THE notion of fuzzy groups defined by A. Rosenfeld[13]

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1 I-Vague Groups Zelalem Teshome Wale Abstract The notions of I-vague groups with membership and non-membership functions taking values in an involutary dually residuated lattice ordered semigroup are introduced which generalize the notions with truth values in a Boolean algebra as well as those usual vague sets whose membership and non-membership functions taking values in the unit interval [0, 1]. Moreover, various operations and properties are established. Keywords Involutary dually residuated lattice ordered semigroup, I-vague set and I-vague group. I. INTRODUCTION THE notion of fuzzy groups defined by A. Rosenfeld[1] is the first application of fuzzy set theory in Algebra. Since then a number of works have been done in the area of fuzzy algebra. M. Demirci[5] studied vague groups. R. Biswas[2] defined the notion of vague groups analogous to the idea of Rosenfeld [1]. H. Khan, M. Ahmad and R. Biswas[8] studied vague groups and made some characterizations. N. Ramakrishna[10] studied vague groups and vague weights. The vague sets of W. L. Gau and D. J. Buehrer[6] and Atanassov s[1] intuitionistic fuzzy sets are mathematically equivalent objects[]. In this paper we prefer the terminology of vague sets as the algebraic study intiated by Biswas[2] follows the terminology of vague sets. K. L. N. Swamy[14], [15], [16] introduced the concept of dually residuated lattice ordered semigroup(in short DRLsemigroup) which is a common abstraction of Boolean algebras and lattice ordered groups. The subclass of DRLsemigroups which are bounded and involutary(i.e having 0 as least, 1 as greatest and satisfying 1-(1-x) = x) which is categorically equivalent to the class of MV-algebras of C. C. Chang[4] and well studied offer a natural generalization of the closed unit interval [0, 1] of real numbers as well as Boolean algebras. Thus, the study of vague sets (t A, f A ) with values in an involutary DRL-semigroup promises a unified study of real valued vague sets and also those Boolean valued vague sets[11]. In his thesis T. Zelalem[19] studied the concept of I-vague sets. In this paper using the definition of I-vague sets, we defined and studied I-vague groups where I is an involutary DRLsemigroup. In this paper we shall recall some basic results in [14], [15], [19] without proof. Moreover, notation, terminology and results of [19] are used in this paper. Throughout this paper, we shall denote the identity element of a group (G,.) by e and the order of an element x of G by O(x). Moreover, for x G, < x > denotes the cyclic group generated by x. Zelalem Teshome: Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia. zelalemwale@yahoo.com Manuscript received October xx, 2011; revised January 11, II. PRELIMINARIES Definition 2.1: [14] A system A = (A, +,, ) is called a dually residuated lattice ordered semigroup(in short DRLsemigroup) if and only if i) A = (A, +) is a commutative semigroup with zero 0 ; ii) A = (A, ) is a lattice such that a + (b c) = (a + b) (a + c) and a + (b c) = (a + b) (a + c) for all a, b, c A; iii) Given a, b A, there exists a least x in A such that b+x a, and we denote this x by a - b (for a given a, b this x is uniquely determined); iv) (a-b) 0 + b a b for all a, b A; v) a - a 0 for all a A. Theorem 2.2: [14] Any DRL-semigroup is a distributive lattice. Definition 2.: [19] A DRL-semigroup A is said to be involutary if there is an element 1( 0)(0 is the identity w.r.t. +) such that (i) a + (1 a) = 1 + 1; (ii) 1 (1 a) = a for all a A. Theorem 2.4: [15] In a DRL-semigroup with 1, 1 is unique. Theorem 2.5: [15] If a DRL-semigroup contains a least element x, then x = 0. Dually, if a DRL-semigroup with 1 contains a largest element α, then α = 1. Throughout this paper let I = (I, +,,,, 0, 1) be a dually residuated lattice ordered semigroup satisfying 1 (1 a) = a for all a I. Lemma 2.6: [19] Let 1 be the largest element of I. Then for a, b I (i) a + (1- a) = 1. (ii) 1 - a = 1 - b a = b. (iii)1 - (a b ) = (1- a) (1- b). Lemma 2.7: [19] Let I be complete. If a α I for every α, then (i) 1 a α = (1 a α ). α α (ii) 1 a α = (1 a α ). α α Definition 2.8: [19] An I-vague set A of a non-empty set G is a pair (t A, f A ) where t A : G I and f A : G I with t A (x) 1 f A (x) for all x G. Definition 2.9: [19] The interval [t A (x), 1 f A (x)] is called the I-vague value of x G and is denoted by V A (x). Definition 2.10: [19] Let B 1 = [a 1, b 1 ] and B 2 = [a 2, b 2 ] be two I-vague values. We say B 1 B 2 if and only if a 1 a 2 and b 1 b 2. Definition 2.11: [19] An I-vague set A = (t A, f A ) of G is said to be contained in an I-vague set B = (t B, f B ) of G written as A B if and only if t A (x) t B (x) and f A (x) f B (x) for all x G. A is said to be equal to B written as A = B if and only if A B and B A. 197

2 Definition 2.12: [19] An I-vague set A of G with V A (x) = V A (y) for all x, y G is called a constant I-vague set of G. Definition 2.1: [19] Let A be an I-vague set of a non empty set G. Let A (α, β) = x G : V A (x) [α, β]} where α, β I and α β. Then A (α, β) is called the (α, β) cut of the I-vague set A. Definition 2.14: [19] Let S G. The characteristic function of S denoted as χ S = (t χs, f χs ), which takes values in I is defined as follows: and t χs (x) = f χs (x) = 1 if x S ; 0 otherwise 0 if x S ; 1 otherwise. χ S is called the I-vague characteristic set of S in I. Thus [1, 1] if x S ; V χs (x) = [0, 0] otherwise. Definition 2.15: [19] Let A = (t A, f A ) and B = (t B, f B ) be I-vague sets of a set G. (i) Their union A B is defined as A B = (t A B, f A B ) where t A B (x) = t A (x) t B (x) and f A B (x) = f A (x) f B (x) for each x G. (ii) Their intersection A B is defined as A B = (t A B, f A B ) where t A B (x) = t A (x) t B (x) and f A B (x) = f A (x) f B (x) for each x G. Definition 2.16: [19] Let B 1 = [a 1, b 1 ] and B 2 = [a 2, b 2 ] be I-vague values. Then (i) isupb 1, B 2 } = [supa 1, a 2 }, supb 1, b 2 }]. (ii) iinfb 1, B 2 } = [infa 1, a 2 }, infb 1, b 2 }]. Lemma 2.17: [19] Let A and B be I-vague sets of a set G. Then A B and A B are also I-vague sets of G. Let x G. From the definition of A B and A B we have (i) V A B (x) = isupv A (x), V B (x)}; (ii) V A B (x) = iinfv A (x), V B (x)}. Definition 2.18: [19] Let I be complete and A i : i } be a non empty family of I-vague sets of G where A i = (t Ai, f Ai ). Then (i) A i = ( t Ai, f Ai ) (ii) A i = ( t Ai, f Ai ) Lemma 2.19: [19] Let I be complete. If A i : i } is a non empty family of I-vague sets of G, then A i and A i are I-vague sets of G. Definition 2.20: [19] Let I be complete and A i = (t Ai, f Ai ) : i } be a non empty family of I vague sets of G. Then for each x G, (i) isupv Ai (x) : i } = [ t Ai (x), (1 f Ai )(x)]. (ii) iinfv Ai (x) : i } = [ t Ai (x), (1 f Ai )(x)]. III. I-VAGUE GROUPS Definition.1: Let G be a group. An I-vague set A of a group G is called an I-vague group of G if (i) V A (xy) iinfv A (x),v A (y)} for all x, y G and (ii)v A (x 1 ) V A (x) for all x G. Lemma.2: If A is an I-vague group of a group G, then V A (x) = V A (x 1 ) for all x G. Proof: Since A is an I-vague group of G, V A (x 1 ) V A (x) for all x G. V A (x) = V A ((x 1 ) 1 ) V A (x 1 ). Hence the lemma follows. Lemma.: If A is an I-vague group of a group G, then V A (e) V A (x) for all x G. Proof: Let x G. V A (e) = V A (xx 1 ) iinfv A (x), V A (x 1 )} = V A (x) for all x G. Therefore V A (e) V A (x) for all x G. Lemma.4: Let m Z. If A is an I-vague group of a group G, then V A (x m ) V A (x) for all x G. Proof: Let m Z. We prove that V A (x m ) V A (x) for all x G. Since V A (e) V A (x) for all x G by lemma., the statement is true for m = 0. First we prove that the lemma is true for positive integers by induction. Since V A (x) V A (x), it is true for m = 1. Assume it is true for m. V A (x m+1 ) = V A (x m x) iinfv A (x m ),V A (x)} = V A (x). Thus V A (x m+1 ) V A (x). Hence the statement is true for non-negative integers. Suppose that m is a negative integer. V A (x m ) = V A ((x 1 ) m ) V A (x 1 ) = V A (x). We have V A (x m ) V A (x). Consequently, V A (x m ) V A (x) for all x G and for every integer m. Hence the lemma follows. Lemma.5: A necessary and sufficient condition for an I-vague set A of a group G to be an I-vague group of G is that V A (xy 1 ) iinfv A (x), V A (y)} for all x, y G. Proof: Let A be an I-vague set of G. Suppose that V A (xy 1 ) iinfv A (x),v A (y)} for all x, y G. Let x G. Then V A (e) = V A (xx 1 ) iinfv A (x), V A (x)} = V A (x). Thus V A (e) V A (x) for all x G. V A (x 1 ) = V A (ex 1 ) iinfv A (e),v A (x)} = V A (x). Thus V A (x 1 ) V A (x) for each x G. Let x, y G. Then V A (xy) = V A (x(y 1 ) 1 ) iinfv A (x),v A (y 1 )} iinfv A (x), V A (y)}. Hence V A (xy) iinfv A (x),v A (y)} for each x,y G, so A is an I-vague group of G. Conversely, suppose that A is an I-vague group of G. Let x, y G. Then V A (xy 1 ) iinfv A (x), V A (y 1 )} = iinfv A (x), V A (y)}. Therefore V A (xy 1 ) iinfv A (x), V A (y)} for all x, y G. Hence the theorem follows. Lemma.6: Let H be a subgroup of G and [γ, δ] [α, β] with α, β,γ,δ I where α β and γ δ. Then the I-vague set A of G defined by [α, β] if x H ; V A (x) = [γ, δ] otherwise is an I-vague group of G. Proof: Let H be a subgroup of G. We prove that the I-vague set A defined as above is an I-vague group of G. 198

3 Let x, y G. If xy 1 H, then V A (xy 1 ) = [α, β]. Hence V A (xy 1 ) iinfv A (x),v A (y)}. If xy 1 / H, then either x / H or y / H. Thus, iinfv A (x), V A (y)} = [γ, δ]. It follows that V A (xy 1 ) iinfv A (x), V A (y)}. Hence V A (xy 1 ) iinfv A (x), V A (y)} for every x, y G. Therefore A is an I-vague group of G. Example: Consider the group (Z, +). Let I be the unit interval [0, 1] of real numbers. Let a b = min 1, a + b}. With the usual ordering (I,,, ) is an involutary DRLsemigroup. Define the I-vague set A of G as follows: V A (x) = [a 1, b 1 ] if x 4Z; [a 2, b 2 ] if x 2Z 4Z; [a, b ] otherwise where [a, b ] [a 2, b 2 ] [a 1, b 1 ] and a i, b i [0, 1] for i = 1, 2,. Then A is an I-vague group of G. We prove that V A (xy 1 ) iinfv A (x), V A (y)} for all x, y G. (i) If xy 1 4Z, then V A (xy 1 ) = [a 1, b 1 ] iinfv A (x), V A (y)}. (ii) If xy 1 2Z 4Z, then there exist x,y Z such that x / 4Z or y / 4Z. This implies iinfv A (x), V A (y)} [a 2, b 2 ] = V A (xy 1 ). (iii) If xy 1 = x y is odd, then one of them must be odd. Hence iinfv A (x), V A (y)} = [a, b ] V A (xy 1 ). Therefore A is an I-vague group of G. Lemma.7: Let H and H G. The I-vague characterstic set of H, χ H is an I-vague group of G iff H is a subgroup of G. Proof: Suppose that H is a subgroup of G. By Lemma.6, χ H is an I-vague group of G. Conversely, suppose that χ H is an I-vague group of G. We show that H is a subgroup of G. Let x, y H. Then V χh (xy 1 ) iinfv χh (x),v χh (y)} = [1, 1]. Hence V χh (xy 1 ) = [1, 1], so xy 1 H. Therefore H is a subgroup of G. Hence the lemma follows. Lemma.8: If A and B are I-vague groups of a group G, then A B is also an I-vague group of G. Proof: Let A and B are I-vague groups of G. Then A B is an I-vague set of G by lemma Now we show that V A B (xy 1 ) iinfv A B (x),v A B (y)} for each x, y G. Let x, y G. Then V A B (xy 1 )= iinfv A (xy 1 ), V B (xy 1 )} iinfiinfv A (x), V A (y)}, iinfv B (x), V B (y)}} = iinfiinfv A (x), V B (x)}, iinf V A (y), V B (y)}} = iinfv A B (x), V A B (y)}. Thus V A B (xy 1 ) iinfv A B (x), V A B (y)} for every x, y G. Therefore A B is an I-vague group of G. Lemma.9: Let I be complete. If A i : i } is a non empty family of I-vague groups of G, then A i is an I- vague group of G. Proof: Let A = A i. Then A is an I-vague set of G by lemma Now we prove that V A (xy 1 ) iinfv A (x),v A (y)} for every x, y G. Let x, y G. Then V A (xy 1 ) = V A i (xy 1 ) = iinfv Ai (xy 1 ) : i } iinfiinfv Ai (x),v Ai (y)} : i } = iinfiinfv Ai (x) : i }, iinfv Ai (y) : i }} = iinfv A (x), V A (y)}. Hence V A (xy 1 ) iinfv A (x), V A (y)} for every x, y G. Therefore A i is an I-vague group of G. Example: Let I = The positive divisors of 0 = 1,2,, 5,6, 10, 15,0} in which x y = The least common multiple of x and y. x y = The greatest common divisor of x and y. x = 0 x. Then I = (I,,,, 1, 0) is a Boolean algebra. Hence it is an involutary DRL-semigroup. Consider the group G = (Z, +). Then H = (2Z, +) and K = (Z, +) are subgroups of G. Define the I-vague groups A and B of G as follows: [15, 0] if x H ; V A (x) = [5, 10] otherwise and V B (x) = [15, 0] if x K; [5, 10] otherwise. Let x = 2 and y =. xy = x + y = 5. V A B (xy) = V A B (5) = isupv A (5), V B (5)} = [5, 10]. V A B (x) = V A B (2) = isupv A (2), V B (2)} = [15, 0]. V A B (y) = V A B () = isupv A (), V B ()} = [15, 0]. iinfv A B (x), V A B (y)} = [15, 0]. But V A B (xy) = [5, 10] < [15, 0] = iinfv A B (x), V A B (y)}. Therefore A B is not an I-vague group of G. The above example shows that the union of two I-vague groups of G is not an I-vague group of G. However we have the following. Lemma.10: Let A be an I-vague group of G and B be a constant I-vague group of G. Then A B is an I-vague group of G. Proof: Let A be an I-vague group of G and B be a constant I-vague group of G. Then A B is an I-vague set of G by lemma We prove that A B is an I-vague group of G. Since B is a constant I-vague group of G, V B (x) = V B (y) for all x, y G. Let x, y G. Then V A B (xy 1 )= isupv A (xy 1 ), V B (xy 1 ) } isupiinfv A (x), V A (y)}, V B (x)} = iinfisupv A (x), V B (x)}, isupv A (y), V B (x)}} = iinfisupv A (x), V B (x)}, isupv A (y), V B (y)}} = iinfv A B (x), V A B (y)}. Thus V A B (xy 1 ) iinfv A B (x), V A B (y)} for all x, y G. Hence A B is an I-vague group of G. Theorem.11: An I-vague set A of a group G is an I- vague group of G if and only if for all α, β I with α β, the I-vague cut A (α, β) is a subgroup of G whenever it is non empty. Proof: Let A be an I-vague set of G. Suppose that A is an I-vague group of G. We prove that A (α, β) is a subgroup of G whenever it is non empty. Let x, y A (α, β). Then V A (x) [α, β] and 199

4 V A (y) [α, β]. Since A is an I-vague group of G, V A (xy 1 ) iinfv A (x), V A (y)} [α, β]. Hence xy 1 A (α, β), so A (α, β) is a subgroup of G. Conversely, suppose that for all α, β I with α β, the non empty set A (α, β) is a subgroup of G. We prove that A is an I-vague group of G. Let x, y G. Suppose that V A (x) = [α, β] and V A (y) = [γ, δ]. Then x A (α, β) and y A (γ, δ). Let iinfv A (x), V A (y)} = [α γ, β δ] = [η, ζ]. It follows that x, y A (η, ζ). Since A (η, ζ) is a subgroup of G, xy 1 A (η, ζ). Thus V A (xy 1 ) [η, ζ]. As a result we have V A (xy 1 ) iinfv A (x), V A (y)}. Therefore A is an I-vague group of G. Hence the theorem follows. Theorem.12: Let A be an I-vague group of a group G. If V A (xy 1 ) = V A (e) for x, y G, then V A (x) = V A (y). Proof: Suppose that V A (xy 1 ) = V A (e) for x, y G. V A (x) = V A (xe) = V A (xy 1 y) iinfv A (xy 1 ), V A (y)}= iinfv A (e), V A (y)} = V A (y). Thus V A (x) V A (y). Since V A (xy 1 ) = V A (yx 1 ), we have V A (y) V A (x). Therefore V A (x) = V A (y). Hence the theorem follows. The following example shows that the converse of the preceeding theorem is not true. Example: Let I be the unit interval [0, 1] of real numbers. Define a b = min 1, a + b}. With the usual ordering (I,,, ) is an involutary DRL-semigroup. Consider G = (Z, +) and H = (Z, +). Let A be the I-vague group of G defined by [ 1 V A (x) = 2, 1] if x H ; [0, 4 ] otherwise. Let x = 2 and y = 1. V A (x) = V A (y) = [0, 4 ] but V A(xy 1 ) = V A (2 1) = V A (1) = [0, 4 ] V A(0). Theorem.1: Let A be an I-vague group of a group G and x G. Then V A (yx) = V A (xy) = V A (y) for all y G iff V A (x) = V A (e). Proof: Let A be an I-vague group of a group G and x G. Suppose that V A (yx) = V A (xy) = V A (y) for all y G. Take y = e. It follows that V A (x) = V A (e). Conversely, suppose that V A (x) = V A (e). We prove that V A (yx) = V A (xy) = V A (y) for all y G. For any y G, V A (y) V A (e) = V A (x). V A (xy) iinfv A (x), V A (y)} = V A (y). Hence V A (xy) V A (y). V A (y) = V A (ey) = V A (x 1 xy) iinfv A (x 1 ), V A (xy)} = iinfv A (x), V A (xy)} = iinfv A (e), V A (xy)} = V A (xy). Thus V A (y) V A (xy). Hence we have V A (xy) = V A (y) Similarly, V A (yx) = V A (y). Therefore V A (yx) = V A (xy) = V A (y). Hence the theorem follows. Lemma.14: Let A be an I-vague group of a group G. Then GV A = x G : V A (x) = V A (e)} is a subgroup of G. Proof: Let A be an I-vague group of G. Since e GV A, GV A and GV A G. Let x, y GV A. We prove that xy 1 GV A. V A (xy 1 ) iinfv A (x), V A (y)} = V A (e). Since V A (e) V A (xy 1 ) for all x, y GV A, V A (xy 1 ) = V A (e). Thus xy 1 GV A. Therefore GV A is a subgroup of G. Lemma.15: Let A be an I-vague group of a group G. If < x > < y > then V A (y) V A (x). Proof: Suppose that < x > < y >. Then x < y >. It follows that x = y m for some m Z. V A (x) = V A (y m ) V A (y). Therefore V A (x) V A (y). The following example shows that the converse of lemma.15 is not true. Example: Let I be the unit interval [0, 1] of real numbers. Let a b = min 1,a + b}. With the usual ordering (I,,, ) is an involutary DRL-semigroup. Let G = The klein-4-group = e, a, b, c}. Define the I-vague set A of G by [ 1 V A (x) = 2, 1] if x < a > ; [0, 4 ] otherwise. Then V A (c) = [0, 4 ] [1 2, 1] = V A(a) but < a > is not a subset of < c >. Definition.16: Let A be an I-vague group of a group G. Image of A is defined as ImA = V A (x) : x G}. Since V A (e) V A (x) for all x G, V A (e) is the greatest element of ImA. Theorem.17: Let A be an I-vague group of a group G. (i) If G is cyclic then ImA has a least element. (ii) If V A (x) V A (y) then < x > < y > and ImA has a least element then G is cyclic. Proof: Let A be an I-vague group of G. (i) Suppose that G is cyclic. Then G = < x > for some x G. We prove that V A (x) is the least element of ImA. Let y G. Then y = x m for some m Z. V A (y) = V A (x m ) V A (x). We have V A (x) V A (y) for every y G. Thus V A (x) is the least element of image of A. Hence ImA has a least element. (ii) Suppose that ImA has a least element say V A (x) for some x G. Let y G. Thus V A (y) V A (x) for all y G. By our condition we have < y > < x >. Since y < y >, y < x >. Hence G < x >. Consequently, we have G = < x >. Therefore G is cyclic. Lemma.18: Let A be an I-vague group of G. Let x, y G. The two conditions i) V A (x) = V A (y) < x >=< y > ii) V A (x) > V A (y) < x > < y > are equivalent to the condition V A (x) V A (y) < x > < y >. Proof: Assume that the two conditions are given. We prove that V A (x) V A (y) < x > < y >. If V A (x) > V A (y), then < x > < y > by (ii). If V A (x) = V A (y), then < x >=< y > by (i). We have < x > < y >. Conversely, assume that V A (x) V A (y) < x > < y >. (i) Suppose that V A (x) = V A (y). V A (x) = V A (y) V A (x) V A (y) and V A (y) V A (x). < x > < y > and < y > < x >. < x >=< y >. (ii) V A (x) > V A (y) V A (x) V A (y) < x > < y >. Thus V A (x) > V A (y) < x > < y >. Therefore (i) and (ii) are equivalent to V A (x) V A (y) < x > < y >. 1940

5 Theorem.19: Let A be an I-vague group of a group G such that the image set of A is given by ImA = I 0 > I 1 >... > I n } and such that (i) V A (x) = V A (y) < x >=< y >; (ii) V A (x) < V A (y) < x > < y >. Then G is a cyclic group of prime power order. Proof: Let A be an I-vague group of a group G. Since ImA =I 0 > I 1 >... > I n }, Im A has a least element. By theorem.17, G is cyclic. It follows that G = Z or G = Z m for some m N. Suppose that G = Z. Consider V A (2) and V A (). If V A (2) = V A (), then < 2 >=< > by (i). But this is not true since 2 / < >. So either V A (2) > V A () or V A () > V A (2). If V A (2) > V A (), then < 2 > < > by (ii). But this is not true since 2/ < >. If V A () > V A (2), then < > < 2 > by (ii). But this is not true since / < 2 >. Therefore G is not isomorphic to Z. Thus G = Z m for some m N. Suppose that m is not a prime power. Then there exist prime numbers p and q such that p q which are factors of m. Consider V A (p) and V A (q). Since ImA = I 0 > I 1 >... > I n }, either V A (p) V A (q) or V A (p) < V A (q). It follows that < p > < q > or < q > < p >, a contradiction. Thus our supposition is false. Therefore m is prime power. Hence the theorem follows. Theorem.20: Let G be a cyclic group of prime power order then there is an I and an I-vague group A of G such that for all x, y G (i) V A (x) = V A (y) < x > = < y >; (ii) V A (x) > V A (y) < x > < y >. Proof: Suppose that G is a cyclic group of order p n where p is prime and n N 0}. We find an I and an I-vague group A of G satisfying (i) and (ii). Step(1) We construct an I and an I-vague set of G. Let I be the unit interval [0, 1] of real numbers. Define a b = min 1, a + b} With the usual ordering (I,,, ) is an involutary DRLsemigroup. Now we construct our I-vague set of G. Let z G. Then O(z) = p i where i = 0, 1,2,..., n. Define A=(t A, f A ) where t A : G I and f A : G I such that t A (z) = a i, f A (z) = b i where a i, b i [0, 1] satisfying a i 1 b i for i = 0, 1, 2,..., n. Choose the intervals I 0, I 1,..., I n in such a way that I 0 > I 1 >... > I n where I i = [a i, 1 b i ]. Then V A (z) = I i. Hence A is an I-vague set of G. We have V A (e) = I 0. Step(2) We show that A is an I-vague group of G. Let x G. O(x) = O(x 1 ) implies V A (x) = V A (x 1 ). To show A is an I-vague group of G it remains to prove that V A (xy) iinfv A (x), V A (y)} for every x, y G. Let x, y G. Since G is a cyclic group of order p n and the order of the subgroup divides the order of the group, O(< x >) = p j, O(< y >) = p k and O(< xy >) = p m for some j, k, m 0,1,...,n} say. Therefore V A (x) = I j, V A (y) = I k and V A (xy) = I m. Moreover, since G is a cyclic group of prime power order, < x > < y > or < y > < x >. If < x > < y > then x, y < y >. Hence < xy > < y >. If < y > < x > then x, y < x >. Hence < xy > < x >. Therefore < xy > < y > or < xy > < x >. Assume that < xy > < x >. It follows that O(< xy >) < O(< x >) or O(< xy >) = O(< x >). If O(< xy >) < O(< x >) then m < j. It follows that I m > I j. Hence V A (xy) = I m iinfi j, I k }= iinfv A (x), V A (y)}. Thus V A (xy) iinfv A (x), V A (y)}. If O(< xy >) = O(< x >) then m = j. Hence I m = I j. V A (xy) = I m iinfi m, I k } =iinfv A (x), V A (y)}. Thus V A (xy) iinfv A (x), V A (y)}. In both cases V A (xy) iinfv A (x), V A (y)} and V A (x) V A (x 1 ) for all x, y G. Thus A is an I-vague group of G. Step() We show that A satisfies the conditions (i) and (ii) of the theorem. (a) Suppose that V A (x) = V A (y) for x, y G. By the definition of A we have O(< x >) = O(< y >). Since G is a cyclic group of prime power order, O(< x >) = O(< y >) implies < x >=< y >. Hence V A (x) = V A (y) < x > = < y >. (b) Suppose that V A (x) > V A (y) for x, y G. Then I j > I k. It follows that j < k. Hence p j < p k, so O(< x >) < O(< y >). 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