RATIO MATHEMATICA 27 (2014) 27-36 ISSN:1592-7415 Some properties of residual mapping and convexity in ∧-hyperlattices Reza Ameria, Mohsen Amiri-Bideshkib, A. Borumand Saeidc aSchool of Mathematics, Statistics and Computer Science, College of Sciences, University of Tehran, P.O. Box 14155-6455, Teheran, Iran, rameri@ut.ac.ir bDepartment of Mathematics, Payame-Noor University, Tehran, Iran, amirimohsen61@yahoo.com cDepartment of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran, arsham@uk.ac.ir Abstract The aime of this paper is the study of residual mappings and con- vexity in hyperlattices. To get this point, we study principal down set in hyperlattices and we give some conditions for a mapping between two hyperlattices to be equivalent with a residual maping. Also, we investigate convex subsets in ∧-hyperlattices. Key words: residuated map, convex, down-set, hyperideal, hy- perfilter. 2000 AMS: 06F35, 03G25. 1 Introduction Hyperalgebras (multialgebra) are generalization of classical algebras that are introduced by F. Marty in the eighth congress of Scandinavian in 1934 [11]. In [4], Ameri and M. M. Zahedi introduced and studied notion of hyperal- gebraic systems. In [2], Ameri and Nozari Studied relationship between the 27 Reza Ameri, Mohsen Amiri-Bideshki, A. Borumand Saeid categories of multialgebra and algebra. C. Pelea and I. Purdea have been proved that complete hyperalgebra can be obtained from a universal algebra and a appropriate congruence on it. Also, Pelea and others studied multial- gebra, direct limit, and identities, for more details see [16, 17, 18, 19]. Hyperalgebras (multialgebra) are generalization of classical algebras that are introduced by F. Marty in the eighth congress of Scandinavian in 1934 [11]. In [4], Ameri and M. M. Zahedi introduced and studied notion of hyperal- gebraic systems. In [2], Ameri and Nozari Studied relationship between the categories of multialgebra and algebra. C. Pelea and I. Purdea have been proved that complete hyperalgebra can be obtained from a universal algebra and a appropriate congruence on it. Also, Pelea and others studied multial- gebra, direct limit, and identities, for more details see [16, 17, 18, 19]. Theory of hyperlattices introduced by Konstantinidou and J. Mittas in 1977[9]. In [10], G. A. Moghani and A. R. Ashrafi proved that in some cases the set of all subhypergroups G has a hyperlattice structure . In [24], X. L. Xin and X. G. Li studied hyperlattices and quotient hyperlattices. In [5], A. Asokku- mar in 2007 proved that under certain conditions, the idempotent elements of a hyperring form a hyperlattice and the orthogonal idempotent elements form a quassi-distributive hyperboolean algebra. In [1], R. Ameri, M. Amiri Bideshki, and A. Borumand Said studied prime hyperfilters (hyperideals) in hyperlattices. Also, they gave some examples of ∧-hyperlattices and dual distributive ∧-hyperlattices. In section 3, down set and residual maps in hyperlattices are studied and some properties of them are given. In section 4, convex subsets of a hyper- lattice and some properties of them are given. 2 Preliminary In this section we give some results of hyperlattices that we need to de- velop our paper. Definition 2.1. [1] Let L be a nonempty set. L is called a ∧− hyperlattice if (i) a ∈ a∧a,a∨a = a, (ii) a∧ b = b∧a,a∨ b = b∨a, (iii) a∧ (b∧ c) = (a∧ b) ∧ c,a∨ (b∨ c) = (a∨ b) ∨ c, 28 Some properties of residual mapping and convexity in ∧-hyperlattices (iv) a ∈ (a∧ (a∨ b)) ∩ (a∨ (a∧ b)), (v) a ∈ a∧ b =⇒ a∨ b = b, for all a,b,c ∈ L. Let A,B ⊆ L. Then: A∧B = ∪{a∧ b|a ∈ A,b ∈ B}; A∨B = {a∨ b|a ∈ A,b ∈ B}. Example 2.2. Let (L,∨,∧) be a lattice and define a⊕ b = {x | x ≤ a∧ b}. Then (L,∨,⊕) is a ∧− hyperlattice. Definition 2.3. [1] Let L be a ∧−hyperlattice. We say that L is bounded If there exist 0, 1 ∈ L, such that 0 ≤ x ≤ 1, for all x ∈ L. We say that 0 is the least element of L and 1 is the greatest element of L. Example 2.4. Let L = {0,a, 1}, and define ∧-hyper operation and ∨- operation on L with tables 3. Then (L,∧,∨) is a bounded ∧-hyperlattice. ∧ 0 a 1 0 {0} {0} {0} a {0} {a, 0} {a, 0} 1 {0} {a, 0} L (a) ∨ 0 a 1 0 0 a 1 a a a 1 1 1 1 1 (b) Table 1 Definition 2.5. [1] Let I and F are nonempty subsets of L. Then: (i) I is called hyperideal if the following conditions hold. (a) If x,y ∈ I, then x∨y ∈ I, (b) If x ∈ I and a ∈ L, such that a ≤ x, then a ∈ I. (ii) F is called hyperfilter if the following conditions hold. (a) If x,y ∈ F , then x∧y ⊆ F, (b) If x ∈ F and a ∈ L, such that x ≤ a, then a ∈ F . (iii) A hyperideal I is called prime if x∧y ∈ I, then x ∈ I or y ∈ I, for all x,y ∈ L. (iv) A hyperfilter F is called prime if x ∈ F or y ∈ F , where (x∧y)∩F 6= ∅, for all x,y ∈ L. 29 Reza Ameri, Mohsen Amiri-Bideshki, A. Borumand Saeid 3 Resedual Mappings in ∧-Hyperlattices In this section, we are going to introduce down-set and resedual mapping in ∧-hyperlattice. Let L be a ∧-hyperlattice. Definition 3.1. Let ∅ 6= A ⊆ L. A is called a down-set, if x ∈ A and y ≤ x, then y ∈ A. Example 3.2. every hyperideal of L is a down-set that is called principal down-set. Example 3.3. Let L = {0,a,b, 1}. ∧ and ∨ are given by Table 2 and 3. ∧ 0 a b 1 0 {0} {0} {0} {0} a {0} {0,a} {0} {0,a} b {0} {0} {0,b} {0,b} 1 {0} {0,a} {0,b} {1} Table 2: ∨ 0 a b 1 0 0 a b 1 a a a 1 1 b b 1 b 1 1 1 1 1 1 Table 3: I = {0,a,b} is a down-set, but it is not a hyperideal. We have a,b ∈ I and a∨ b = 1 /∈ I. Let x ∈ L and x↓ = {y ∈ L|y ∈ x∧y}. Proposition 3.4. ∀x ∈ L,x↓ is a down set. x↓ is called a principal down-set. Proposition 3.5. Let L be a dual distributive ∧-hyperlattice. Then every principal down-set is a hyperideal. Proof. If A ⊆ L and a∨ b ⊆ A, for all a,b ∈ L, then A is called join-closed. 30 Some properties of residual mapping and convexity in ∧-hyperlattices Corollary 3.6. Let I ⊆ L. Then I is an ideal if and only if I is a down-set and it is a join-closed set. Proposition 3.7. Let L and K be hyperlattice. If f : L −→ K is a isotone map and A ⊆ L is a down-set, then f(A) is a down-set. Proof. Since A is a down-set, there exists x ∈ L such that A = x↓. It is sufficient set f(A) = f(x)↓. Let L and K be hyperlattices and f : L −→ K is a mapping. We define two map f→ and f← that f→ is called direct image map and f← is called inverse image map. f→ : P(L) −→ P(K) is defined by f→(X) = {f(x)|x ∈ L}, for all X ⊆ L, and f← : P(K) −→ P(L) is defined by f←(Y ) = {x{∈ L|f(x) ∈ Y} for all Y ⊆ K. Definition 3.8. A mapping F : L −→ K is called residuated if the inverse image under F of every principal down-set of K is a principal down-set of L. Example 3.9. Let L be a ∧-hyperlattice and A ⊆ L. We define fA : P(L) −→ P(L) by fA(B) = A∩B, for all B ∈ P(L). Then fA is a residuated and residual g is given by gA(C) = C ∪A′, where that A′ = L\A. Example 3.10. Let L be a ∧-hyperlattice. Mapping f : P(L) −→ P(L) that is defined by f(A) = A, for all A ∈ P(L), is a residuated mapping. Theorem 3.11. Let L and K be two hyperlattices. A mapping f : L −→ K is a residuated iff f is a is isotone and there exists an isotone mapping g : K −→ L such that gof ≥ idL and fog ≤ idK . Proof. For all x ∈ L, x ∈ f←[f(x)↓]. If y ≤ x, then y ∈ f←[f(x)↓]. We have: f(x)↓ = {y|y ≤ f(x)} and f←[f(x)↓] = {t ∈ L|f(t) ∈ f(x)↓}. y ∈ f←[f(x)↓], so f(y) ≤ f(x). Then f is isotone. By assumption we have (∀y ∈ K)(∃x ∈ L) such that f←(y↓) = x↓. Now, for every given y ∈ K, this element x is clearly unique. So we can define a mapping g : K −→ L by g(y) = x. Since f← is isotone, it follow that so is g. For this mapping g, we have: g(y) ∈ g(y)↓ = x↓ = f←(y↓). So, f[g(y)] ≤ y, for all y ∈ K and therefore fog ≤ idK. Also, x ∈ f←[f(x)↓] = g[f(x)]↓, so that x ≤ g[f(x)], for all x ∈ L, and therefore gof ≥ id L . Conversely, Since g is isotone, we have: f(x) ≤ y =⇒ x ≤ g[f(x)]. 31 Reza Ameri, Mohsen Amiri-Bideshki, A. Borumand Saeid Also, we have: x ≤ g(y) =⇒ f(x) ≤ f[g(x)] ≤ y. It follows from these observations that f(x) ≤ y iff x ≤ g(y) and therefore f←(y↓) = g(y)↓. Proposition 3.12. The residual of f is unique. Proof. Suppose that g and g′ are residual of f. Then we have: g = idLog ≤ (g′of)og = g′o(fog) ≤ g′oidK = g′. Similarly, g′ ≤ g, then g = g′. We shall denote residual of f, by f+. Proposition 3.13. Mapping f : L =⇒ K is residuated iff for every y ∈ K, there exists g(y) = maxf←(y↓) = max{x ∈ L|f(x) ≤ y}. Moreover, f+of ≥ idL and fof + ≤ idK . Definition 3.14. Let f : L −→ K be a residuated mapping. Then f is called range closed if Im(f) is a down-set of K. Example 3.15. Let L be a ∧-hyperlattice with a top element 1. Given a ∈ L, consider the mapping fa : L −→ L given by: fa(x) = fa is residuated. Clearly, Im(fa) is the down-set a ↓ of L then fa is a range closed. Remark 3.16. In Example 3.15, L must have top element 1. Example 3.17. Let N be the set of natural numbers. We define ∧-hyperoperation and ∨ operation by: a∧ b = {m ∈ N|m ≤ min{a,b}}; a∨ b = max{a,b},foralla,b ∈ N. Then (L,∧,∨) is a ∧-hyperlattice. Consider f : N −→ N by f(x) = x, for all x ∈ N. f is a residated mapping, but it is not range closed. Theorem 3.18. Let f : L −→ K be a residuated mapping. Then f = f+ iff f2 = idL. Proof. =⇒ It is obvious. ⇐= Since f is residuated, then f2 = idL. By f2 = idL, we have fof ≤ idL and fof ≥ idL. So f = f+. Theorem 3.19. Let L and K be two ∧-hyperlattices and Let L has a top element 1. If f : L −→ K be a residuated mapping, then the following statements are equivalent. 32 Some properties of residual mapping and convexity in ∧-hyperlattices (i) f is range closed. (ii) forally ∈ K inf{y,f(1)} there exists and it equal to ff+(y). Proof. (i → ii):We have f+(y) ≤ 1, for all y ∈ L and by isotonic f, ff+(y) ≤ f(1). Also ff+(y) ≤ y, for all y ∈ K. So ff+(y) is a lower bound of f(1) and y. We must show that ff+(y) is the greatest lower bound of f(1) and y. Suppose that x ∈ K is such that x ≤ y and x ≤ f(1). By (i), we have x = f(z), for some z ∈ L and f(z) ≤ y; Since f+ is isotone, f+f(x) ≤ f+(y). We have z ≤ f+f(x), so z ≤ f+(y). By isotonic f, f(z) ≤ ff+(y), Then x ≤ ff+(y). Thus inf{y,f(1)} = ff+(y). (ii → i): We claim that Im(f) = f(1)↓. We have x ≤ 1, for all x ∈ L, then f(x) ≤ f(1), for all x ∈ L. So Im(f) ⊆ f(1)↓. Let y ∈ K be such that y ≤ f(1). Then by (ii), ff+(y) = inf{y,f(1)} = y. We Know ff+(y) ∈ Im(f), so y ∈ Im(f). Thus f(1)↓ ⊆ Im(f). Therefore Im(f) = f(1)↓. Proposition 3.20. Let f : L −→ K and g : K −→−→ M be residual map. Then gof so is, also (gof)+ = f+og+. 4 Convexity In ∧-hyperlattice In this section, we are going to introduce convex subsets in ∧-hyperlattices and we are going to give some properties of convex subsets. Proposition 4.1. Let F ⊆ L. Then F is a hyperfilter of L, if and only if (i) a,b ∈ F implies that a∧ b ∈ F . (ii) ∀a ∈ F and ∀x ∈ L, a∨x ∈ F . Proof. Since F is a filter, ∀a,b ∈ F , a ∧ b ∈ F . We know a ≤ a ∨ x, then a∨x ∈ F. So (i) and (ii) hold. Conversely, Let a ∈ F and a ≤ x. So, a ∨ x = x, by (ii) a ∨ x ∈ F , then x ∈ F . Proposition 4.2. Every hyperfilter of a ∧-hyperlattice L is a ∧-subhyperlattice. Remark 4.3. Converse of the above proposition does not hold. Consider hyperlattice in the Example 3.2. A = {0,a} is a subhyperlattice. We have a ≤ 1 and 1 /∈ A, then A is not a filter. Remark 4.4. Every hyperideal of L is not a subhyperlattice. Also, every subhyperlattice is not an ideal. 33 Reza Ameri, Mohsen Amiri-Bideshki, A. Borumand Saeid Definition 4.5. Let ∅ 6= K ⊆ L. We say K to be convex subset, if a,b ∈ K and c ∈ L such that a ≤ c ≤ b, then c ∈ K. Example 4.6. Consider hyperlattice L in Example 3.2. Then A = {0,a} is a convex subset, but B = {0, 1} is not a convex subset. we have 0 ≤ a ≤ 1 and a /∈ B. Proposition 4.7. Every hyperideal (hyperfilter) of L is a convex subset of L. Remark 4.8. Every convex subset of L is not a hyperideal (a filter). Con- sider hyperlattice L in Example 3.2. Then K = {a,b, 1} is a convex subset, but it is not a hyperideal (0 /∈ K). Also, K is not a hyperfilter (a∧ b = {0} and 0 /∈ K). Theorem 4.9. Let L has a bottom element 0 and let K be a convex subset of L. If K is a chain and 0 ∈ K, then K is a hyperideal of L. Remark 4.10. In Example 4.9, K must be a chain; also K must contain bottom element 0. Example 4.11. Let L be hyperlattice in Example3.2 (i) K1 = {a,b, 0} is a convex subset, but it is a not chain(a,b are not comparable). Since a∨ b /∈ K1, K1 is not a hyperideal. (ii) K2 = {a,b, 1} is a convex subset, but it is not a hyperideal (0 /∈ K2). Example 4.12. Consider hyperlattice L in Example 3.17. ThenK = {2, 3, 4, ..., 10} is a convex subset; Since K does not has bottom element 1, it is not a hy- perideal. Proposition 4.13. Every principal down-set of L is a convex subset. Theorem 4.14. Let I be a hyperideal and F be a hyperfiler of L, such that I ∩ F 6= ∅, then I ∩ F is a convex sub-hyperlattice if and only if for all a,b ∈ I ∩F , a∧ b ⊆ I. Proposition 4.15. If Ki, ∀i ∈ I is a convex sub-hyperlattice of L, then ∩i∈IKi is so. Theorem 4.16. Let K1 and K2 be convex sub-hyperlattices of L and let 0 ∈ K1 ∩ K2. Then K1 ∪ K2 is a convex sub-hyperlattice if and only if K1 ⊆ K2 or K2 ⊆ K1. 34 Some properties of residual mapping and convexity in ∧-hyperlattices Proof. Let K1 ∪K2 be a convex subhyperlattice, but K1 * K2 or K2 * K1. So, there exist a,b ∈ L, such that a ∈ K1 \ K2 and K1 \ K2. 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