Homomorphisms on The Monoid of Fuzzy Implications

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1 Homomorphisms on The Monoid of Fuzzy mplications Nageswara Rao Vemuri and Balasubramaniam Jayaram Department of Mathematics ndian nstitute of Technology Hyderabad Yeddumailaram, A.P {ma10p001, Abstract n this work we propose and study a particular type of lattice and semigroup homomorphisms on the monoid (, ) of the set of all fuzzy implications proposed in [1]. We show that the subclass of neutral implications which generate homomorphisms of the defined form and the set of such homomorphisms themselves form abelian groups, suggesting that the investigated homomorphisms form the group of inner semigroup homomorphisms. Finally, investigating the images of the studied homomorphisms, we present some natural partitions on and orderings on these equivalence classes. Our investigations have led us to obtain a group structure on a subset of. Note that, to the best of the authors knowledge, this is the first work to present such a rich algebraic structure on the set of all fuzzy implications. Keywords Fuzzy implication, monoid, homomorphism, center, idempotent element, group of inner homomorphisms.. NTRODUCTON Fuzzy mplications are defined as a generalization of the classical implication to the multivalued setting. These operators have many applications in areas like approximate reasoning, control theory, decision making, expert systems, fuzzy logic etc, see [2], [3] for more details. Hence there is an essential need to generate new implications satisfying various properties. The different ways of obtaining fuzzy implications found in the literature, so far, can be largely categorized based on the underlying operators from where they are generated: (i) from other fuzzy logic connectives (ii) from monotone functions over the unit interval [0, 1] and (iii) from given fuzzy implications. The last of these generation processes also has an interesting fall out: Considering fuzzy implications as real functions on [0, 1] 2, this process defines a binary operation on the set of all fuzzy implications, which will be denoted by. Thus, this process allows us to study the algebraic structures that arise on. A. Motivation for the work Recently, in [1], we had proposed the following novel method of generating fuzzy implications from a given pair of fuzzy implications, J: ( J)(x, y) = (x, J(x, y)), x, y [0, 1]. (1) Further, it was shown that this method gives rise to richer algebraic structures on the set than were obtained before in [4], [5], [6]. n fact, in [1], it was shown that (, ) is a monoid. While the main emphasis in [1] was to study the analytical properties (like powers, closures, convergence) of this construction, in this work we concentrate on the algebraic aspects of this construction. n algebra, especially in the study of groups three important concepts arise, viz., Normal subgroups, Homomorphisms and the group of inner automorphisms. As a first step in this direction, we would like to study the homomorphic images of (, ) and the group of inner homomorphisms. B. Main contributions of the work n this work we study functions g K : of the type g K () = K for some fixed K. We give a complete characterization and representation of fuzzy implications K that lift the above functions g K to homomorphisms on (, ). Based on our results, we obtain for the first time a group structure on a subset of. We show that neutral implications which generate homomorphisms of the defined form and the set of such homomorphisms themselves contain abelian groups, suggesting that the investigated homomorphisms contain the group of inner homomorphisms. Finally, investigating the images of the studied homomorphisms, we present some natural partitions on and orderings on these equivalence classes.. PRELMNARES We assume that the reader is familiar with the classical results concerning basic fuzzy logic connectives, but to make this work self-contained, we present some notations, concepts and results employed in the rest of the work. A. Fuzzy mplications As noted before fuzzy implications are a generalization of classical implications to fuzzy logic. There are many acceptable definitions in the literature. Here, its definition is given from [5], [7]. Definition 1. A binary function on [0, 1] is called a fuzzy implication if it is such that (i) is decreasing in the first variable and increasing in the second variable, (ii) (0, 0) = (1, 1) = 1 and (1, 0) = 0.

2 As fuzzy implications are a generalization of classical implication to fuzzy logic, there are many properties that can be generalized to fuzzy logic. n the following definition, one such important property among them is given. Definition 2. [5] A fuzzy implication is said to satisfy the left neutrality property (NP) if (1, y) = y, y [0, 1]. (NP) n the following, we recall few basic definitions from algebra. For more details, please see any modern algebra book, for instance, [8], [9], [10]. Definition 3. A nonempty set G with a closed binary operation on G is called a semigroup if is associative in G. n addition if there is a unique element e G such that a e = a = e a, for all a G, then (G, ) is called a monoid and the unique element e is called the identity of the monoid. n a monoid G, for each a G, if there exists a unique b G s.t. a b = b a = e then (G, ) is called a group. The following example of a group will be required later. Example 4. For x, y [0, 1[ define of x, y as { x + y, if x + y < 1, x y = x + y 1, otherwise. t is easy to see that ([0, 1[, ) is a group. Definition 5. A monoid (M, ) is called a lattice ordered monoid (l-monoid ) if (M,,, ) is a lattice and is compatible with the lattice operations,. t is denoted by (M,,,, ). Definition 6. The center of the monoid (M, ) is defined and denoted by Z M = {z M x z = z x, for all x M}. A monoid is called abelian if its center is itself. Definition 7. An element e M is called idempotent if e 2 = e. The set of all idempotents of a monoid (M, ) is denoted by M. Definition 8. Let (L, L,, ), (M, M,, ) be two lattices. A map f : L M is called a lattice homomorphism from L to M if f preserves the lattice operations. Definition 9. Let (S, ), (T, ) be two algebraic structures. A map f : (S, ) (T, ) is called a homomorphism if f(a b) = f(a) f(b), a, b S. Definition 10. A bijective homomorphism from a group to itself is called an automorphism. Definition 11. Let G be a group and let a G. A map f a : G G defined by f a (x) = axa 1, for all x G is called an inner automorphism. The set of all inner automorphisms forms a group and it is denoted by nn(g).. MONOD STRUCTURE ON. As noted in the ntroduction, in [1], a few novel generating methods of fuzzy implications from a given pair of fuzzy implications were proposed. Among the algebraic structures thus obtained in [1], semigroup and monoid are the most natural. n this section we recall the proposed operation, structure (, ) and some relevant results presented in [1]. Definition 12. [1] For, J, define J as ( J)(x, y) = (x, J(x, y)), x, y [0, 1]. (1) Theorem 13. [1] J is an implication on [0, 1], i.e., J. Theorem 14. [1] (, ) forms a monoid, whose identity element is given by { 1, if x = 0, D (x, y) = y, otherwise. From Theorem in [5], it is clear that (,,, ) is a lattice. Hence we also have the following richer structure on. Lemma 15. The pentuple (,,,, ) is an l-monoid. For more details, like the properties that preserves, powers of elements under the operation, closures of the subsets of, please see [1]. V. HOMOMORPHSMS ON (, ) n algebra, especially in the study of groups three important concepts arise, viz., Normal subgroups, Homomorphisms and the group of inner automorphisms. Since, we are in the setting of monoids where invertibility of all the elements is not available, the concept of normal subgroups and hence a partition of the set with a possibility of unique representation based on these partitions is not possible. However, we would like to study the homomorphic images of (, ) and the group of inner homomorphisms. Towards this end, we define a function g K as follows. Definition 16. For any fixed K, define g K : (, ) (, ) by g K () = K,. Clearly g K is well defined. Since there are two structures on the set (namely, lattice and monoid), we investigate K, for which the map g K can be lifted to a homomorphism on these structures. A. Lattice Homomorphisms on (,,,, ). Since is an l-monoid (see Lemma 15), we have the following result. Proposition 17. For every K, the map g K is a lattice homomorphism. Proof: Let K. Let, J and x, y [0, 1]. Then, g K ( J)(x, y) = (( J) K)(x, y) = ( J)(x, K(x, y)) = max((x, K(x, y)), J(x, K(x, y))) = max(( K)(x, y), (J K)(x, y)) = (g K () g K (J))(x, y)

3 Similarly, one can prove that g K ( J) = g K () g K (J),, J. Thus g K is a lattice homomorphism. B. Semigroup Homomorphisms on (, ). n general, for every K, the map g K need not be a semigroup homomorphism (s.g.h). The nature of K affects whether the map g K is an s.g.h or not. Example 18. For instance, the Łukasiewicsz implication LK does not give an s.g.h. When (x, y) = KD (x, y) = max(1 x, y) and J(x, y) = RC (x, y) = 1 x+xy and x = 0.4, y = 0.2, we observe that ( J LK )(0.4, 0.2) = 0.92, while, ( LK J LK )(0.4, 0.2) = 1. n the following, we investigate the necessary conditions that K should satisfy for g K to be an s.g.h. Proposition 19. Let K be fixed arbitrarily. Then the following statements are equivalent: (i) g K is an s.g.h. (ii) J K = K J K for all J. Proof: (i) (ii): Let K and g K be an s.g.h. Then for all, J, we have J K = K J K. f we take = D, the identity in (, ) in the above equation, it follows that J K = K J K for all J. (ii) (i): Let K be s.t J K = K J K for all J. This directly implies that J K = K J K for every, since every is a well-defined function on. Thus g K is an s.g.h. From Proposition 19, the following result is straight forward. Remark 20. Let K be s.t g K is an s.g.h. Then (i) K 2 = K. Thus K n = K for all n N (ii) K n J K = J K, for all n N. C. Characterization of K such that g K is an s.g.h. From Example 18, we observe that g K need not be an s.g.h. for every K. Here in this subsection we give the characterizations of K s for which g K will be an s.g.h. on. First we characterize g K when the range of K is trivial, i.e., K(x, y) {0, 1} for all x, y [0, 1]. Theorem 21. Let K be s.t the range of K is trivial. Then the following statements are equivalent: (i) (ii) Proof: g K is an s.g.h. K = K δ for some δ ]0, 1] where { 1, if x < 1 or (x = 1 and y δ), K δ (x, y) = 0, otherwise. (i) (ii) = (ii). Let g K be an s.g.h. Claim: K(x, y) = 1, for all x [0, 1[ and for all y [0, 1]. f x = 0, it is trivial that K(x, y) = 1 for all y [0, 1]. Let 0 < x < 1. Suppose for some y 0 [0, 1), K(x, y 0 ) < 1. i.e., K(x, y 0 ) = 0. Since g K is an s.g.h, it follows that J K = K J K for all J. Now, (J K)(x, y 0 ) = J(x, K(x, y 0 )) = J(x, 0), (K J K)(x, y 0 ) = K(x, J(x, K(x, y 0 ))) = K(x, J(x, 0)). Since the range of K is trivial, J(x, 0) {0, 1} for all J. This gives a contradiction if we take a J s.t J(x, 0) / {0, 1}. Thus K(x, y) = 1, for all x < 1. Now for x = 1, y [0, 1], we have either K(x, y) = 0 or K(x, y) = 1. Let us define δ = sup{y [0, 1] K(1, y) = 0} Let us take K s.t K(1, y) is right continuous. Then for y δ, K(1, y) = 1 and for y < δ, K(1, y) = 0. Thus K = K δ. = (i). t can be verified easily. Remark 22. Note that in the proof of Theorem 21 we have chosen K δ such that it is right-continuous in the second variable, when x = 1. However, if we choose K δ such that it is left-continuous in the second variable at x = 1, i.e., K δ (1, y) = 1 when y > δ and K δ (1, y) = 0 when y δ, it can be easily verified that g K δ is still an s.g.h. This particular choice was made to conform to the tradition in the literature of requiring right-continuity in the second variable, as in the case of implications from which the deresiduum is constructed. n fact, it can be shown that the K δ characterizes exactly the set of all right annihilators in w.r.to. The proof of this result is omitted owing to space constraints. Lemma 23. Let A be the set of all right annihilators of. Then A = K δ = {K δ δ ]0, 1]} From Definitions 6, 7, the center Z and the set of all idempotent elements of the monoid (, ) are defined respectively as Z = { J = J, J }, = { = }. The following lemma which plays an important role when dealing with the implications K having (NP) gives a relation between the above two sets in (, ). Lemma 24. The center Z of the monoid (, ) is contained in the set, i.e., Z.

4 Proof: Let K Z. We need to show that K 2 = K. i.e., K(x, K(x, y)) = K(x, y), x, y [0, 1]. Suppose for some x 0 ]0, 1], y 0 [0, 1[ that Thus K(x 0, β) = α. Claim: β / {0, 1}. α = K(x 0, K(x 0, y 0 )) K(x 0, y 0 ) = β. Proof of the claim: Let β = 0 i.e., K(x 0, y 0 ) = 0 and hence K(x 0, K(x 0, y 0 )) = K(x 0, 0) = α 0. Then ( 0 K)(x 0, y 0 ) = 0 (x 0, K(x 0, y 0 )) = 0 (x 0, 0) = 0 (K 0 )(x 0, y 0 ) = K(x 0, 0 (x 0, y 0 )) = K(x 0, 0) = α 0. Thus 0 K K 0, contradicting the fact K Z. Thus β 0. Let β = 1. i.e., K(x 0, y 0 ) = 1. Then it implies that K(x 0, K(x 0, y 0 ) = 1, contradicting our assumption K(x 0, y 0 ) K(x 0, K(x 0, y 0 ). Thus β 1. Claim: α 1. Proof of the claim: Let α = 1. i.e., K(x 0, K(x 0, y 0 ) = α = 1. We have already proved that β 0, 1. Now define β by 1, if x = 0 or y = 1, β (x, y) = 0, if x = 1 and y = 0, β, otherwise. Now, β (x 0, K(x 0, y 0 )) = β and K(x 0, β (x 0, y 0 )) = K(x 0, β) = α = 1. Thus β (x 0, K(x 0, y 0 )) K(x 0, β (x 0, y 0 )), a contradiction to the fact that K Z. Thus α 1. Now, β (x 0, K(x 0, β)) = β (x 0, α) = β and K(x 0, β (x 0, β)) = K(x 0, β) = α. Thus β (x 0, K(x 0, β)) K(x 0, β (x 0, β)), a contradiction to the fact that K Z. Thus K and hence Z. Remark 25. n Lemma 24, the inclusion is strict. For example, 1 but 1 / Z. To see this let us take 0 given by { 1, if x = 0 or y = 1, 0 (x, y) = 0, otherwise. At x = 1, y = 0.4 we observe that ( 1 0 )(1, 0.4) = 1 (1, 0 (1, 0.4)) = 1 (1, 0) = 0 ( 0 1 )(1, 0.4) = 0 (1, 1 (1, 0.4)) = 0 (1, 1) = 1 (2) Of course, it is strightforward to see that 1 1 = 1, i.e., 1. Similarly, one can observe that GD, 0 but GD, 0 / Z. Based on Lemma 24, we have a first partial characterization of K such that g K is an s.g.h. Lemma 26. f K Z then g K is an s.g.h. Proof: Let K Z. Then, from Lemma 24, it follows that K. Let, J. Now, g K () g K (J) = ( K) (J K) = ( K) (K J) = (K K) J = (K J) = (J K) = ( J) K = g K ( J). Thus g K is an s.g.h. n fact, as we show in the following that the converse of Lemma 26 is also true for neutral implications, i.e., that K that satisfy (NP). Before proving this fact, we need the following result which gives a complete characterisation of all K for which g K is an s.g.h. Proposition 27. Let the range of K be nontrivial and g K be an s.g.h. Then there exist ɛ 0, ɛ 1 [0, 1] such that the vertical section K(1,.) has the following form: 0, if y [0, ɛ 0 ), 0 or ɛ 0, if y = ɛ 0, K(1, y) = y, if y (ɛ 0, ɛ 1 ), (3) ɛ 1 or 1, if y = ɛ 1, 1, if y (ɛ 1, 1]. Proof: Let K be s.t the range of K is nontrivial and g K is an s.g.h. Let us define two real numbers ɛ 0, ɛ 1 in the following way: ɛ 0 = sup{t [0, 1] K(1, t) = 0}, ɛ 1 = inf{t [0, 1] K(1, t) = 1}. Since K(1, 0) = 0 and K(1, 1) = 1, we have that 0 ɛ 0 ɛ 1 1. Further, since, g K is an s.g.h, we see that for all J the following equality should hold for all y [0, 1]: (i) (ii) (J K)(1, y) = (K J K)(1, y), (4) i.e., = J(1, K(1, y)) = K(1, J(1, K(1, y))). From the definition of ɛ 0, ɛ 1 above, it is clear that K(1, y) = 0 whenever 0 y < ɛ 0 and K(1, y) = 1, whenever ɛ 1 < y 1. Let ɛ 0 < y < ɛ 1. We claim that K(1, y) = y. f not, let there be a ɛ 0 < y 0 < ɛ 1 such that K(1, y 0 ) = y y 0. Let us choose a J such that J(1, y ) = y 0. Then, we have LHS (4) = J(1, K(1, y 0 )) = J(1, y ) = y 0, RHS (4) = K(1, J(1, K(1, y 0 ))) = K(1, J(1, y )) = K(1, y 0 ) = y,

5 from whence we obtain that g K is not an s.g.h., a contradiction. Thus K(1, y) = y whenever ɛ 0 < y < ɛ 1. (iii) Note that since ɛ 0, ɛ 1 are only the infimum and supremum of these sets, which are intervals due to the monotonicity of K in the second variable, they may not belong to these intervals themselves. n other words, K(1, ɛ 0 ) 0 and K(1, ɛ 1 ) 1. a) Clearly, if ɛ 0 = max{t [0, 1] K(1, t) = 0}, then K(1, ɛ 0 ) = 0. b) However, if ɛ 0 / {t [0, 1] K(1, t) = 0} then clearly 0 < K(1, ɛ 0 ) = δ. We claim that δ = ɛ 0. On the contrary, let δ ɛ, then, once again, one can choose a J such that J(1, δ) = ɛ 0. Then, we have LHS (4) = J(1, K(1, ɛ 0 )) = J(1, δ) = ɛ 0, RHS (4) = K(1, J(1, K(1, ɛ 0 ))) = K(1, J(1, δ)) = K(1, ɛ 0 ) = δ, from whence we obtain that g K is not an s.g.h., a contradiction. Thus K(1, ɛ 0 ) = ɛ 0. c) A similar proof as above shows that if ɛ 1 {t [0, 1] K(1, t) = 1} then K(1, ɛ 1 ) = 1, while if ɛ 1 / {t [0, 1] K(1, t) = 1} then K(1, ɛ 1 ) = ɛ 1. On the contrary, let K(1, y 0 ) = 0 for some y 0 ]0, 1[. Then, on the one hand, and on the other hand, ( 1 K)(1, y 0 ) = 1 (1, K(1, y 0 )) = 1 (1, 0) = 0, (K 1 )(1, y 0 ) = K(1, 1 (1, y 0 )) = K(1, 1) = 1, which contradicts the fact K Z. Thus for any y 0 ]0, 1[, K(1, y 0 ) 0. Similarly, by taking 0 instead of 1, above we can show that for any y 0 ]0, 1[, K(1, y 0 ) 1. From Proposition 27, we see that this is equivalent to stating ɛ 0 = 0 and ɛ 1 = 1 and hence it follows that K must have (NP). D. Representations of K satisfying (NP) s.t g K is an s.g.h. We define below a special class of implications satisfying (NP). Definition 31. For ɛ [0, 1[ define { 1, if x ɛ, K ɛ (x, y) = y, otherwise, Note that K ɛ, for all ɛ [0, 1] and sup K ɛ = WB where (5) From Proposition 27 we see that if ɛ 0 = 0 and ɛ 1 = 1, then K has (NP). Further, we have the following result: WB (x, y) = { 1, if x < 1, y, if x = 1. (6) Corollary 28. Let the range of K be nontrivial and g K be an s.g.h. Then either K has (NP) or the vertical section K(1,.) is discontinuous. Lemma 29. f K Z, then the range of K is nontrivial. Proof: Let K Z. Suppose that the range of K is trivial. Since K Z, from Lemma 26 it follows that g K is an s.g.h. Again from Theorem 21, it follows that K = K δ for some δ ]0, 1]. Choose two real numbers δ, δ ]0, 1] s.t δ < δ < δ. Let = β as defined in the equation (2) with β = δ. Then ( K δ )(1, δ ) = (1, K δ (1, δ )) = 1 but (K δ )(1, δ ) = K δ (1, (1, δ )) = K δ (1, δ ) = 0, contradicting that K Z. Thus the range of K is nontrivial. Here we claim that δ 1. f δ = 1, then K = K δ will be of the form { 1, if x < 1 or y = 1 K(x, y) = 0, if x = 1 & y 1 Now it is easy to see that ( 1 K)(1, 0.2) = 0 where as (K 1 )(1, 0.2) = 1, proving that K / Z, a contradiction to the fact K Z. Thus δ 1 and such δ, δ do exist. Proposition 30. Let K Z. Then K has (NP). Proof: Let K Z. From Lemma 26, it follows that g K is an s.g.h and also from Lemma 29, it follows that range of K is nontrivial. To prove that K has (NP), from Proposition 27 it suffices to show that K(1, y) 0 or 1 for any y ]0, 1[. For notational convenience, we denote the set of all such K ɛ implications by K ɛ = { = K ɛ for some ɛ [0, 1[}. K + ɛ = { = K ɛ for some ɛ [0, 1[} WB. The following results lists a few properties of the implications from the set K + ɛ. Proposition 32. The following properties hold true. (i) ɛ 1 < ɛ 2 K ɛ1 K ɛ2 (ii) K ɛ1 K ɛ2 = K max(ɛ1,ɛ 2) = K ɛ2 K ɛ1 (iii) (iv) ɛ 1 < ɛ 2 g Kɛ1 (K ɛ2 ) = g Kɛ2 (K ɛ1 ) = K ɛ2 g Kɛ () = g (K ɛ ), for all (v) ɛ 1 < ɛ 2 g Kɛ2 () g Kɛ1 (). Proposition 33. (K + ɛ, ) is a commutative submonoid of (, ). n the following we present some results relating to the sets K ɛ, Z and. Lemma 34. K + ɛ Z. Proof: n Lemma 24, we proved that Z. So here it is enough to show that K + ɛ Z. Now if K + ɛ then { 1, if x ɛ, (x, y) = y, otherwise.

6 for some ɛ [0, 1[ or = WB. For any J and K ɛ, we have { 1, if x ɛ, ( J)(x, y) = (J )(x, y) = J(x, y), otherwise. showing that Z. f = WB, then { 1, if x < 1, ( J)(x, y) = (J )(x, y) = J(x, y), otherwise. for all J. Thus WB Z. n fact, the opposite inclusion ( i.e., Z K + ɛ ) is also true, a fact that we prove in Lemma 36. Now, we are ready to give a complete characterization and representation of K satisfying (NP) for which g K will be an s.g.h. Theorem 35. Let K satisfy (NP). The following statements are equivalent: (i) g K is an s.g.h. (ii) K K + ɛ. (i) Proof: = (ii): Let g K be an s.g.h for some K. Since K has (NP) the range of K is [0, 1]. Let α < 1 be chosen arbitrarily. Then there exists some x 0 ]0, 1], y 0 [0, 1[, s.t K(x 0, y 0 ) = α < 1. We keep K fixed, vary J and investigate the equivalence J K = K J K. When J = 0, we have (J K)(x 0, y 0 ) = 0 (x 0, K(x 0, y 0 )) = 0 (x 0, α) = 0, (K J K)(x 0, y 0 ) = K(x 0, 0 (x 0, K(x 0, y 0 ))) = K(x 0, 0). Since g K is an s.g.h., K(x 0, 0) = 0. Hence, if K(x 0, y 0 ) = α < 1, then K(x 0, 0) = 0. Now, (J K)(x 0, 0) = J(x 0, K(x 0, 0)) = J(x 0, 0) and (K J K)(x 0, 0) = K(x 0, J(x 0, K(x 0, 0))) = K(x 0, J(x 0, 0)). Now let us, once again, choose J as in (2) with β = y 0. Thus we have J(x 0, 0) = y 0 and hence y 0 = J(x 0, 0) = K(x 0, J(x 0, 0)) = K(x 0, y 0 ) = α. = α = y 0. Since α is chosen arbitrarily, we have K(x 0, y) = y, y [0, 1]. (7) Let x = inf{x K(x, y) = y, for all y} 0. Note that the infimum exists because K has (NP). Claim: K(s, y) = 1, for any s [0, x [ and for all y [0, 1]. Proof of the claim: On the contrary, let us suppose that 1 > K(s, y 0 ) = y 1 > y 0 for some y 0, y 1. Now, J(s, K(s, y 0 )) = J(s, y 1 ), K(s, J(s, K(s, y 0 ))) = K(s, J(s, y 1 )). Once again, choosing a J as in (2) with β = y 0, we have J(s, y 1 ) = y 0 and K(s, J(s, y 1 )) = K(s, y 0 ) = y 1, = J(s, K(s, y 0 )) K(s, J(s, K(s, y 0 ))), i.e., g K is not an s.g.h., a contradiction. Thus K(s, y) = 1, for all s [0, x [. Now the question is what value should one assign to K(x, y). Since it is customary to assume leftcontinuity of fuzzy implications in the first variable, we let K(x, y) = 1. Note that letting K(x, y) = y also gives a K such that g K is a homomorphism. From the above claim and (7) we see that every K is of the form (5) for some ɛ [0, 1) or K = WB. (ii) = (i): Follows from Lemmata 34 and 26. Lemma 36. Let K Z. Then K K + ɛ i.e.,z K + ɛ. Proof: Let K Z. From Lemma 26 it follows that g K is an s.g.h and also from Lemma 29 it follows that range of K is nontrivial. Further, from Proposition 30 we know that K has (NP). Again from Theorem 35 it follows that K K + ɛ. Corollary 37. Z = K + ɛ. Proof: n Lemma 34 we proved that Z K + ɛ. From Lemma 36 it follows that Z K + ɛ. From Corollary 37, it follows that the center Z is nothing but the set of all implications of the form K ɛ for some ɛ [0, 1) or WB. For the characterization of some well-known families of fuzzy implications satisfying = (i.e., ) of (, ), please see [11]. Using Corollary 37 we have the following complete result. Theorem 38. Let K be s.t K has (NP). Then the following statements are equivalent: (i) g K is an s.g.h. (ii) K Z (iii) K K + ɛ. V. THE GROUP OF (NNER)HOMOMORPHSMS ON (, ) While the above sections have detailed the existence of s.g.h on the monoid (, ), it is not immediately apparent why one should consider the particular definition of g K as given in Definition 16. n the following we make a few observations to throw some light on the choice of this particular function. From abstract algebra, one can notice that the kernel of a homomorphism partitions the whole set (Groups, Rings, Vector spaces, etc). Typically, one resorts to finding the normal sugroups or the ideals to help us in this quest. The following classical result from the theory of groups goes much further in terms of characterizing the partitions so obtained.

7 Theorem 39 ([9]). Let G be a group, Z G its center and let nn(g) denote the group of all inner automorphisms on G. Then G = nn(g). Z G The above result states that Z G, the center of the group G partitions the whole group G into equivalence classes, called cosets, and the group of all equivalence classes is isomorphic to the group of inner automorphisms nn(g) on G. From the results presented above and noting the important role played by the center Z of the monoid (, ), one can only surmise that the set {g K K K ɛ } acts as the group of inner semigroup homomorphisms for the monoid (, ). n fact, as is shown below, we see that the above set itself forms a group. Towards this end, let us denote by H the following set of s.g.h of the type g K : H = {g Kɛ K ɛ K ɛ and ɛ [0, 1[ }. (8) We show below that K ɛ does form a group. The group presented in Example 4 plays an important role in the following, which we recall for the convenience of the readers. Example 4. For x, y [0, 1[ define of x, y as { x + y, if x + y < 1, x y = x + y 1, otherwise. Definition 40. For any K ɛ1, K ɛ2 K ɛ, define K ɛ1 K ɛ2 = K ɛ1 ɛ 2. The proof of the following result is straight forward. Lemma 41. (K ɛ, ) is an abelian group. Definition 42. For any g Kɛ1, g Kɛ2 H define the composition on H as g Kɛ1 g Kɛ2 = g Kɛ1 (g Kɛ2 ). i.e., (g Kɛ1 g Kɛ2 )() = g Kɛ1 (g Kɛ2 ()) = g Kɛ1 ( K ɛ2 ) = K ɛ2 K ɛ1. Note that if ɛ 1 < ɛ 2, then g Kɛ1 g Kɛ2 = g Kɛ2. n general, we have g Kɛ1 g Kɛ2 = g Kɛ, where ɛ = max(ɛ 1, ɛ 2 ). We first note that H forms a commutative monoid under function composition. Once again, the proof of the following result is straight forward. Lemma 43. The set (H, ) is a commutative monoid. Now we present the main result of this section that H indeed can be shown to form a group under the following operation. Definition 44. For ɛ 1, ɛ 2 [0, 1) we have g Kɛ1, g Kɛ2 H. Define g Kɛ1 g Kɛ2 = g Kɛ1 ɛ 2. Lemma 45. (H, ) is an abelian group. Theorem 46. The groups ([0, 1[, ), (K ɛ, ), (H, ) are all isomorphic. Proof: The maps defined by ɛ K ɛ, ɛ g Kɛ from [0, 1[ to K ɛ, [0, 1[ to H respectively are isomorphisms. V. ALGEBRAC STRUCTURES ON THE HOMOMORPHC MAGES OF g K So far, we have seen the algebraic structures on the subsets of that are obtained by characterizing s.g.h of the type g K. n this section, we study the range sets of these s.g.h and highlight the structures that exist on them. Most importantly, we show that they give rise to a natural partition on the set of all fuzzy implications. Towards this end, we fix the following notation: For any ɛ [0, 1], g Kɛ : R ɛ, where R ɛ = Range(g Kɛ ). Note that when ɛ = 1, we fix K ɛ = WB. A. Algebraic structures on R ɛ Since (,,,, ) is an l-monoid (see Lemma 15), the following are true for all, J and K ɛ K ɛ. ( K ɛ ) (J K ɛ ) = ( J) K ɛ, ( K ɛ ) (J K ɛ ) = ( J) K ɛ. The following results are easy to see. Lemma 47. For any ɛ [0, 1], R ɛ is a sublattice of. Lemma 48. For any ɛ [0, 1], (R ɛ, ) is a submonoid of (, ) with identity K ɛ. Theorem 49. (i) The range sets of g Kɛ form a telescopic sequence of sets, i.e., if 0 = ɛ 0 < ɛ 1 <... < ɛ n <... < ɛ = 1 then { WB } = R ɛ... R ɛn... R ɛ1 R ɛ0 =. (ii) R ɛi is a sub-lattice of R ɛj whenever i j. (iii) = and R ɛi = { WB }. i=0 R ɛi i=0 (iv) For any ɛ 1, ɛ 2 [0, 1], we have R ɛ1 R ɛ2 = R min(ɛ1,ɛ 2) and R ɛ1 R ɛ2 = R max(ɛ1,ɛ 2). Theorem 50. Let us denote the set of all R ɛ for ɛ [0, 1] by R = {R ɛ = Range(g Kɛ ) ɛ [0, 1]}. Then (R,,,, { WB }, ) is a Complete (bounded) Lattice with the singleton set { WB } and the whole of acting as the smallest and the largest elements of R, respectively. B. nduced partition on Based on the results obtained above, in this section, a partition on the set of all fuzzy implications is discussed and a total ordering on these equivalence classes is defined. Towards this end, we define a real number γ for every in the following way. Definition 51. Define γ : [0, 1] as γ() := γ = sup{x [0, 1] (x, 0) = 1},.

8 For, J, define a relation γ by γ J γ = γ J. Now it is easy to check that γ is an equivalence relation on and thus γ partitions into equivalence classes. Let γ denote the set of all equivalence classes, i.e., γ = {[] }. Definition 52. For [], [J] γ, define [] γ [J] if γ γ J. ( ) Theorem 53. γ, γ is a totally ordered set. C. Partition on : An Equivalent Definition n this section, we give another but equivalent formulation of the partition and the ordering obtained above. Definition 54. Define λ(): [0, 1] by λ() = λ = sup{α [0, 1] R α }. Define a relation λ on by λ J λ() = λ(j). t is easy to see that λ is an equivalence relation on. Also this equivalence relation partitions the whole set into equivalence classes. Let λ be the set of all equivalence classes w. r. t the relation λ. Definition 55. For [], [J] λ define [] λ [J] if λ λ J. Theorem 56. ( λ, λ ) is a totally ordered set. Theorem 57. [] γ = [] λ, for any. V. CONCLUDNG REMARKS n this work we have proposed and discussed a particular type of lattice homomorphisms that exist on the monoid (, ) of fuzzy implications proposed in [1]. Further, we have investigated and given a complete characterization and representation of those fuzzy implications that lift the above functions to semigroup homomorphisms (s.g.h) on (, ). We have shown that neutral implications which generate homomorphisms of the defined form and the set of such s.g.h themselves form abelian groups, suggesting that the investigated s.g.h s form the group of inner homomorphisms. Another interesting fall out of the above investigation is that we obtain a group structure on a subset of. Note that, to the best of the authors knowledge, this is the first work to present such a rich algebraic structure on. We have only investigated fuzzy implications K that have continuous vertical sections at x = 1 and ensure that g K is a homomorphism. We intend to take up the investigation of K s whose vertical section is non-continuous in the near future. t is not immediately clear how one may define different homomorphisms g on the monoid (, ) other than the one given in Definition 16 without in some form knowing the representation of the fuzzy implication that g acts on. However, such attempts are important to know if and what kind of correspondences exist between the monoid (, ) and some well known monoids in the literature. Note also that while functions of the type g K () = K and g K () = K K might appear different from the one given in Definition 16, it is easy to see that if K Z they reduce to the function g K. However, whether the converse is true is yet to be ascertained. While we have obtained a group structure on a subset of, the group operation was different from the binary operation that makes a monoid. Thus, it would be both interesting and revealing to investigate group-theoretic structures that may exist in the monoid (, ). This study is already under way and we intend to present these results in a future work. V. ACKNOWLEDGEMENTS The first author would like to acknowledge CSR HRDG, NDA for its fellowship (09/1001/(0008)/2011-EMR-) and the second author would like to thank the Department of Science and Technology, NDA, project SERB/F/2862/ for its partial support. REFERENCES [1] N. R. Vemuri and B. Jayaram, Fuzzy implications: Novel generation process and the consequent algebras, in PMU (2), ser. Communications in Computer and nformation Science, S. Greco, B. Bouchon- Meunier, G. Coletti, M. Fedrizzi, B. Matarazzo, and R. R. Yager, Eds., vol Springer, 2012, pp [2] R. R. Yager, On some new classes of implication operators and their role in approximate reasoning, nf. Sci., vol. 167, no. 1-4, pp , [3] Lotfi.A.Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, Systems, Man and Cybernetics, EEE Transactions on, vol. SMC-3, no. 1, pp , [4] M. Baczyński, J. Drewniak, and J. Sobera, Semigroups of fuzzy implications, Tatra Mt. Math Publ, vol. 21, pp , [5] M. Baczyński and B. Jayaram, Fuzzy mplications, ser. Studies in Fuzziness and Soft Computing. Springer, 2008, vol [6] J. Drewniak, nvariant fuzzy implications, Soft Comput., vol. 10, no. 10, pp , [7] J. C. Fodor and M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic, [8] J. M. Howie, ntroduction to Semigroup Theory. Academic Press, [9]. N. Herstein, Topics in Algebra. Wiley, [10] J. J. Rotman, An ntroduction to the Theory of Groups. Springer, [11] Y. Shi, D. Ruan, and E. E. Kerre, On the characterizations of fuzzy implications satisfying (x, y) = (x, (x, y)), nf. Sci., vol. 177, no. 14, pp , 2007.