Applied and Computational Mathematics
Volume 6, Issue 1, February 2017, Pages: 45-53

Strict Left (Right)-Conjunctive Left (Right) Semi-Uninorms and Implications Satisfying the Order Property

Zhudeng Wang1, *, Yuan Wang2, Keming Tang2

1School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, People's Republic of China

2College of Information Engineering, Yancheng Teachers University, Yancheng, People's Republic of China

Email address:

(Zhudeng Wang)
(Yuan Wang)
(Keming Tang)

*Corresponding author

To cite this article:

Zhudeng Wang, Yuan Wang, Keming Tang. Strict Left (Right)-Conjunctive Left (Right) Semi-Uninorms and Implications Satisfying the Order Property. Applied and Computational Mathematics. Vol. 6, No. 1, 2017, pp. 45-53. doi: 10.11648/j.acm.20170601.13

Received: January 8, 2017; Accepted: January 19, 2017; Published: February 23, 2017


Abstract: We firstly give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation. Then, we lay bare the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation. Finally, we reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary ˅-distributive left (right) semi-uninorms and lower approximation right arbitrary ˄-distributive implications which satisfy the order property.

Keywords: Fuzzy Logic, Fuzzy Connective, Left (Right) Semi-Uninorm, Implication, Strict Left (Right)-Conjunctive


1. Introduction

In fuzzy logic systems (see [1-2]), connectives "and", "or" and "not" are usually modeled by t-norms, t-conorms, and strong negations on  (see [3]), respectively. Based on these logical operators on , the three fundamental classes of fuzzy implications on , i.e., R-, S-, and QL-implications on , were defined and extensively studied. But, as was pointed out by Fodor and Keresztfalvi [4], sometimes there is no need of the commutativity or associativity for the connectives "and" and "or". Thus, many authors investigated implications based on some other operators like weak t-norms [5], pseudo t-norms [6], pseudo-uninorms [7], left and right uninorms [8], semi-uninorms [9], aggregation operators [10] and so on.

Uninorms, introduced by Yager and Rybalov [11], and studied by Fodor et al. [12], are special aggregation operators that have proven useful in many fields like fuzzy logic, expert systems, neural networks, aggregation, and fuzzy system modeling. This kind of operation is an important generalization of both t-norms and t-conorms and a special combination of t-norms and t-conorms. But, there are real-life situations when truth functions cannot be associative or commutative. By throwing away the commutativity from the axioms of uninorms, Mas et al. introduced the concepts of left and right uninorms on  in [13] and later in a finite chain in [14], and Wang and Fang [8, 15] studied the left and right uninorms on a complete lattice. By removing the associativity and commutativity from the axioms of uninorms, Liu [9] introduced the concept of semi-uninorms, and Su et al. [16] discussed the notions of left and right semi-uninorms, on a complete lattice. On the other hand, it is well known that a uninorm (semi-uninorm)  can be conjunctive or disjunctive whenever  or 1, respectively. This fact allows us to use uninorms in defining fuzzy implications.

Constructing fuzzy connectives is an interesting topic. Recently, Jenei and Montagna [17] introduced several new types of constructions of left-continuous t-norms, Wang [18] laid bare the formulas for calculating the smallest pseudo-t-norm that is stronger than a binary operation and the largest implication that is weaker than a binary operation, Su et al. [16] studied the constructions of left and right semi-uninorms on a complete lattice, Su and Wang [19] investigated the constructions of implications and coimplications on a complete lattice. and Wang et al. [20-22] studied the relations among implications, coimplications and left (right) semi-uninorms, on a complete lattice. Moreover, Wang et al. [23-24] investigated the constructions of conjunctive left (right) semi-uninorms, disjunctive left (right) semi-uninorms, strict left (right)-disjunctive left (right) semi-uninorm, implications and coimplications satisfying the neutrality principle.

This paper is a continuation of [16, 19, 23-24]. Motivated by these works in [16, 19, 23-24], we will further focus on this issue and investigate constructions of the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms and the upper and lower approximation implications which satisfy the order property.

This paper is organized as follows. In Section 2, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation. In Section 3, we lay bare the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation. In Section 4, we reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary -distributive left (right) semi-uninorms and lower approximation right arbitrary -distributive implications which satisfy the order property, and find out some conditions such that the lower approximation strict left (right)-conjunctive left (right) semi-uninorm of a binary operation and upper approximation implication, which satisfies the order property, of left (right) residuum of the binary operation satisfy the generalized modus ponens rule.

The knowledge about lattices required in this paper can be found in [25].

Throughout this paper, unless otherwise stated, always represents any given complete lattice with maximal element 1 and minimal element 0; stands for any index set.

2. Strict Conjunctive Left and Right Semi-Uninorms

In this section, we firstly recall some necessary concepts about the strict conjunctive left (right) semi-uninorms on a complete lattice.

Definition 2.1 (Su et al. [16]). A binary operation on  is called a left (right) semi-uninorm if it satisfies the following two conditions:

(U1) there exists a left (right) neutral element, i.e., an element  () satisfying  ( for all ,

(U2) is non-decreasing in each variable.

Clearly,  and hold for any left (right) semi-uninorm  on .

If a left (right) semi-uninorm  is associative, then is the left (right) uninorm [8, 15] on .

If a left (right) semi-uninorm  with the left (right) neutral element  () has a right (left) neutral element  (), then . Let . Here,  is the semi-uninorm [9].

For any left (right) semi-uninorm  on ,  is said to be left-conjunctive and right-conjunctive if  and , respectively.  is said to be conjunctive if both  and  since it satisfies the classical boundary conditions of AND.

 is said to be strict left-conjunctive and strict right- conjunctive if  is conjunctive and for any  and , respectively.

Definition 2.2 (Wang and Fang [8]). A binary operation  on  is called left (right) arbitrary -distributive if

(1)

left (right) arbitrary -distributive if

(2)

If a binary operation  is left arbitrary -distributive (- distributive) and also right arbitrary -distributive (-distributive), then  is said to be arbitrary -distributive (-distributive).

For the sake of convenience, we introduce the following symbols:

: the set of all strict left-conjunctive left semi-uninorms with the left neutral element on ;

: the set of all strict right-conjunctive right semi-uninorms with the right neutral element  on ;

: the set of all strict left-conjunctive left arbitrary -distributive left semi-uninorms with the left neutral element  on ;

: the set of all strict right-conjunctive right arbitrary -distributive right semi-uninorms with the right neutral element  on .

Below, we illustrate these notions by means of two examples.

Example 2.1. Let ,

where  and  are elements of . By Example 2 and Theorem 8 in [20], we see that  and  are two join-semilattices with the greatest element . When  and , it is straightforward to verify that  is the smallest element of .

Moreover, assume that .  is left arbitrary -distributive and the smallest element of .

Example 2.2. Let ,

where  and  are elements of . By Example 3 and Theorem 8 in [20], we see that  and  are two join-semilattices with the greatest element .

 Similarly, When  and ,  is the smallest element of . Moreover, if, then  is the smallest element of .

Constructing aggregation operators is an interesting work. Recently, Jenei and Montagna [17] introduced several new types of constructions of left-continuous t-norms, Su et al. [16] studied the constructions of left and right semi-uninorms on a complete lattice, and Wang et al. [23-24] investigated the constructions of conjunctive left (right) semi-uninorms and disjunctive left (right) semi-uninorms on a complete lattice. Now, we continue this work and give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.

It is easy to verify that  for any nonempty subset  of . If  and , then  is a complete lattice with the smallest element  and greatest element  by Example 2.1. Thus, for a binary operation  on , if there exists  such that , then

(3)

is the smallest strict left-conjunctive left semi-uninorm that is stronger than  on , we call it the upper approximation strict left-conjunctive left semi-uninorm of  and write as ; if there exists  such that , then

(4)

is the largest strict left-conjunctive left semi-uninorm that is weaker than  on , we call it the lower approximation strict left-conjunctive left semi-uninorm of  and write as .

Similarly, we introduce the following symbols:

 the upper approximation strict right-conjunctive right semi-uninorm of ;

: the lower approximation strict right-conjunctive right semi-uninorm of ;

: the upper approximation strict left-conjunctive left arbitrary -distributive left semi-uninorm of ;

: the lower approximation strict left-conjunctive left arbitrary -distributive left semi-uninorm of ;

: the upper approximation strict right-conjunctive right arbitrary -distributive right semi-uninorm of ;

: the lower approximation strict right-conjunctive right arbitrary -distributive right semi-uninorm of .

Definition 2.3 (Su et al. [16]). Let  be a binary operation on . Define the upper approximation aggregator  and the lower approximation aggregator  of  as follows:

(5)

(6)

Theorem 2.1 (Su et al. [16]). Let . Then the following statements hold:

(7)

 and

(8)

 and  are non-decreasing in its each variable.

If is non-decreasing in its each variable, then

(9)

Theorem 2.2. Let .

(1).   If  is left (right) arbitrary -distributive, then  is left (right) arbitrary -distributive.

(2).   If  is left (right) arbitrary -distributive, then    is left (right) arbitrary -distributive.

Proof. We only prove that statement (1) holds.

If  is left arbitrary -distributive, then  is non-decreasing in its first variable,

(10)

(11)

i.e.,  is left arbitrary -distributive.

Similarly, we can show that  is right arbitrary -distributive when  is right arbitrary -distributive.

The theorem is proved.

Below, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.

Theorem 2.3. Suppose that ,  and .

(1).   If , then ;

if , then .

(2).   If ,  and  is left arbitrary -distributive, then

(12)

Moreover, if  is non-decreasing in its second variable, then .

Proof. Assume that  and . Then  and  are, respectively, the smallest and greatest elements of  by Example 2.1.

(1) Let . If , then , . Thus,

(13)

It implies that and  for any . If , then  and so , i.e.,  is strict left-conjunctive. By Theorem 2.1(3) and the monotonicity of , we can see that  is non-decreasing in its each variable. So, . If  and , then  and . Therefore,

(14)

Let . If , then

 and .      (15)

Thus,  and  for any  and  is strict left-conjunctive. By Theorem 2.1(3) and the monotonicity of , we know that  is non-decreasing in its each variable. So, . If  and , then  and . Therefore,

(16)

(2) When ,  and  are, respectively, the smallest and greatest elements of  by Example 2.1. Let . If , then  by statement (1). Noting that  is left arbitrary -distributive, we can see that  is also left arbitrary -distributive by Theorem 2.2(1). Thus,  is left arbitrary -distributive and . By the proof of statement (1), we have that .

Moreover, if  is non-decreasing in its second variable, then  by Theorem 2.1(4) and so

(17)

The theorem is proved.

Similarly, for calculating the upper and lower approximation strict right-conjunctive right semi-uninorms of a binary operation, we have the following theorem.

Theorem 2.4. Suppose that ,  and .

(1).   If , then ; if , then .

(2).   If ,  and   is right arbitrary -distributive, then

(18)

Moreover, if  is non-decreasing in its first variable, then .

3. Implications Satisfying the Order Property

Recently, Su and Wang [19] have studied the constructions of implications and coimplications and Wang et al. [23-24] further investigated the constructions of implications and coimplications satisfying the neutrality principle on a complete lattice. This section is a continuation of [19,23,24]. We will study the constructions of the upper and lower approximation implications which satisfy the order property.

Definition 3.1 (Baczynski and Jayaram [26], Bustince et al. [27], De Baets and Fodor [28], Fodor and Roubens [1]).  An implication  on  is a hybrid monotonous (with decreasing first and increasing second partial mappings) binary operation that satisfies the corner conditions  and .

An implication  is said to satisfy the order property with respect to  (w.r.t. e, for short) when  if and only if  for any .

Note that for any implication  on , due to the monotonicity, the absorption principle holds, i.e.,  for any .

For the sake of convenience, we introduce the following symbols:

: the set of all implications which satisfy the order property w. r. t. e on ;

: the set of all right arbitrary -distributive implications which satisfy the order property w. r. t. e on .

Clearly,  and  are all meet-semilattices.     Definition 3.2. Let  be a binary operation on . Define  as follows:

(19)

(20)

Here,  and  are, respectively, called the left and right residuum of the binary operation .

When  is non-decreasing in each variable, it is easy to see that  and  are all decreasing in the first variable and increasing in the second one by Definition 3.2.

Example 3.1. For some left and right semi-uninorms in Examples 2.1-2.2, a simple computation shows that

where  and  are elements of . By the virtue of Theorem 8 in [20], we see that  is the smallest element of both  and .

When  and , it is easy to see that  is the greatest element of .

Moreover, assume that .  is the greatest element of .

Similar conclusions hold for  and .

It is easy to verify that if , then

(21)

When  and , we see that  is also a complete lattice with the smallest element  and greatest element  by Example 3.1. Thus, for a binary operation  on , if there exists  such that , then

(22)

is the smallest implication that is stronger than  and satisfies the order property w. r. t.  on . Here, we call it the upper approximation implication, which satisfies the order property w. r. t. , of  and write as . Similarly, if there exists  such that , then

(23)

is the largest implication that is weaker than  and satisfies the order property w. r. t.  on . Here, we call it the lower approximation implication, which satisfies the order property w. r. t. , of  and write as .

Likewise, for a binary operation  on , we may introduce the following symbols:

: the upper approximation implication, which satisfies the order property w. r. t. , of ;

: the lower approximation implication, which satisfies the order property w. r. t. , of ;

(): the upper approximation right arbitrary -distributive implication, which satisfies the order property w. r. t.  (), of ;

(): the lower approximation right arbitrary -distributive implication, which satisfies the order property w.r. t.  (), of .

Definition 3.3 (see Su and Wang [19]). Let  be a binary operation on . Define the upper approximation implicator  and the lower approximation implicator  of  as follows:

(24)

(25)

Theorem 3.1 (see Su and Wang [19]). Let . Then the following statements hold:

(26)

 and

(27)

 and  are hybrid monotonous.

If  is are hybrid monotonous, then .

Theorem 3.2. Let .

(1).   If is right arbitrary -distributive, then  is also right arbitrary -distributive,

(28)

(29)

(2).   If  is right arbitrary -distributive, then  is also right arbitrary -distributive.

(3).   If is left arbitrary -distributive, then,

(30)

(31)

Proof. We only prove that statement (1) holds.

Assume that  is a right arbitrary -distributive binary operation on . Clearly,  is also right arbitrary -distributive. By Definition 3.3, the monotonicity of  and , and the right residual principle, we have that

(32)

(33)

Thus, . Similarly, we have that

(34)

(35)

(36)

If , let , then

(37)

So,  for any , i.e., . Moreover, we know that  is right arbitrary -distributive and hence

(38)

The theorem is proved.

Below, we give out the formulas for calculating the upper and lower approximation implications which satisfy the order property.

Theorem 3.3. Suppose that ,  and .

(1)  If , then ;

if , then .

(2)  If ,  and  is right arbitrary -distributive, then

(39)

Moreover, if  is non-decreasing in its first variable, then .

Proof. Assume that  and. Then  and  are, respectively, the smallest and greatest elements of by Example 3.1.

(1) If , let , then  and

(40)

Thus,  and . If , then ; if , then  and so , i.e.,  satisfies the order property w. r. t. . By Theorem 3.1(3) and the hybrid monotonicity of , we know that  is hybrid monotonous. So, . If  and , then  and . Therefore,

(41)

If , let , then ,

(42)

Thus, we can prove in an analogous way that  and .

(2) When , and  are, respectively, the smallest and greatest elements of by Example 3.1. Let . If , then  by statement (1). Noting that  is right arbitrary -distributive, we can see that  is also right arbitrary -distributive by Theorem 3.2(2). So,  is right arbitrary -distributive, i.e., . By the proof of statement (1), we know that .

Moreover, if  is non-decreasing in its first variable, then  by Theorem 3.1(4) and so

(43)

The theorem is proved.

Analogous to Theorem 3.3, we have the following theorem.

Theorem 3.4. Suppose that ,  and .

(1)  If , then ;

if , then .

(2)  If ,  and  is right arbitrary -distributive, then

(44)

Moreover, if  is non-decreasing in its first variable, then .

4. The Relations Between Strict (Right)-Conjunctive Left (Right) Semi-Uninorms and Implications

In this section, we reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary -distributive left (right) semi-uninorms and lower approximation right arbitrary -distributive implications which satisfy the order property.

Theorem 4.1. Suppose that ,  and .

(1)  If ,  and  is left arbitrary -distributive, then

(45)

(2)  If ,  and   is right arbitrary -distributive, then

(46)

Proof. We only prove the statement (1) holds.

Assume that  and  is left arbitrary -distributive. Then it follows from Theorem 4.6 in [8] and Definition 3.2 that  and  is right arbitrary -distributive. Thus,  by Theorem 3.3(2). Moreover, it follows from Theorems 2.2(1) and 2.3(2) and the left residual principle that

(47)

i.e., . By Theorem 3.2(3), we know that . Therefore,

(48)

The theorem is proved.

Finally, we give out some conditions such that the lower approximation strict left (right)-conjunctive left (right) semi-uninorm of a binary operation and upper approximation implication, which satisfies the order property, of left (right) residuum of the binary operation satisfy the GMP rule.

Theorem 4.2. Suppose that ,  and .

(1)  If ,  and  is non-decreasing in its second variable and left arbitrary -distributive and  and  are comparable (see [25]) when , then  and  satisfy the GMP rule in the form

(49)

(2)  If ,  and  is non-decreasing in its first variable and right arbitrary -distributive and  and  are comparable (see [25]) when , then  and satisfy the GMP rule in the form

(50)

Proof. We only prove the statement (1) holds.

Assume that ,  is non-decreasing in its second variable and left arbitrary -distributive. Then, , ,  is non-increasing in its first variable by Definition 3.2 and right arbitrary -distributive by Theorem 4.6 in [8],  by Theorem 3.1(4),

(51)

By the virtue of Theorem 2.3(2), we see that

(52)

By Example 3.1 and Theorem 3.3(1), we know that

Thus,

When , noting that  and  are comparable, we see that

So, when ,

(53)

Therefore,  for all , i.e.,  and  satisfy the GMP rule.

The theorem is proved.

5. Conclusions and Future Works

Constructing fuzzy connectives is an interesting topic. Recently, Su et al. [16] studied the constructions of left and right semi-uninorms, and Wang et al. [19-20, 22, 24] investigated the constructions of implications and coimplications on a complete lattice. In this paper, motivated by these works, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation; lay bare the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation; reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary -distributive left (right) semi-uninorms and lower approximation right arbitrary -distributive implications which satisfy the order property.

In a forthcoming paper, we will further investigate the constructions of left (right) semi-uninorms and coimplications on a complete lattice.

Acknowledgements

This work is supported by Jiangsu Provincial Natural Science Foundation of China (BK20161313), Science Foundation of Yancheng Teachers University (16YCKLQ006) and the National Natural Science Foundation of China (61379064).


References

  1. J. Fodor and M. Roubens, "Fuzzy Preference Modelling and Multicriteria Decision Support", Theory and Decision Library, Series D: System Theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers, Dordrecht, 1994.
  2. G. J. Klir and B. Yuan, "Fuzzy Sets and Fuzzy Logic, Theory and Applications", Prentice Hall, New Jersey, 1995.
  3. E. P. Klement, R. Mesiarand E. Pap, "Triangular Norms", Trends in Logic-Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.
  4. J. Fodor and T. Keresztfalvi, "Nonstandard conjunctions and implications in fuzzy logic", International Journal of Approximate Reasoning, 12, 69-84, 1995.
  5. J. Fodor, "Srict preference relations based on weak t-norms", Fuzzy Sets and Systems, 43, 327-336, 1991.
  6. Z. D. Wang and Y. D. Yu, "Pseudo-t-norms and implication operators on a complete Brouwerian lattice", Fuzzy Sets and Systems, 132, 113-124, 2002.
  7. Y. Su and Z. D. Wang, "Pseudo-uninorms and coimplications on a complete lattice", Fuzzy Sets and Systems, 224, 53-62, 2013.
  8. Z. D. Wang and J. X. Fang, "Residual operators of left and right uninorms on a complete lattice", Fuzzy Sets and Systems, 160, 22-31, 2009.
  9. H. W. Liu, "Semi-uninorm and implications on a complete lattice", Fuzzy Sets and Systems, 191, 72-82, 2012.
  10. Y. Ouyang, "On fuzzy implications determined by aggregation operators", Information Sciences, 193, 153-162, 2012.
  11. R. R. Yager and A. Rybalov, "Uninorm aggregation operators", Fuzzy Sets and Systems, 80, 111-120, 1996.
  12. J. Fodor, R. R. Yager and A. Rybalov, "Structure of uninorms", Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems, 5, 411-427, 1997.
  13. M. Mas, M. Monserrat and J. Torrens, "On left and right uninorms", Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems, 9, 491-507, 2001.
  14. M. Mas, M. Monserrat and J. Torrens, "On left and right uninorms on a finite chain", Fuzzy Sets and Systems, 146, 3-17, 2004.
  15. Z. D. Wang and J. X. Fang, "Residual coimplicators of left and right uninorms on a complete lattice", Fuzzy Sets and Systems, 160, 2086-2096, 2009.
  16. Y. Su, Z. D. Wang and K. M. Tang, "Left and right semi-uninorms on a complete lattice", Kybernetika, 49, 948-961, 2013.
  17. S. Jenei and F. Montagna, "A general method for constructing left-continuous t-norms", Fuzzy Sets and Systems, 136, 263-282, 2003.
  18. Z. D. Wang, "Generating pseudo-t-norms and implication operators", Fuzzy Sets and Systems, 157, 398-410, 2006.
  19. Y. Su and Z. D. Wang, "Constructing implications and coimplications on a complete lattice", Fuzzy Sets and Systems, 247, 68-80, 2014.
  20. X. Y. Hao, M. X. Niu and Z. D. Wang, "The relations between implications and left (right) semi-uninorms on a complete lattice", Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems, 23, 245-261, 2015.
  21. M. X. Niu, X. Y. Hao and Z. D. Wang, "Relations among implications, coimplications and left (right) semi-uninorms", Journal of Intelligent and Fuzzy Systems, 29, 927-938, 2015.
  22. Z. D. Wang, "Left (right) semi-uninorms and coimplications on a complete lattice", Fuzzy Sets and Systems, 287, 227-239, 2016.
  23. X. Y. Hao, M. X. Niu, Y. Wang and Z. D. Wang, "Constructing conjunctive left (right) semi-uninorms and implications satisfying the neutrality principle", Journal of Intelligent and Fuzzy Systems, 31, 1819-1829, 2016.
  24. Z. D. Wang, M. X. Niu and X. Y. Hao, "Constructions of coimplications and left (right) semi-uninorms on a complete lattice", Information Sciences, 317, 181-195, 2015.
  25. G. Birkhoff, "Lattice Theory", American Mathematical Society Colloquium Publishers, Providence, 1967.
  26. M. Baczynski and B. Jayaram, "Fuzzy Implication", Studies in Fuzziness and Soft Computing, Vol. 231, Springer, Berlin, 2008.
  27. H. Bustince, P. Burillo and F. Soria, "Automorphisms, negations and implication operators", Fuzzy Sets and Systems, 134, 209-229, 2003.
  28. B. De Baets and J. Fodor, "Residual operators of uninorms", Soft Computing, 3, 89-100, 1999.

Article Tools
  Abstract
  PDF(266K)
Follow on us
ADDRESS
Science Publishing Group
548 FASHION AVENUE
NEW YORK, NY 10018
U.S.A.
Tel: (001)347-688-8931