In this paper using strongly duplexes we introduce a new class of (semi)hypergroups. The associated (semi)hypergroup from a strongly duplex is called duplex (semi)hypergroup. Two computer programs written in MATLAB show that the two groups $Z_{2n}$ and $Z_{n}times Z_{2}$ produce a strongly duplex and its associated hypergroup is a complementary feasible hypergroup

Fatemeh Alizadeh

Morteza Jafarpour

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Journal of Algebraic SystemsVol. 2, No. 2, (2014), pp 83-96ASSOCIATED (SEMI)HYPERGROUPS FROMDUPLEXESM. JAFARPOUR∗ AND F. ALIZADEHAbstract. In this paper using strongly duplexes we introduce anew class of (semi)hypergroups. The associated (semi)hypergroupfrom a strongly duplex is called duplex (semi)hypergroup. Twocomputer programs written in MATLAB show that the two groupsZ2n and Zn × Z2 produce a strongly duplex and its associatedhypergroup is a complementary feasible hypergroup.1. IntroductionHyperstructure theory was founded in 1934 at the 8th congress ofScandinavian Mathematicians. Marty [11] introduced the hypergroupnotion as a generalization of groups and then he proved its utility insolving some problems of groups, algebraic functions and rational frac-tions. Surveys of the theory can be found in the books of Corsini [2],Davvaz and Leoreanu-Fotea [7], Corsini and Leoreanu [4] and Vou-giouklis [15, 16]. Now this field of modern algebra is widely studiedfrom the theoretical and applied viewpoints because of their applica-tions to many subjects of pure and applied mathematics. Some appli-cations can be used in the following areas: geometry, graphs, fuzzy sets,cryptography, automata, lattices, binary relations, codes, and artificialintelligence, see for example [1, 3, 5, 6, 8, 9, 12].Jean-Louis Loday introduced the notion of dimonoid [10]. Dimonoidsare a tool to study Leibniz algebras. A dimonoid is a set equipped withMSC(2010): Primary: 20N20Keywords: Duplexes, semihypergroups, complementary feasible (semi)hypergroups.Received: 21 November 2013, Revised: 15 May 2014.∗Corresponding author .8384 JAFARPOUR AND ALIZADEHtwo binary associative operations satisfying some axioms. If the opera-tions of a dimonoid coincide, then the dimonoid becomes a semigroup.T. Pirashvili [13] introduced the notion of a duplex which generalizesthe notion of a dimonoid. In this paper we make a connection betweenduplexes and hyperstructures. We introduce strongly duplexes and wedefine and study a new class of semihypergroups. Two computer pro-grams written in MATLAB help us to see that the two groups Z2n(cyclic group of order 2n) and Zn × Z2 produce a strongly duplex andits associated hypergroup is a complementary feasible hypergroup.2. PreliminariesLet us briefly recall some basic notions and results about groups andhypergroups; for a comprehensive overview of this subject, the readeris refereed to [2, 4, 7, 14].For a non-empty set H and a natural number n, we denote by P∗(H)the set of all non-empty subsets of H and Hn = H ×H × ...×H.Definition 2.1. A non-empty set H, endowed with a mapping calledhyperoperation, • : H2 −→ P∗(H) is named hypergroupoid. A hyper-groupoid which verifies the following conditions:(i): (x• y)• z = x• (y • z), for all x, y, z ∈ H (the associativity)(ii): x•H = H = H •x, for all x ∈ H (the reproduction axiom)is called hypergroup. In particular, an associative hypergroupoid iscalled a semihypergroup and a hypergroupoid that verifies the repro-duction axiom is called a quasihypergroup.If A and B are non-empty subsets of H, then A •B = ∪a∈Ab∈Ba • b.Definition 2.2. A duplex is a set D equipped with two associativeoperations · : D2 → D and ◦ : D2 → D. We will denote by (D, ·, ◦) theset duplex with two associative operations (·) and (◦). Moreover, a mapf : D → D′ from a duplex D to another duplex D′ is a homomorphism,provided that f(x · y) = f(x) · f(y) and f(x ◦ y) = f(x) ◦ f(y).Definition 2.3. A duplex (D, ·, ◦) is called commutative if two opera-tions (·) and (◦) are commutative.Definition 2.4. Let (D, ·, ◦) be a duplex. A strong duplex is a type ofduplex that follows the condition:{(a · b) ◦ c, (a ◦ b) · c} = {a ◦ (b · c), a · (b ◦ c)},for all (a, b, c) ∈ D3.ASSOCIATED (SEMI)HYPERGROUPS FROM DUPLEXES 853. (Semi)hypergroups derived from strongly duplexesLet m be a natural number and (Zm,+) be the cyclic group of theset of residue classes modulo m. Moreover, suppose x is a real numberand [x] is the integer part of x. Then we have the following.Theorem 3.1. Let n be a positive integer and Hn = {0, 1, · · · , 2n− 1}be the set of natural numbers less than 2n. The pair (Hn,⊕) is a groupwhich is isomorphic to (Z2n,+), where ⊕ is defined by:x⊕ y = x+ y − 2n[(x+ y)/2n], ∀(x, y) ∈ H2n.Proof. Let f : Hn → Z2n by x 7→ x be a map. It is obvious that f isonto and one-to-one. We shall show that f is homomorphism. Thereare two cases:(1) If x+ y < 2n then [(x+ y)/2n] = 0 thus we have:f(x ⊕ y) = f(x + y − 2n[(x + y)/2n]) = f(x + y) = x+ y =x+ y = f(x) + f(y).(2) If x+ y ≥ 2n then [(x+ y)/2n] = 1 hence:f(x ⊕ y) = f(x + y − 2n[(x + y)/2n]) = f(x + y − 2n) =x+ y − 2n = x+ y + 2n = x+ y = f(x) + f(y).Therefore (Hn,⊕) is a group isomorphic to (Z2n,+). Symbolically,(Hn,⊕) ∼= (Z2n,+). □Example 3.2. Let H4 = {0, 1, 2, 3, 4, 5, 6, 7} be a set. The group(H4,⊕) can be shown as follows:⊕ 0 1 2 3 4 5 6 70 0 1 2 3 4 5 6 71 1 2 3 4 5 6 7 02 2 3 4 5 6 7 0 13 3 4 5 6 7 0 1 24 4 5 6 7 0 1 2 35 5 6 7 0 1 2 3 46 6 7 0 1 2 3 4 57 7 0 1 2 3 4 5 6Hence (H4,⊕) ∼= (Z8,+).Theorem 3.3. Let n be a natural number and Hn = {0, 1, · · · , 2n−1}.We define an operation ⊗ on Hn by:x⊗ y =x+ y − n[(x+ y)/n], if 0 ≤ x < n, 0 ≤ y < nx+ y − n[(x+ y)/n] + n, if 0 ≤ x < n, n ≤ y < 2nx+ y − n[(x+ y)/n], if n ≤ x < 2n, n ≤ y < 2nx+ y − n[(x+ y)/n] + n, if n ≤ x < 2n, 0 ≤ y < n86 JAFARPOUR AND ALIZADEHThen (Hn,⊗) is a group and (Hn,⊗) ∼= (Z2 × Zn,+).Proof. Suppose that Z2×Zn = {(x, y)|x ∈ Z2, y ∈ Zn}. Now considerthe mapg : Hn −→ Z2 × Zng(x) ={(0̄, x) if 0 ≤ x < n(1̄, x) if n ≤ x < 2nWe prove that g(x ⊗ y) = g(x) + g(y). To end this first suppose that0 ≤ x < n, 0 ≤ y < n so 0 ≤ x+ y < 2n.(1) If 0 ≤ x + y < n, then [(x + y)/n] = 0, thus g(x ⊗ y) =g(x+y−n[(x+y)/n]) = g(x+y) = (0̄, x+ y) = (0̄, x)+(0̄, y) =g(x) + g(y).(2) If n ≤ x + y < 2n then [(x + y)/n] = 1, and so g(x ⊗ y) =g(x + y − n[(x + y)/n]) = g(x + y − n) = (0̄, x+ y − n) =(0̄, x) + (0̄, y) = g(x) + g(y).Now let 0 ≤ x < n, n ≤ y < 2n. Then we have two cases:(3) If n ≤ x+ y < 2n, then [(x+ y)/n] = 1 so we have g(x⊗ y) =g(x + y − n[(x + y)/n] + n) = g(x + y − n + n) = g(x + y) =(1̄, x+ y) = (0̄, x) + (1̄, y) = g(x) + g(y).(4) If 2n ≤ x + y < 3n, then [(x + y)/n] = 2, hence g(x ⊗ y) =g(x+y−n[(x+y)/n]+n) = g(x+y−2n+n) = g(x+y−n) =(1, x+ y) = (0̄, x) + (1, y) = g(x) + g(y).Consider n ≤ x < 2n, n ≤ y < 2n.(5) If 2n ≤ x + y < 3n, then [(x + y)/n] = 2, thus g(x ⊗ y) =g(x + y − n[(x + y)/n]) = g(x + y − 2n) = (0̄, x+ y − 2n) =(1̄, x) + (1̄, y) = g(x) + g(y).(6) If 3n ≤ x+ y < 4n, then [(x+ y)/n] = 3, hence:g(x⊗y) = g(x+y−n[(x+y)/n]) = g(x+y−3n) = (0, x+ y − 3n) =(1̄, x) + (1̄, y) = g(x) + g(y).Finally, if n ≤ x < 2n, 0 ≤ y < n, the proof is similar above. Therefore(Hn,⊗) ∼= (Z2 × Zn,+) and the proof is completed. □ASSOCIATED (SEMI)HYPERGROUPS FROM DUPLEXES 87Example 3.4. Let H4 = {0, 1, 2, 3, 4, 5, 6, 7}. We can show the Cayleytable of the group (H4,⊗) as follow:⊗ 0 1 2 3 4 5 6 70 0 1 2 3 4 5 6 71 1 2 3 0 5 6 7 42 2 3 0 1 6 7 4 53 3 0 1 2 7 4 5 64 4 5 6 7 0 1 2 35 5 6 7 4 1 2 3 06 6 7 4 5 2 3 0 17 7 4 5 6 3 0 1 2In this case we have (H4,⊗) ∼= (Z2 × Zn,+).For a given n the following program written in MATLAB, shows thatthe duplex (Hn,⊕,⊗), where two hyperoperations ⊕ and ⊗ are intro-duced in Theorems (3.1) and (3.3) is a strongly duplex. We could checkthat our claim is true for the natural numbers less than 230; of coursewe guess that (Hn,⊕,⊗) is a strongly duplex for all natural numbers.The program works in this way which first for a given n as the inputof (Hn,⊕), (Hn,⊗) group’s tables entitled a and b which every one isisomorphic to (Z2n,+) and (Z2×Zn,+) groups, respectively. Then weconsider table k with 7 columns that the first row shows x, the secondy and the third z. In 4th to 7th rows which are related to x, y and z,equations (x ⊕ y) ⊗ z, (x ⊗ y) ⊕ z, x ⊕ (y ⊗ z) and x ⊗ (y ⊕ z) areobtained and finally the associativity of (Hn,⊕,⊗) are checked.n=input(’please inter n=’);a=zeros(2*n);for x=0:2*n-1for y=0:2*n-1a(x+1,y+1)=x+y-2*n*(floor((x+y)/(2*n)));endendab=zeros(2*n);for x=0:n-1for y=0:n-1b(x+1,y+1)=x+y-n*(floor((x+y)/n));endendfor x=n:2*n-188 JAFARPOUR AND ALIZADEHfor y=0:n-1b(x+1,y+1)=x+y-n*(floor((x+y)/n))+n;endendfor x=n:2*n-1for y=n:2*n-1b(x+1,y+1)=x+y-n*(floor((x+y)/n));endendfor x=0:n-1for y=n:2*n-1b(x+1,y+1)=x+y-n*(floor((x+y)/n))+n;endendbf=(2*n)3;k=zeros(f,7);i=1;for x=0:(2*n)-1for y=0:(2*n)-1for z=0:(2*n)-1d1=a(x+1,y+1);c1=b(1+d1,z+1);d2=b(x+1,y+1);c2=a(1+d2,z+1);d3=b(y+1,z+1);c3=a(x+1,1+d3);d4=a(y+1,z+1);c4=b(x+1,1+d4);k(i,1)=x;k(i,2)=y;k(i,3)=z;k(i,4)=c1;k(i,5)=c2;k(i,6)=c3;k(i,7)=c4;i=i+1;endendendkASSOCIATED (SEMI)HYPERGROUPS FROM DUPLEXES 89ans1 = 1;for i=1:fif (k(i, 4) == k(i, 5) == k(i, 6) == k(i, 7))elseif (k(i, 4) == k(i, 6) , k(i, 5) == k(i, 7))elseif (k(i, 4) == k(i, 7) , k(i, 5) == k(i, 6))elseans1 = 0;break;endendif ans1 == 1disp(’the duplex (Hn, ·, ◦) is strong duplex’)elsedisp(’the duplex (Hn, ·, ◦) is not strong duplex’)endProposition 3.5. Let (D, ·, ◦) be a strong duplex. Then (D,⊙) is asemihypergroup, where the hyperoperation ⊙ is defined on D as follows:x⊙ y = {x · y, x ◦ y}, for all (x, y) ∈ D2.Proof.(x⊙ y)⊙ z = ∪t∈x⊙yt⊙ z= (x · y)⊙ z ∪ (x ◦ y)⊙ z= {(x · y) · z, (x · y) ◦ z, (x ◦ y) · z, (x ◦ y) ◦ z}.On the other handx⊙ (y ⊙ z) = ∪t∈y⊙zx⊙ t= x⊙ (y · z) ∪ x⊙ (y ◦ z)= {x · (y · z), x ◦ (y · z), x · (y ◦ z), x ◦ (y ◦ z)}.Since (D, ·, ◦) is a strong duplex, we have:{(x ◦ y) · z, (x · y) ◦ z} = {x ◦ (y · z), x · (y ◦ z)}.Therefore (x⊙ y)⊙ z = x⊙ (y ⊙ z). □The associated semihypergroup (D,⊙) is called duplex semihyper-group.90 JAFARPOUR AND ALIZADEHExample 3.6. (H4,⊙) is a hypergroup, where ⊙ is defined as follows:⊙ 0 1 2 3 4 5 6 70 0 1 2 3 4 5 6 71 1 2 3 0, 4 5 6 7 0, 42 2 3 0, 4 1, 5 6 7 0, 4 1, 53 3 0, 4 1, 5 2, 6 7 0, 4 1, 5 2, 64 4 5 6 7 0 1 2 35 5 6 7 0, 4 1 2 3 0, 46 6 7 0, 4 1, 5 2 3 0, 4 1, 57 7 0, 4 1, 5 2, 6 3 0, 4 1, 5 2, 6Proposition 3.7. The duplex semihypergroup (D,⊙) is commutativeif and only if (D, ·, ◦) is commutative duplex.Example 3.8. For every natural number n, (Hn,⊕,⊗) is a commuta-tive duplex.Definition 3.9. Let ∗ be a hyperoperation on H such that x ∗ y ̸= H,for all (x, y) ∈ H2. We define the hyperoperation ∗c on H as follows:x ∗c y = H ∖ (x ∗ y).From now on we call ∗c, the complementary hyperoperation of ∗ on H.Definition 3.10. A (semi)hypergroup (H, ∗) is called a complementaryfeasible (semi)hypergroup if (H, ∗c) is a (semi)hypergroup.Example 3.11. Let H = {0, 1, 2, 3}. Then the semihypergroup (H, ∗),where the hyperoperation (∗) is as follows:∗ 0 1 2 30 0 1 2, 3 31 1 0, 1 2, 3 2, 32 2 2, 3 0, 1 0, 13 2, 3 2, 3 0, 1 0, 1is not a complementary feasible semihypergroup.Example 3.12. Every group is a complementary feasible hypergroup.For a given n, the following program written in MATLAB showsthat the hypergroup (Hn,⊙), where the hyperoperation ⊙ introducedin Proposition (3.5) is complementary feasible for the natural numbersless than 31; of course we also guess (Hn,⊙) is a complementary feasi-ble hypergroup for all natural numbers such that n ≥ 2. This programworks in this way for a given n Cayley tables (Hn,⊕), (Hn,⊗) enti-tled a and b then the Cayley table (Hn,⊙c) entitled c. The functionsASSOCIATED (SEMI)HYPERGROUPS FROM DUPLEXES 91compute and compute1 obtain (x ⊙c y) ⊙c z and x ⊙c (y ⊙c z), for all(x, y, z) and the function equal checks the associativity.n=input(’please inter n=’);a=zeros(2*n);for x=0:2*n-1for y=0:2*n-1a(x+1,y+1)=x+y-2*n*(floor((x+y)/(2*n)));endendab=zeros(2*n);for x=0:n-1for y=0:n-1b(x+1,y+1)=x+y-n*(floor((x+y)/n));endendfor x=n:2*n-1for y=0:n-1b(x+1,y+1)=x+y-n*(floor((x+y)/n))+n;endendfor x=n:2*n-1for y=n:2*n-1b(x+1,y+1)=x+y-n*(floor((x+y)/n));endendfor x=0:n-1for y=n:2*n-1b(x+1,y+1)=x+y-n*(floor((x+y)/n))+n;endendbd=zeros(1,2*n);for i=1:2*nd(i)=i-1;endc=zeros(2*n,(2*n-1)*(2*n));for i=1:2*ns=1;for j=1:2*nk=a(i,j);92 JAFARPOUR AND ALIZADEHk1=b(i,j);for t=1:2*nif (d(t)∼=k && d(t)∼=k1)c(i,s)=d(t);s=s+1;endendif a(i,j) =b(i,j)c(i,s)=-1;s=s+1;endendendcT=zeros(1,(2*n)∧3);y=1;f=zeros(1,2*n-1);h=zeros(1,(2*n-1)*(2*n-1));h1=zeros(1,(2*n-1)*(2*n-1));for i=1:2*nfor j=1:2*nfor k=1:2*nh=compute(c,i,j,k,n);h1=compute1(c,i,j,k,n);if (equal(h,h1,n))T(y)=1;y=y+1;elseT(y)=0;y=y+1;endendendendif(isequal(T,ones(1,(2*n)∧3)))disp(’the hypergroup (Hn,⊙) is a complementary feasible’);elsedisp(’the hypergroup (Hn,⊙) is not a complementary feasible’);endIn this step the function compute calculate a⊙(b⊙c), for all (a, b, c) ∈(Hn)3.ASSOCIATED (SEMI)HYPERGROUPS FROM DUPLEXES 93function h=compute(c,i,j,k,n)t=k*(2*n-1);t1=t-(2*n-1)+1;s=1;for m=t1:t;f(s)=c(j,m);s=s+1;ends1=1;for m=1:(s-1)g=f(m)+1;if(g =0)t=g*(2*n-1);t1=t-(2*n-1)+1;for x=t1:th(s1)=c(i,x);s1=s1+1;endendendIn this step the function compute1, calculate (a ⊙ b) ⊙ c for all(a, b, c) ∈ (Hn)3.function h1=compute1(c,i,j,k,n) t=j*(2*n-1);t1=t-(2*n-1)+1;s=1;for m=t1:t;f(s)=c(i,m);s=s+1;ends1=1;for m=1:(s-1)g=f(m)+1;if(g =0)t=k*(2*n-1);t1=t-(2*n-1)+1;for x=t1:th1(s1)=c(g,x);s1=s1+1;end94 JAFARPOUR AND ALIZADEHendendIn this step the function equals shows the equality between two sets(a⊙ b)⊙ c and a⊙ (b⊙ c), for all (a, b, c) ∈ (Hn)3.function e=equal(h,h1,n)r(1)=h(1);k=2;find=0;for i=2:length(h)find=0;if(h(i) =-1)for j=1:k-1if (h(i)==r(j))find=1;break;endendif(find==0)r(k)=h(i);k=k+1;endendendr1(1)=h1(1);k=2;find=0;for i=2:length(h1)find=0;if(h1(i) =-1)for j=1:k-1if (h1(i)==r1(j))find=1;break;endendif(find==0)r1(k)=h1(i);k=k+1;endendASSOCIATED (SEMI)HYPERGROUPS FROM DUPLEXES 95endr2=sort(r);r3=sort(r1);if (isequal(r2,r3))e=1;elsee=0;end4. conclusionIn this paper we introduce and analyze a new class of (semi)hypergroupsthat derived from strongly duplexes. Several properties of the derived(semi)hypergroups are investigated. This research can be continued inthe study of some particular classes of (semi)hypergroups, for instance.Two open problems are raised that they can be incipience of a newwork. The problems are:1) Let n be a natural number. Then (Hn,⊕,⊗) is a strongly duplex.2) Let n ≥ 2 be a natural number. Then (Hn,⊙) is a complementaryfeasible hypergroup.References1. Z. Bavel, J. Grzymala and K. Soo Hong, On the conectivity of the product ofautomata, Fundam. Informatics, 7(2) (1984), 225-265.2. P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, Tricesimo, 1993.3. P. Corsini and I. Cristea, Fuzzy grade of i.p.s. hypergroups of order 7, IranianJournal of Fuzzy Systems, 1(2) (2004), 15-32.4. P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Aca-demical Publications, Dordrecht, 2003.5. I. Cristea, Hyperstructures and fuzzy sets endowed with two membership func-tions, Fuzzy Sets and Systems, 160 (2009), 1114-1124.6. I. Cristea, M. Jafarpour, S. Sh. Mousavi and A. Soleymani, Enumerationof Rosenberg hypergroups, Computer and mathematics with applications, 60(2010), 2753-2763.7. B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, Interna-tional Academic Press, USA, 2007.8. D. Freni, Strongly transitive geometric spaces: Applications to hypergroups andsemigroups theory, Comm. Algebra, 32(8) (2004) 969-988.9. V. Leoreanu-Fotea and B. Davvaz, n-hypergroups and binary relations, Europ.J. Combin., 29 (2008), 1207-1218.10. J.L. Loday, Dialgebras, In: Dialgebras and related operads, Lecture Notes inMath. 1763, SpringerVerlag, Berlin, 2001, pp.766.11. F. Marty, Sur une Generalization de la Notion de Groupe, 8th Congress Math.Scandenaves, Stockholm, Sweden, (1934), 45-49.96 JAFARPOUR AND ALIZADEH12. C. Massouros and J. Mittas, Languages-automata and hypercompositionalstructure, Alg. Hyperstructure and Appl. (T. Vougiouklis, ed.) Proc. 4th In-ter. Congress Xanthi, Greece 1990, World Scientific, Singapore (1991), 137-147.13. T. Pirashvili, Sets with two associative operations, Central European Journalof Mathematics, 2 (2003), 169-183.14. J. J. Rotman, An introduction to the theory of groups, volume 148 of GraduateTexts in Mathematics. Springer-Verlag, New York, fourth edition, (1995).15. T. Vougiouklis, Groups in hypergroups, Combinatorics ’86 (Trento, 1986), 459–467, Ann. Discrete Math, 37, North-Holland, Amsterdam, 1988.16. T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press,Palm Harbor, FL, 1994.Morteza JafarpourDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.Email: m.j@mail.vru.ac.irFatemeh AlizadehDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.Email: falizadeh123@yahoo.com

ASSOCIATED (SEMI)HYPERGROUPS FROM DUPLEXES

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ASSOCIATED (SEMI)HYPERGROUPS FROM DUPLEXES

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