RATIO MATHEMATICA 29 (2015) 25-40 ISSN:1592-7415 Fundamental hoop-algebras R. A. Borzooei1, H. R. Varasteh2, K. Borna2 1 Department of Mathematics, Shahid Beheshti University, Tehran, Iran borzooei@sbu.ac.ir 2 Faculty of Mathematics and Computer Science, Kharazmi University, Tehran, Iran varastehhamid@gmail.com, borna@khu.ac.ir Abstract In this paper, we investigate some results on hoop algebras and hyper hoop-algebras. We construct a hoop and a hyper hoop on any countable set. Then using the notion of the fundamental relation we define the fundamental hoop and we show that any hoop is a fundamental hoop and then we con- struct a fundamental hoop on any non-empty countable set. Keywords: hoop algebras, hyper hoop algebras, (strong) regular rela- tion,fundamental relations. 2000 AMS subject classifications: 20N20, 14L17, 97H50, 03G25,06F35. doi:10.23755/rm.v29i1.20 1 Introduction Hoop-algebras are naturally ordered commutative residuated integral monoids were originally introduced by Bosbach in [7] under the name of complementary semigroups. It was proved that a hoop is a meet-semilattice. Hoop-algbras then investigated by Büchi and Owens in an unpublished manuscript [8] of 1975, and they have been studied by Blok and Ferreirim[2],[3], and Aglianò et.al.[1]. The study of hoops is motivated by researchers both in universal algebra and algebraic logic.In recent years, hoop theory was enriched with deep structure theorems. Many of these results have a strong impact with fuzzy logic. Particularly, from the structure theorem of finite basic hoops one obtains an elegant short proof of 25 R. A. Borzooei, H. R. Varasteh, K. Borna the completeness theorem for propositional basic logic(see Theorem 3.8 of [1]) introduced by Hájek in [13]. The algebraic structures corresponding to Hájek’s propositional (fuzzy) basic logic, BL-algebras, are particular cases of hoops and MV-algebras, product algebras and Gödel algebras are the most known classes of BL-algebras. Recent investigations are concerned with non-commutative gener- alizations for these structures. Hypersructure theory was introduced in 1934[15], by Marty. Some fields of applications of the mentioned structures are lattices, graphs, coding, ordered sets, median algebra, automata, and cryptography[9]. Many researchers have worked on this area. The authors applied hyper structure theory on hyper hoop and intro- duced and studied hyper hoop algebras in [17]and[16]. In this paper, we investigate some new results on hoop-algebras and hyper hoop-algebras. We construct a hoop and a hyper hoop on any countable set. Then using the notion of the fundamental relation we define the fundamental hoop. 2 Preliminaries First, we recall following basic notions of the hypergroup theory from[10]: Let A be a non-empty set. A hypergroupoid is a pair (A,�), where � : A × A −→ P(A) −{∅} is a binary hyperoperation on A. If associativity low holds, then (A,�) is called a semihypergroup, and it is said to be commutative if � is commutative. An element 1 ∈ A is called a unit, if a ∈ 1 � a ∩ a � 1, for all a ∈ A and is called a scaler unit, if 1�a = a�1 = {a}, for all a ∈ A. Note that if B,C ⊆ A, then we consider B �C by B �C = ⋃ b∈B,c∈C (b� c). (See [10]) Definition 2.1. [3] A hoop-algebra or briefly hoop is an algebra (A,�,→,1) of type (2,2,0) such that, (HP1): (A,�,1) is a commutative monoid and for all x,y,z ∈ A, (HP2): x → x = 1, (HP3): (x � y) → z = x → (y → z) and (HP4): (x → y) �x = (y → x) �y. On hoop A we define ”x ≤ y” if and only if x → y = 1. It is easy to see that ≤ is a partial order relation on A. Definition 2.2. [17] A hyper hoop-algebra or briefly, a hyper hoop is a non- empty set A endowed with two binary hyperoperations �,→ and a constant 1 such that, for all x,y,z ∈ A satisfying the following conditions, (HHA1) (A,�,1) is a commutative semihypergroup with 1 as the unit, (HHA2) 1 ∈ x → x, (HHA3) (x → y)�x = (y → x)�y, (HHA4) x → (y → z) = (x�y) → z, (HHA5) 1 ∈ x → 1, 26 Fundamental hoop-algebras (HHA6) if 1 ∈ x → y and 1 ∈ y → x then x = y, (HHA7) if 1 ∈ x → y and 1 ∈ y → z then 1 ∈ x → z. In the sequel we will refer to the hyper hoop (A,�,→,1) by its universe A. On hyper hoop A, we define x ≤ y if and only if 1 ∈ x → y. If A is a hyper hoop, it is easy to see that ≤ is a partial order relation on A. Moreover, for all B,C ⊆ A we define B � C iff there exist b ∈ B and c ∈ C such that b ≤ c and define B ≤ C iff for any b ∈ B there exists c ∈ C such that b ≤ c. A hyper hoop A is bounded if there is an element 0 ∈ A such that 0 ≤ x, for all x ∈ A. Proposition 2.3. In any hyper hoop (A,�,→,1), if x � y and x → y are sin- gletons, for any x,y ∈ A, then (A,�,→,1) is a hoop. Then hyper hoops are a generalization of hoops and every hoop is a trivial hyper hoop. Proposition 2.4. [17] Let A be a hyper hoop. Then for all x,y,z ∈ A and B,C,D ⊆ A, the following hold, (HHA8) x�y � z ⇔ x ≤ y → z, (HHA9) B �C � D ⇔ B � C → D, (HHA10) z → y ≤ (y → x) → (z → x), (HHA11) z → y � (x → z) → (x → y), (HHA12) 1�1 = {1}. Notations: Let R be an equivalence relation on hyper hoop A and B,C ⊆ A. Then BRC, BRC and BRC denoted as follows, (i) BRC if there exist b ∈ B and c ∈ C such that bRc, (ii) BRC if for all b ∈ B there exists c ∈ C such that bRc and for all c ∈ C there exists b ∈ B such that bRc, (iii) BRC if for all b ∈ B and c ∈ C, we have bRc. Remark 2.5. It is clear that BRC and CRD imply that BRD, for all B,C,D ⊆ A. Definition 2.6. [17] Let R be an equivalence relation on hyper hoop A. Then R is called a regular relation on A if and only if for all x,y,z ∈ A, (i) if xRy, then x�zRy �z, (ii) if xRy, then x → zRy → z and z → xRz → y, (iii) if x → yR{1} and y → xR{1}, then xRy. Definition 2.7. [17] Let R be an equivalence relation on hyper hoop A. Then R is called a strong regular relation on A if and only if, for all x,y,z ∈ A, (i) if xRy, then x�zRy �z, (ii) if xRy, then x → zRy → z and z → xRz → y, 27 R. A. Borzooei, H. R. Varasteh, K. Borna Theorem 2.8. [17] Let R be a regular relation on hyper hoop A and AR be the set of all equivalence classes respect to R, that is AR = {[x]|x ∈ A}. Then ( A R ,⊗, ↪→, [1]) is a hyper hoop, which is called the quotient hyper hoop of A respect to R, where for all [x], [y] ∈ AR , [x]⊗ [y] = {[t]|t ∈ x�y} and [x] ↪→ [y] = {[z]|z ∈ x → y} Theorem 2.9. [17] Let R be a strong regular relation on hyper hoop A. Then (AR ,⊗, ↪→, [1]) is a hoop which is called the quotient hoop of A respect to R. Theorem 2.10. [4] Let X and Y be two sets such that |X| = |Y |. If (Y,≤,0) is a well-ordered set, then there exists a binary order relation on X and x0 ∈ X, such that (X,≤,x0) is a well-ordered set. Lemma 2.11. [14] Let X be an infinite set. Then for any set {a,b}, we have |X ×{a,b}| = |X|. 3 Constructing of hoops In this section, we show that we can construct a hoop on any non-empty count- able set. Lemma 3.1. Let A and B be two sets such that |A| = |B|. If A is a hoop, then we can construct a hoop on B by using of A. Proof. Since |A| = |B|, there exists a bijection ϕ : A → B. For any b1,b2 ∈ B. We define the binary operations �B and →B on B by, b1 �B b2 = ϕ(a1 �A a2) and b1 →B b2 = ϕ(a1 →A a2) where b1 = ϕ(a1), b2 = ϕ(a2) and a1,a2 ∈ A. It is easy to show that �B and →B are well-defined. Moreover, for any b ∈ B we define 1B as 1B = ϕ(1A). Now, by some modification we can show that (B,�B,→B,1B) is a hoop.2 Lemma 3.2. For any k ∈ N, we can construct a hoop on Wk = {0,1,2,3, ...,k− 1}. Proof. Let k ∈ N. We define the operations ”�” and ”→”, on Wk as follows, for all a,b ∈ Wk, a� b= { 0 if a + b ≤ k −1, a + b−k + 1 otherwise a → b = { k −1 if a ≤ b, k −1−a + b otherwise 28 Fundamental hoop-algebras Now, we show that (Wk,�,→,k −1) is a hoop, (HP1): Since, + is commutative, hence � is commutative. Now, we show that � is associative on Wk. For all a,b,c ∈ Wk, Case 1: If a + b ≤ k −1 and b + c ≤ k −1, then (a� b)� c = (0)� c = 0 and a� (b� c) = a�0 = 0 and so (a� b)� c = a� (b� c). Case 2: If a + b > k − 1 and b + c ≤ k − 1, since a + b + c ≤ 2(k − 1) and so a + b + c−k + 1 ≤ k−1, we get (a�b)�c = (a + b−k + 1)�c = 0. On the other hand, a� (b� c) = a�0 = 0 and then (a� b)� c = a� (b� c). Case 3: If a+b > k−1 and b+c > k−1, then (a�b)�c = (a+b−k+1)�c and a�(b�c) = a�(b+c−k+1). If a+b+c ≤ 2k then (a�b)�c = a�(b�c) = 0 and if a + b + c > 2k then (a� b)� c = a� (b� c) = a + b + c−2k + 2. Case 4: Let a + b ≤ k −1 and b + c > k −1. This case is similar to the Case 2. Now, we have 0�k−1 = 0 and if 0 6= a ∈ Wk, we have a+ (k−1) > k−1 and so a� (k −1) = a + k −1−k + 1 = a. Then (k −1) is the identity of (Wk,�) and so (Wk,�,k −1) is a commutative monoid. (HP2): It is clear that, for all a ∈ Wk, a → a = k −1. (HP3): Let a,b,c ∈ Wk. We show that (a� b) → c = a → (b → c). Case 1: If a + b ≤ k −1 and a ≤ b ≤ c, then (a� b) → c = 0 → c = k −1 and a → (b → c) = a → (k −1) = k −1. Hence, (a� b) → c = a → (b → c). Case 2: If a + b ≤ k − 1 and a ≤ c < b, (a � b) → c = 0 → c = k − 1 and since k − 1 − b + c ≥ a, a → (b → c) = a → (k − 1 − b + c) = k − 1. Hence, (a� b) → c = a → (b → c). Case 3: If a + b ≤ k −1 and b ≤ a ≤ c, then (a� b) → c = 0 → c = k −1 and a → (b → c) = a → (k −1) = k −1. Hence, (a� b) → c = a → (b → c). Case 4: If a + b ≤ k −1 and b ≤ c < a, then (a� b) → c = 0 → c = k −1 and a → (b → c) = a → (k −1) = k −1. Hence, (a� b) → c = a → (b → c). Case 5: If a + b ≤ k −1 and c ≤ b ≤ a, then (a� b) → c = 0 → c = k −1. On the other hand since a+b ≤ k−1, we get a+b−c ≤ k−1, a ≤ (k−1−b+c) and a → (k−1−b+c) = k−1. Then a → (b → c) = a → (k−1−b+c) = k−1. Hence, (a� b) → c = a → (b → c). Case 6: If a+b ≤ k−1 and c ≤ a < b, then (a�b) → c = 0 → c = k−1. On the other hand since a+b ≤ k−1, we get a+b−c ≤ k−1, a ≤ (k−1−b+c) and a → (k−1−b+c) = k−1. Then a → (b → c) = a → (k−1−b+c) = k−1. Hence, (a� b) → c = a → (b → c). Case 7: Let a + b > k − 1 and a ≤ b ≤ c. Since a ≤ b ≤ c, we get a + b− c ≤ a ≤ k − 1 and so a + b − k + 1 ≤ c. Then (a � b) → c = (a + b − k + 1) → c = k − 1. On the other hand, a → (b → c) = a → (k − 1) = k − 1. Hence, (a� b) → c = a → (b → c). Case 8: Let a+b > k−1 and a ≤ c < b. Since a ≤ c < b we get a+b−c ≤ b ≤ k−1 and so a+b−k +1 ≤ c. Then (a�b) → c = (a+b−k +1) → c = k−1. On the other hand, since k − 1 − b + c ≥ c ≥ a, we get a → (b → c) = a → 29 R. A. Borzooei, H. R. Varasteh, K. Borna (k −1− b + c) = k −1. Hence, (a� b) → c = a → (b → c). Case 9: Let a+b > k−1 and b ≤ a ≤ c. Since b ≤ a ≤ c, we get a+b−c ≤ a ≤ k−1 and so a+b−k +1 ≤ c. Then (a�b) → c = (a+b−k +1) → c = k−1. On the other hand since k − 1 − b + c ≥ c ≥ a, we get a → (b → c) = a → (k −1− b + c) = k −1. Hence, (a� b) → c = a → (b → c). Case 10: Let a + b > k − 1 and b ≤ c < a. Since b ≤ c < a, we get a + b− c ≤ a ≤ k − 1 and so a + b − k + 1 ≤ c. Then (a � b) → c = (a + b − k + 1) → c = k − 1. On the other hand a → (b → c) = a → (k − 1) = k − 1. Hence, (a� b) → c = a → (b → c). Case 11: If a+b > k−1 and c ≤ b ≤ a, then (a�b) → c = (a+b−k +1) → c and a → (b → c) = a → (k − 1 − b + c). Hence, if a + b − c ≤ k − 1, then (a � b) → c = a → (b → c) = k − 1 and if a + b − c > k − 1, then (a� b) → c = a → (b → c) = 2k −2−a− b + c. Case 12: If a+b > k−1 and c ≤ a < b, then (a�b) → c = (a+b−k + 1) → c and a → (b → c) = a → (k − 1 − b + c). Hence, if a + b − c ≤ k − 1, then (a � b) → c = a → (b → c) = k − 1 and if a + b − c > k − 1, then (a� b) → c = a → (b → c) = 2k −2−a− b + c (HP4): Now, we show that (a → b)�a = (b → a)� b, for all a,b ∈ Wk. Case 1: If a ≤ b, then (a → b) � a = (k − 1) � a = a and (b → a) � b = (k−1−b+a)�b = k−1−b+a+b−k+1 = a. Hence, (a → b)�a = (b → a)�b. Case 2: If a > b, then (a → b)�a = (k−1−a+b)�a = k−1−a+b+a−k+1 = b and (b → a)� b = (k −1)� b = b. Hence, (a → b)�a = (b → a)� b. Therefore, (Wk,�,→,k −1) is a hoop.2 Theorem 3.3. Let A be a finite set. Then there exist binary operations � and → and constant 1 on A, such that (A,�,→,1), is a hoop. Proof. Let A be a finite set. Then, there exists k ∈ N such that |A| = |Wk|. Now, by Lemma 3.2, (Wk,�,→,1) is a hoop and so by Lemma 3.1, there exist binary operations � and →, and constant 1 on A , such that (A,�,→,1) is a hoop.2 Lemma 3.4. Let 1 < n ∈ Q. Then there exist binary operations � and → on E = Q∩ [1,n], such that (E,�,→,n) is a hoop. Proof. For any 1 < n ∈ E, we define the binary operations � and → on E as follows, for all a,b ∈ E, a� b= { 1 if ab ≤ n, ab n otherwise a → b = { n if a ≤ b, nb a otherwise Clearly, � and → are well-defined on E. Now, we show that (E,�,→,n) is a hoop. 30 Fundamental hoop-algebras (HP1): For all a ∈ E, if a 6= 1, since an > n we have a�n= n�a = an n = a and if a = 1, we have a�n = 1 �n = 1 = a. Then n is the identity element of (E,�). Now, we show that � is associative on E. Let a,b,c ∈ E, Case 1: If ab ≤ n and bc ≤ n, then (a� b) � c = 1 � c = 1. On the other hand a� (b� c) = a� (1) = 1. Then (a� b)� c = a� (b� c). Case 2: If ab ≤ n and bc > n, then (a� b) � c = 1 � c = 1. On the other hand b � c = bc n and then a � (b � c) = a � (bc n ). Since abc n = ab n c ≤ c ≤ n, we get a� (b� c) = 1 and so (a� b)� c = a� (b� c). Case3: If ab > n and bc > n, then (a � b) � c = (ab n ) � c. On the other hand a � (b � c) = a � (bc n ). If abc n ≤ n, then (a � b) � c = a � (b � c) = 1 and if abc n > n, then (a�b)�c = a� (b�c) = abc n2 . Hence, (a�b)�c = a� (b�c). Case 4: Let ab > n and bc ≤ n. This case is similar to the Case 2. It is clear that, for all a,b ∈ E, a�b = b�a. Hence, (E,�,n) is a commutative monoid. (HP2): It is clear that, for all a ∈ E, we have a → a = n. (HP3): For all a,b,c ∈ E, we have the following cases, Case 1: If b ≤ c and ab ≤ n, then a → (b → c) = a → n = n and (a � b) → c = 1 → c = n. Then a → (b → c) = (a� b) → c. Case 2: If b ≤ c and ab > n, then a → (b → c) = a → n = n and since a n < 1 , we get ab n < b ≤ c and so (a � b) → c = ab n → c = n. Then a → (b → c) = (a� b) → c. Case 3: If b > c and ab ≤ n, since ab ≤ n ≤ nc and so a ≤ nc b , then a → (b → c) = a → nc b = n. On the other hand, (a � b) → c = 1 → c = n. Then a → (b → c) = (a� b) → c. Case 4: If b > c and ab > n, then a → (b → c) = a → nc b and (a � b) → c = ab n → c. We have, a ≤ nc b if and only if ab n ≤ c, and so a → (b → c) = (a�b) → c. HP4: For all a,b ∈ E, we have the following cases, Case 1: If a ≤ b, then a � (a → b) = a � n = an n = a and b � (b → a) = b� na b = bna bn = a and so a� (a → b) = b� (b → a). Case 2: If a > b, then a � (a → b) = a � nb a = anb an = b and b � (b → a) = b�n = bn n = b and so a� (a → b) = b� (b → a). Therefore, (E,�,→,n) is a hoop.2 Theorem 3.5. Let A be an infinite countable set. Then there exist binary opera- tions � and → and constant 1 on A, such that (A,�,→,1) is a hoop. Proof. Let A be an infinite countable set and E = Q∩[1,n]. Then by Lemma 3.4, (E,�,→,1) is an infinite countable hoop and |A| = |E|. Hence, by Lemma 3.1, there exist binary operations � and → and constant 1, such that (A,�,→,1) is a hoop.2 Corollary 3.6. For any non-empty countable set A, we can construct a hoop on A. 31 R. A. Borzooei, H. R. Varasteh, K. Borna Proof. Let A be a non-empty countable set. Then, A is a finite set, or an infinite countable set . Then by the Theorems 3.3 and 3.5, the proof is clear.2 4 Constructing of some hyper hoops In this section first we show that the Cartesian product of hoops is a hyper hoop and then we construct a hyper hoop by any non-empty countable set. Theorem 4.1. Let (A,�A,→A,1A) and (B,�B,→B,1B) be two hoops. Then there exist hyperoperations �, → and constant 1 on A × B such that (A × B,�,→,1) is a hyper hoop. Proof. For any (a1,b1),(a2,b2) ∈ A×B, we define the binary hyperopera- tions �, → on A×B by, (a1,b1)� (a2,b2) = {(a1 �A a2,b1),(a1 �A a2,b2)}, (a1,b1) → (a2,b2) = { {(a1 →A a2,b2),(a1 →A a2,1B)} if b1 = b2, {(a1 →A a2,b2)} otherwise and constant 1 = (1A,1B). It is easy to show that the hyperoperations are well- defined. Now, we show that (A×B,�,→,1) is a hyper hoop. (HHA1): Since �A , is associative and commutative, we get � is associative and commutative. Moreover, for all (a,b) ∈ A × B, we have (a,b) � (1A,1B) = {(a�A 1A,b),(a�A 1A,1B)} 3 (a,b). Then (A×B,�,→,1) is a commutative semihypergroup with 1 as the unit, where 1 = (1A,1B). (HHA2): For all (a,b) ∈ A×B, we have (a,b) → (a,b) = {(a →A a,b),(a →A a,1B)} = {(a →A a,b),(1A,1B)}3 (1A,1B) = 1 (HHA3): For all (a1,b1),(a2,b2) ∈ A×B, we have the following cases, Case 1: If b1 6= b2, then, ((a1,b1) → (a2,b2))� (a1,b1) = {(a1 → a2,b2)}� (a1,b1) = {((a1 → a2)�A a1,b1),((a1 → a2)�A a1, b2)} = {((a2 → a1)�A a2,b1),((a2 → a1)�A a2, b2)} = ((a2,b2) → (a1,b1))� (a2,b2) 32 Fundamental hoop-algebras Case 2: If b1 = b2, then, ((a1,b1) → (a2,b2))� (a1,b1) = {(a1 → a2,b2),(a1 → a2,1B)}� (a1,b1) = {((a1 → a2)�A a1,b1),((a1 → a2)�A a1, b2),((a1 → a2)�A a1,1B)} = {((a2 → a1)�A a2,b1),((a2 → a1)�A a2, b2),((a2 → a1)�A a2,1B)} = ((a2,b2) → (a1,b1))� (a2,b2) (HHA4): For all (a1,b1),(a2,b2),(a3,b3) ∈ A×B, we have the following cases, Case 1: If b1 = b2 = b3, (a1,b1) → ((a2,b2) → (a3,b3)) = (a1,b1) →{((a2 →A a3),b3),((a2 →A a3), 1B)} = {(a1 →A (a2 →A a3),1B),(a1 →A (a2 →A a3),b3)} = {((a1 �A a2) →A a3,1B),((a1 �A a2) →A a3),b3)} = ((a1,b1)� (a2,b2)) → (a3,b3) Case 2: If b1 6= b2 = b3, (a1,b1) → ((a2,b2) → (a3,b3)) = (a1,b1) →{((a2 →A a3),b3),((a2 →A a3), 1B)} = {(a1 →A (a2 →A a3),1B),(a1 →A (a2 →A a3),b3)} = {(a1 �A a2) →A (a3,1B),((a1 �A a2) →A a3),b3)} = ((a1,b1)� (a2,b2)) → (a3,b3) Case 3: If b1 = b2 6= b3, (a1,b1) → ((a2,b2) → (a3,b3)) = (a1,b1) →{((a2 →A a3),b3)} = {a1 →A (a2 →A a3),b3)} = {((a1 �A a2) →A a3,b3)} = ((a1,b1)� (a2,b2)) → (a3,b3) 33 R. A. Borzooei, H. R. Varasteh, K. Borna Case 4: If b1 6= b2 6= b3, (a1,b1) → ((a2,b2) → (a3,b3)) = (a1,b1) →{((a2 →A a3),b3)} = {(a1 →A (a2 →A a3),b3)} = {((a1 �A a2) →A a3,b3)} = ((a1,b1)� (a2,b2)) → (a3,b3) (HHA5): For all (a,b) ∈ A×B, we have the following cases, Case 1: If b = 1B, then (a,b) → (1A,1B) = {(a → 1A,1B),(a → 1A,b → 1B)} = {(1A,1B)}3 (1A,1B). Case 2: If b 6= 1B, then (a,b) → (1A,1B) = {(a → 1A,1B)} = {(1A,1B)} 3 (1A,1B). (HHA6): For all (a1,b1),(a2,b2) ∈ A×B, if (1A,1B) ∈ (a1,b1) → (a2,b2) and (1A,1B) ∈ (a2,b2) → (a1,b1), then we have the following cases, Case 1: If b1 6= b2, then (1A,1B) ∈ {(a1 →A a2,b2)} and (1A,1B) ∈ {(a2 →A a1,b1)}. Hence, 1A = a1 →A a2 and 1A = a2 → a1 and 1B = b1 = b2. Since A is a hoop, we get a1 = a2 and so (a1,b1) = (a2,b2) Case 2: If b1 = b2, then (1A,1B) ∈ {(a1 →A a2,b2),(a1 →A a2,1B)} and (1A,1B) ∈ {(a2 →A a1,b1),(a2 →A a1,1B)}. Hence 1A = a1 →A a2 and 1A = a2 → a1. Since A is a hoop, we get a1 = a2 and by assumption, we have b1 = b2. So (a1,b1) = (a2,b2). (HHA7): For all (a1,b1),(a2,b2),(a3,b3) ∈ A × B, let (1A,1B) ∈ (a1,b1) → (a2,b2) and (1A,1B) ∈ (a2,b2) → (a3,b3). Then we consider the following cases: Case 1: If b1 = b2 = b3, then (1A,1B) ∈ {(a1 →A a2,1B),(a1 →A a2,b2)} and (1A,1B) ∈ {(a2 →A a3,1B),(a2 →A a3,b3)}. Hence 1A = a1 →A a2 and 1A = a2 → a3. Since A is a hoop, we get 1A = a1 →A a3. Hence, (a1,b1) → (a3,b3) = {(a1 →A a3,b3),(a1 →A a3,1B)} = {(1A,b3),(1A,1B)}3 (1A,1B). Case 2: If b1 6= b2 = b3, then (1A,1B) ∈ {(a1 →A a2,b2)} and (1A,1B) ∈ {(a2 →A a3,1B),(a2 →A a3,b3)}. Hence 1A = a1 →A a2 and 1A = a2 → a3 and b2 = b3 = 1B. Since A is a hoop, we get 1A = a1 →A a3. Hence, (a1,b1) → (a3,b3) = {(a1 →A a3,b3)} = {(1A,1B)}3 (1A,1B). Case 3: Let b1 = b2 6= b3. Then proof is similar to the Case 2. Case 4: If b1 6= b2 6= b3, then (1A,1B) ∈ {(a1 →A a2,b2)} and (1A,1B) ∈ {(a2 →A a3,b3)}. Hence, 1A = a1 →A a2 and 1A = a2 → a3 and b2 = b3 = 1B. Since A is a hoop, we get 1A = a1 →A a3. Hence, (a1,b1) → (a3,b3) = {(a1 →A a3,b3)} = {(1A,1B)}3 (1A,1B). Therefore,(A×B,�,→,1) is a hyper hoop, where 1 = (1A,1B).2 Lemma 4.2. Let A and B be two sets such that |A| = |B|. If (A,�A,→A,1A) is a hyper hoop, then there exist hyperoperations �B , →B and constant 1B on B, such that (B,�B,→B,1B) is a hyper hoop and (A,�A,→A,1A) ∼= (B,�B,→B,1B). 34 Fundamental hoop-algebras Proof. Since |A| = |B|, then there exists a bijection ϕ : A → B . For any b1,b2 ∈ B, there exist a1,a2 ∈ A such that b1 = ϕ(a1) and b2 = ϕ(a2). Then we define the hyperoperations �B,→B on B by, b1 �B b2 = {ϕ(a)|a ∈ a1 �a2}, and b1 →B b2 = {ϕ(a)|a ∈ a1 → a2}. It is easy to show that �B,→B are well-defined and (B,�B,→B,1B) is a hyper hoop, where 1B = ϕ(1A). Now, we define the map θ : (A,�A,→A,1A) → (B,�B,→B,1B) by θ(x) = ϕ(x). Since ϕ is a bijection then θ is a bijection and it is easy to see that θ is a homomorphism and so it is an isomorphism.2 Corollary 4.3. For any non-empty countable set A and any hoop B, we can con- struct a hyper hoop on A×B. Proof. By Corollary 3.6, we can construct a hoop on A and by Theorem 4.1, we can construct a hyper hoop on A×B.2 Corollary 4.4. Let A be an infinite countable set. We can construct a hyper hoop on A. Proof. Let A be an infinite countable set. Then by Corollary 3.6, we can construct a hoop on A. Now, By Theorem 3.3, for arbitrary elements x,y not belonging to A, we can define operations � and → on the set {x,y}, such that ({x,y},�,→) is a hoop. Then by Theorem 4.1, we can construct a hyper hoop on A×{x,y}. Then by Lemma 2.11 and 4.2, there exists a hyper hoop on A.2 5 Fundametal hoops In this section we apply the β∗ relation to the hyper hoops and obtain some results. Then we show that any hoop is a fundamental hoop. Let (A,�,→,1) be a hyper hoop and U(A) denote the set of all finite com- binations of elements of A with respect to � and →. Then, for all a,b ∈ A, we define aβb if and only if {a,b}⊆ u, where u ∈ U(A), and aβ∗b if and only if there exist z1, ...,zm+1 ∈ A with z1 = a,zm+1 = b such that {zi,zi+1} ⊆ ui ⊆ U(A), for i = 1, ...,m (In fact β∗ is the transitive closure of the relation β). Theorem 5.1. Let A be a hyper hoop. Then β∗ is a strong regular relation on A. Proof. Let aβ∗b, for a,b ∈ A. Then there exist x1, ...,xn+1 ∈ A with x1 = a,xn+1 = b and ui ∈ U(A) such that {xi,xi+1} ⊆ ui, for 1 ≤ i ≤ n. Let zi ∈ xi → c, for all 1 ≤ i ≤ n + 1,c ∈ A. Then we have, {zi,zi+1}⊆ (xi → c)∪ (xi+1 → c) ⊆ ui → c ⊆ U(A), for all 1 ≤ i ≤ n. Hence, z1β∗zn+1, where z1 ∈ a → c and zn+1 ∈ b → c and so a → cβ∗b → c. Similarly, we can show that c → aβ∗c → b. Now, by the same way we can prove 35 R. A. Borzooei, H. R. Varasteh, K. Borna that aβ∗b implies a�cβ∗b�c, for all c ∈ A. Hence, β∗ is a strong regular relation on A.2 Corollary 5.2. Let A be a hyper hoop. Then ( A β∗ ,⊗, ↪→) is a hoop, where ⊗ and ↪→ are defined by Theorem 2.8. Proof. By Theorem 2.9 the proof is clear.2 Theorem 5.3. Let A be a hyper hoop. Then the relation β∗ is the smallest equiv- alence relations γ defined on A such that the quotient A γ is a hoop with operations γ(x)⊗γ(y) = γ(t) : t ∈ x�y and γ(x) ↪→ γ(y) = γ(z) : z ∈ x → y where γ(x) is equivalence class of x with respect to the relation γ. Proof. By Corollary 5.2, A β∗ is a hoop. Now, let γ be an equivalence relation on A such that A γ is a hoop. Let xβy, for x,y ∈ A and π : A → A γ be the natural projection such that π(x) = γ(x). It is clear that π is a homomorphism of hyper hoops. Then there exists u ∈ U(A) such that {x,y} ⊆ u. Since π is a homomor- phism of hyper hoops, we get |π(u)| = |γ(u)| = 1. Since {π(x),π(y)} ⊆ π(u) and |π(u)| = 1, we get π(x) = π(y) and so γ(x) = γ(y) i.e. xγy. Hence, β ⊆ γ. Now, let aβ∗b, for a,b ∈ A. Then there exist x1, ...,xn+1 ∈ A, such that a = x1βx2, ...,βxn+1 = b. Since β ⊆ γ, we get a = x1γx2, ...,γxn+1 = b. Then since γ is a transitive relation on A, we get aγb and so β∗ ⊆ γ.2 Corollary 5.4. The relation β∗ is the smallest strong regular relation on hyper hoop A. Proof. The proof is straightforward.2 Lemma 5.5. If A1 and A2 are two hyper hoops, then the Cartesian product A1 × A2 is a hyper hoop with the unit (1A1,1A2) by the following hyperoperations, for (x1,y1),(x2,y2) ∈ A1 ×A2, (x1,y1)� (x2,y2) = {(a,b)|a ∈ x1 �x2,b ∈ y1 �y2}, (x1,y1) → (x2,y2) = {(a′,b′)|a′ ∈ x1 → x2,b′ ∈ y1 → y2} Proof. The proof is straightforward.2 Lemma 5.6. Let A1 and A2 be two hyper hoops. Then, for a,c ∈ A1 and b,d ∈ A2, we have (a,b)β∗A1×A2(c,d) if and only if aβ ∗ A1 c and bβ∗A2d. Proof. We know that u ∈ U(A1 ×A2) if and only if there exist u1 ∈ U(A1) and u2 ∈ U(A2) such that u = u1 × u2. Then (a,b)β∗A1×A2(c,d) if and only if there exist u1 ∈ U(A1) and u2 ∈ U(A2) such that {(a,b),(c,d)}⊆ u1×u2 if and only if {a,c}⊆ u1 and {b,d}⊆ u2 if and only if aβ∗A1c and bβ ∗ A2 d.2 36 Fundamental hoop-algebras Theorem 5.7. Let A1 and A2 be two hyper hoops. Then A1×A2β∗ A1×A2 ∼= A1 β∗ A1 × A2 β∗ A2 . Proof. Let ϕ : A1×A2 β∗ → A1 β∗ A1 × A2 β∗ A2 be defined by ϕ(β∗(x,y)) = (β∗A1(x),β ∗ A2 (y)), where β∗ = β∗A1×A2 By Lemma 5.5, A1×A2 β∗ is well-define. It is clear that ϕ is onto. By Lemma 5.6, we have β∗(x1,y1) = β∗(x2,y2) if and only if β∗A1(x1) = β ∗ A2 (x2) and β∗A2(y1) = β ∗ A2 (y2), for any (x1,y1),(x2,y2) ∈ A1 ×A2. So, ϕ is well defined and one to one. Also, by considering the hyperoperations ⊗ and ↪→ defined in Theorem 2.8, we have, ϕ(β∗(x1,y1) ↪→ β∗(x2,y2)) = ϕ({β∗(a,b)|a ∈ x1 → x2,b ∈ y1 → y2}) = {ϕ(β∗(a,b))|a ∈ x1 → x2,b ∈ y1 → y2} = {(β∗A1(a),β ∗ A2 (b))|a ∈ x1 → x2,b ∈ y1 → y2} = (β∗A1(x1) ↪→ β ∗ A1 (x2),β ∗ A2 (y1) ↪→ β∗A2(y2)) = (β∗A1(x1),β ∗ A2 (y1)) ↪→ (β∗A1(x2),β ∗ A2 (y2)) = ϕ(β∗(x1,y1)) ↪→ ϕ(β∗(x2,y2)) Similarly, we can show that ϕ(β∗(x1,y1)⊗β∗(x2,y2)) = ϕ(β∗(x1,y1))⊗ϕ(β∗(x2, y2)). Moreover, it is clear that ϕ(β∗(1A1,1A2)) = (β ∗(1A1),β ∗(1A2)). Hence, ϕ is an isomorphism.2 Corollary 5.8. Let A1,A2, ....,An be hyper hoops. Then, A1×A2×....×An β∗ A1×A2×....×An ∼= A1 β∗1 × A2 β∗2 × .......× An β∗n Proof. The proof is straightforward.2 Theorem 5.9. Let A and B be two sets such that |A| = |B|. If (A,�A,→A,1A) is a hyper hoop, then there exist hyperoperations �B and →B and constant 1B on B such that (B,�B,→B,1B) is a hyper hoop and (A,�A,→A,1A) β∗ A ∼= (B,�B,→B,1b) β∗ B . Proof. Since |A| = |B|, then by Lemma 4.2, there exist binary hyper- operations �B and →B, such that (B,�B,→B,1B) is a hyper hoop. More- over, there exists an isomorphism f : (A,�A,→A,1A) → (B,�B,→B,1B), such that f(1A) = 1B. Now, we define ϕ : (A,�A,→A,1A) β∗ A → (B,�B,→B,1B) β∗ B by ϕ(β∗A(x)) = β ∗ B(f(x)). Since f is an isomorphism, ϕ is onto. Let y1 , y2 ∈ B. Then there exist a1,a2 ∈ A such that b1 = f(a1) and b2 = f(a2). Then β∗A(a1) = β ∗ A(a2) iff a1β ∗ Aa2 iff there exists u ∈ U(A) such that {a1,a2} ⊆ u iff there existes f(u) ∈ U(B) : {f(a1),f(a2)} ⊆ f(u) iff β∗B(b1) = β ∗ B(b2) iff β∗B(f(a1)) = β ∗ B(f(a2)). Then ϕ is well-defined and one to one. Also, by consid- 37 R. A. Borzooei, H. R. Varasteh, K. Borna ering the hyperoperations ⊗ and ↪→ defined in Theorem 2.8, we have, ϕ(β∗A(a1)⊗β ∗ A(a2)) = ϕt∈a1�a2(β ∗ A(t)) = β ∗ t∈a1�a2(f(t)) = β∗t′∈f(a1�a2)(t ′) = β∗t′∈f(a1)�f(a2)(t ′) = β∗B(f(a1))⊗β ∗ B (f(a2)) = ϕ(β∗A(a1))⊗ϕ(β ∗ A(a2)) By the same way, we can show that ϕ(β∗A(a1) ↪→ β ∗ A(a2)) = ϕ(β ∗ A(a1)) ↪→ ϕ(β ∗ A(a2)) Since f is an isomorphism, we get ϕ(β∗A(1A)) = β ∗ B(f(1A)) = β ∗ B(1B). Hence, ϕ is an isomorphism.2 Definition 5.10. Let A be a hoop algebra. Then A is called a fundamental hoop, if there exists a nontrivial hyper hoop B, such that B β∗ B ∼= A Theorem 5.11. Every hoop is a fundamental hoop. Proof. Let A be a hoop. Then by Theorem 4.1, for any hoop B, A × B is a hyper hoop. By considering the hyperoperations � and → defined in Theorem 4.1, we get that any finite combination u ∈ U(A×B) is the form of u = {(a,xi)|a ∈ A,xi ∈ B}. Hence, for any (a1,b1),(a2,b2) ∈ A×B, (a1,b1)β ∗(a2,b2) ⇔∃u ∈ U(A×B) such that {(a1,b1),(a2,b2)}⊆ u ⇔ a1 = a2 Hence, for any (a,b) ∈ A×B, β∗(a,b) = {(a,x)|x ∈ B}. Now, we define the map ψ : A×B β∗ → A by, ψ(β∗(a,b)) = a. It is clear that, β∗(a1,b1) = β ∗(a2,b2) ⇔ a1 = a2 ⇔ ψ(β∗(a1,b1)) = ψ(β∗(a2,b2)). Then, ψ is well defined and one to one. In the following, we show that ψ is a homomorphism. For this we have, ψ(β∗(a1,b1)⊗β∗(a2,b2)) = ψ(β∗(u,v)) : (u,v) ∈ (a1,b1)� (a2,b2) = ψ(β∗(u,v)) : (u,v) ∈{((a1 �a2),b1),((a1 � a2),b2)} = {u|u ∈ a1 �a2} = a1 �a2 = ψ(β∗(a1,b1))�ψ(β∗(a2,b2)) 38 Fundamental hoop-algebras and similarly, for the operation ↪→, we have the following cases, Case 1: If b1 6= b2, then, ψ(β∗(a1,b1) ↪→ β∗(a2,b2)) = ψ(β∗(u,v)) : (u,v) ∈ (a1,b1) → (a2,b2) = ψ(β∗(u,v)) : (u,v) ∈{((a1 → a2),b2)} = {u|u ∈ a1 → a2} = a1 → a2 = ψ(β∗(a1,b1)) → ψ(β∗(a2,b2)) Case 2:If b1 = b2, then, ψ(β∗(a1,b1) ↪→ β∗(a2,b2)) = ψ(β∗(u,v)) : (u,v) ∈ (a1,b1) → (a2,b2) = ψ(β∗(u,v)) : (u,v) ∈{((a1 → a2),b2),((a1 → a2),1B)} = {u|u ∈ a1 → a2} = a1 → a2 = ψ(β∗(a1,b1)) → ψ(β∗(a2,b2)) Clearly, ψ(β∗(1A,1B) = 1A and ψ is onto. Therefore, ψ is an isomorphism i.e. A×B β∗ ∼= A and so A is fundamental.2 Corollary 5.12. For any non-empty countable set A, we can construct a funda- mental hoop on A. Proof. By Corollary 3.6 and Theorem 5.11 the proof is clear.2 References [1] P. Aglianò, I. M. A. Ferreirim, F. 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