, if the operator f also fulfills (C4) , X ff L , then with the Ltopology o f , X L . for every i.e., f is the interior operator w.r.t 2) If the operator f on satisfying the follwing conditions: X L (C1') 00, XX f (C2') , X L (C3') ,, X ff L then a) when ,, LLL yxyxyL 1, LX L is an Ltopology on X, moreover, if the operator f also fulfills (C4) , X ff L : and XX LL is a bijection, then with the Ltopology 1 , , X L L i.e. , f is the left closure operator w.r.t 1 ; b) when ,, RRR yxyxyL 2, RR X L : is also an Ltopology on X, moreover if (C4) holds and XX LL is a bijection, then with the Ltopology 2 , , X R L 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 [9]. 2) Clearly, 1 0,1. XX If 12 1 ,, then 121 2 1212 12 , LLL LLLL L ff ff 121 i.e., 1, jjJ . If then
Z. D. WANG ET AL. Copyright © 2012 SciRes. APM 44 . LREFERENCES LL JjjJ j LL jJj ff jJ j jJ j f . LL jJ j [1] K. Blount and C. Tsinakis, “The Structure of Residuated Lattices,” International Journal of Algebra and Compu tation, Vol. 13, No. 4, 2003, pp. 437461. Combing with (C2'), we have that [2] N. Galatos, P. Jipsen, T. Kowalski and H. One, “Residu ated Lattices: An Algebraic Glimpse at Substructural Lo gics,” Elsevier, Amsterdam, 2007. jJ j f 1jJ Thus, j 1 and so is an Ltopology on X. For any , X L [3] 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. 57255738. doi:10.1016/j.ins.2007.07.014 1 1 1 , , ,, LL LL L LL ff f . [4] Z. D. Wang and J. X. Fang, “On vFilters and Normal vFilters of a Residuated Lattice with a Weak vtOpera tor,” Information Sciences, Vol. 178, No. 17, 2008, pp. 34653473. doi:10.1016/j.ins.2008.04.003 [5] U. Hohle, “Commutative, Residuated LMonoids,” In: U. Hohle and E. P. Klement, Eds., NonClassical Logics and Their Applications to Fuzzy Subsets, Kluwer Academic Publishers, Boston, Dordrecht, 1995, pp. 53106. i.e., L ff Moreover, if (C4) holds and : X L X L is a bijection, then 1 ,. X LL fL f , [6] 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. 278296. f Therefore, i.e., f is the left closure op erator w.r.t [7] Z. D. Wang and Y. D. Yu, “PseudotNorms and Implica tion Operators on a Complete Brouwerian Lattice,” Fuzzy Sets and Systems, Vol. 132, No. 1, 2002, pp. 113124. doi:10.1016/S01650114(01)00210X 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 [8] 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. 2231. doi:10.1016/j.fss.2008.03.001 2 . 3. Acknowledgements This work is supported by Science Foundation of Yancheng Teachers University (11YSYJB0201). [9] Y. M. Liu and M. K. Luo, “Fuzzy Topology,” World Scientific Publishing, Singapore, 1997.
