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Öğe The results on some Arf numerical semigroups with multiplicity 8(İksad Publications, 2019-12-16) Süer, Meral; İlhan, Sedat; Karakaş, İbrahimIn this paper, we will give some results about Frobenius number, Apery set, type, genus and determine number of Arf numerical semigroup S such that m S() 8 = and CS C ( ) 0, 2,3, 4,5,6,7 ( mod8).Öğe Some results on Arf numerical semigroups with multiplicity 8(Harran Üniversitesi, 2017-05) Süer, Meral; İlhan, SedatA numerical semigroup is a subset of the set of nonnegative integers (denoted here by ¥ ) closed under addition, containing the zero element and with finite complement in ¥ . Note also that up to isomorphism the set of numerical semigroups classify the set of all submonoids of ( , ) ¥ + . Let S be submonoid of ¥ , the condition of having finite complement in ¥ is equivalent to saying that the greatest common divisor (gcd for short) of its elements is one. Those positive integers which do not belong to S are called gaps of S . The number of gaps of S is called the genus of S and it is denoted by G S( ) . The largest gaps of S is F S( ) if S is different from ¥ . m S s S s ( ) min : 0 = Î > { } are called multiplicity of S , respectively. Also, n S Card F S S ( ) 0,1,2,..., ( ) = Ç ({ } ) is called the number determine of S . If a Î ¥ and a S Ï , then a is called gap of S . We denote the set of gaps of S , by H S( ) , i.e, H S S ( ) \ = ¥ .The G S Card H S ( ) ( ( )) = is called the genus of S . Also, It known that G S F S n S ( ) ( ) 1 ( ) = + - . A numerical semigroup S is called Arf if x y z S + - Î for all x y z S , , Î , where x y z ³ ³ . This definition was first given by C. Arf in 1949. In the study, we intend to examine the Arf numerical semigroup with multiplicity eight and fixed conductor. We will also able to compute the notable elements and special sets of these numerical semigroups.Öğe On the numerical semigroups with generated by two elements with multiplicity 3(Harran Üniversitesi, 2017-05) Süer, Meral; İlhan, Sedat; Çelik, AhmetThroughout this study, we assume that ¥ and ¢ be the sets of nonnegative integers and integers, respectively. The subset S of ¥ is a numerical semigroup if 0 Î S , x + y Î S, for all x, y Î S , and Card(¥ \S)< ¥ ( this condition is equivalent to gcd(S)= 1 , gcd(S)= greatest common divisor the element of S ) . Let S be a numerical semigroup, then F(S) = max(¢ \S) and m(S) = min{s Î S: s > 0} are called Frobenius number and multiplicity of S , respectively. Also, n(S) = Card ({0,1,2,...,F(S)}ÇS)is called the number determine of S . If S is a numerical semigroup such that 1 2 , ,..., r S = < a a a > , then we observe that { } 1 2 0 , 2 1 , ,..., 0, , ,..., , ( ) 1, ... r n n S a a a s s s s s F S - = < > = = = + ® where 1 , ( ) i i s s n n S + < = , and the arrow means that every integer greater than F(S) + 1 belongs to S , for i = 1,2,...,n = n(S) . If a Î ¥ and a Ï S , then a is called gap of S . We denote the set of gaps of S , by H(S) , i.e, H(S) = ¥ \S .The G(S) = Card(H(S)) is called the genus of S . Also, It is known that G(S) = F(S) + 1- n(S) . Let S be a numerical semigroup andm Î S ,m > 0 . Then Ap(S,m) xS :x mS is called Apery set of S according to m . A numerical semigroup S is Arf if a+ b- c Î S , for all a,b,c Î S such that a ³ b ³ c. The intersection of any family of Arf numerical semigroups is again an Arf numerical semigroup. Thus, since ¥ is an Arf numerical semigroup, one can consider the smallest Arf numerical semigroup containing a given numerical semigroup. The smallest Arf numerical semigroup containing a numerical semigroup S is called the Arf closure of S , and it is denoted by Arf (S) . In this presentation, we will give some results about gaps, the determine number, Apery set and Arf closure of S numerical semigroup such that S = 3, x .Öğe On Arf closures of some numerical semigroups generated by three elements(Harran Üniversitesi, 2017-05) Süer, Meral; İlhan, Sedat; Çelik, AhmetA numerical semigroup is a subset of the set of nonnegative integers (denoted here by ¥ ) closed under addition, containing the zero element and with finite complement in ¥ . A numerical semigroup is a set of the form 1, 2 , , 1 2 ... p p S = u u ¼ u = u ¥ + u ¥ + + u ¥ where are positive integers, such that ( ) 1 2 gcd , , , 1 p u u ¼ u = . The condition is saying that S has finite complement in ¥ (where short gcd is the greatest common divisor),( Barucci, Dobbs & Fontana, 1997; .Fröberg, Gottlieb & Haggkvist, 1987; . Rosales &Garcia-Sanchez, 2009). If S is a numerical semigroup, then F(S) = max(¢ \S) is called Frobenius number of S . Any numerical semigroup write this form { } 1 2 0 1 2 , ,..., 0 , , ,..., ( ) 1, ... n n S = < a a a > = = s s s s = F S + ® . Where “® ” means that every integer greater than F(S)+ 1 belongs to the set. A numerical semigroup S is called Arf if x + y - z Î S for all x, y, z Î S , where x ³ y ³ z . The smallest Arf numerical semigroup containing a numerical semigroup S is called the Arf closure of S , and it is denoted by Arf (S) . Arf numerical semigroup and their applications to algebraic error corerecting codes have been a special interest in recent times (Brass-Amaros, 2004; Campillo, Farran & Munuera, 2000). The families of Arf numerical semigroups are related with the problem solution of singularities in curve.In this presentation, we will give some results between numerical semigroups and theirs Arf closure. Also, we will obtain some relation for Arf closure of these numerical semigroups.Öğe On a class of Arf numerical semigroups(Fırat Üniversitesi, 2016-05) Süer, Meral; İlhan, SedatA subset S of N is called a numerical semigroup if S is closed under addition and S has element 0 and N\S is finite where N denotes the set of nonnegative integers. In this study, we are interested two subclass of maximal embedding dimension numerical semigroups, which are those semigroups having the Arf property and saturated numerical semigroups. We introduce a new class of both Arf property and saturated numerical semigroups with multiplicity four. We consider numerical semigroups minimally generated by {4, k, k+1, k+2}. Where k is an integer greater than or equal to 5 and k is congruent to 1 (modulo 4). We prove that all these semigroups are both numerical semigroups with Arf property and saturated numerical semigroup. There is not any formulas to calculate invariants as Frobenius number, gaps, n(S) and genus of S even for numerical semigroup with multiplicity four. But this invariants have been calculated by imposing some conditions on elements of the numerical semigroup S. We calculate the Frobenius number, the genus and the set of gaps of each of these numerical semigroups. Additionally, we give a relation between the set of pseudo- Frobenius numbers and the set of all fundamental gaps of these numerical semigroups.Öğe Some results in triply–generated telescopic numerical semigroups(Harran Üniversitesi, 2017-05) Süer, Meral; İlhan, SedatIn this study, we assume that ¥ = {0,1,2,...,n,...}, S Í ¥ and ¢ integer set. S is a numerical semigroup if a + b Î S, for a,b Î S , 0Î S and ¥ \S is finite. Let S be a numerical semigroup and X Ì ¥ . X is called minimal system of generators of S if S = < X > and there is not any subset Y Ì X such that S = < Y > . = < > 1 2 3 S s ,s ,s is called a triply-generated telescopic numerical semigroup if Î < > 1 2 3 , s s s d d where = 1 2 d gcd(s s ) (Here, gcd(S)= greatest common divisor the element of S ). Let S be a numerical semigroup, then m(S) = min{x Î S: x > 0} is called as multiplicity of S . The number F(S) = max(¢ \S) is called Frobenius number of S . A numerical semigroup S can be written the form { } - = < > = = = + ® 1 2 0 1 2 1 , ,..., 0, , ,..., , ( ) 1, ... n n n S a a a s s s s s F S , where + < i i 1 s s , n = n(S) and the arrow means that every integer greater than F(S)+ 1 belongs to S for i = 1,2,...,n = n(S) . If x Î ¥ and x Ï S , then x is called gap of S . We denote the set of gaps ofS , H(S) = ¥ \S . The number G(S) = #(H(S)) is called the genus ofS . Let S be a numerical semigroup and m Î S ,m> 0. Then Ap(S,m) x S :x mS is called Apery set of S according to m . Ap(S,m) is formed by the smallest elements of S belonging to the different congruence classes mod m. Thus # (Ap(S,m)) = m and the Frobenius number of S equal to max(Ap(S,m)) - m ( where #(A) stands for Cardinality (A) ). The number n(S) = #({0,1,2,...,F(S)}ÇS) is called determine of numerical semigroup S . Also, It is known that G(S) = F(S)+ 1- n(S) . In this study, we will give some results about gaps, fundamental gaps and the determine number of S triply-generated telescopic numerical semigroups with arbitrary multiplicity.
