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Öğe On Arf numerical semigroups(SCIK Publishing Corporation, 2016) Süer, Meral; İlhan, SedatIn this study, we obtain an Arf semigroup by means of a sequence. We also establish some results on the Arf semigroupÖğe On telescopic numerical semigroup families with embedding dimension 3(Erzincan Üniversitesi, 2019-03-24) Süer, Meral; İlhan, SedatIn this study, the set of all telescopic numerical semigroups families with embedding dimension three is obtained for some fixed multiplicity by some parameters. Also, some invariants of these families are calculated in term of their generatorsÖğe 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 On a family of saturated numerical semigroups with multiplicity four(TÜBİTAK, 2017-01-16) Süer, Meral; İlhan, SedatIn this study, we will give some results on Arf numerical semigroups of multiplicity four generated by {4, k, k + 1, k + 2} where k is an integer not less than 5 and k ≡ 1(mod 4).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 the saturated numerical semigroups(Open Mathematics, 2016-11) Süer, Meral; İlhan, SedatIn this study, we characterize all families of saturated numerical semigroups with multiplicity four. We also present some results about invariants of these 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.Öğe On the one half of an Arf numerical semigroup(SCIK Publishing Corporation, 2015) Süer, Meral; İlhan, SedatIn this study, we will characterize the Arf numerical semigroup that is computed in a quotient of an Arf numerical semigroup by an positive integer. We will also obtain the one half of this special Arf numerical semigroup and give some results about this numerical semigroup.
