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Öğe Bazı saturated sayısal yarıgruplar üzerine(Dicle Üniversitesi, 2016-12) Süer, Meral; İlhan, Sedat; Çelik, AhmetÖğe All arf or saturated numerical semigroups with multiplicity 7(Dicle Üniversitesi, 2016-12) Süer, MeralÖğe 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 The special gaps of some Arf numerical semigroups(IKSAD Publishing House, 2018-05-01) Süer, Meral; Yalçın, Burak YasinThe concept of special gap of a numerical semigroup is used to by the problem of finding the set of all numerical semigroups containining a given numerical semigroup. In this study, we will find the specific gaps of some Arf numerical semigroups families and all numerical semigroups containining them.Öğe Saturated numerical semigroups with multiplicity four(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. A numerical semigroup S is saturated if the following condition holds: s, s1,s2, …,sk belongs to S are such that s1 < s or equal to s, for all 1 < I < k or i=1 and i=k , and c1,c2,…,ck belongs to are such that c1s1+c2s2+…+cksk > 0 or equal to 0, then s+ c1s1+c2s2+…+cksk belongs to S. The frobenius number of S is the maximum integer not belonging to S, which is denoted by F(S). H(S)= N\S is the set of the elements gaps of S, and the cardinality elements of H(S) is called genus of S, and denoted by g(S). It is said that an integer x is a Pseudo-Frobenius number if x+s belongs to S for s > 0, s belongs to S and x belongs to \S. In this study, we will characterize the all families of Saturated numerical semigroups with multiplicity four. These numerical semigroups generated by 4,k,k 1,k 2 for k>5 or k=5, k=1(mod4), and 4,k,k 2,k 3 for k > 7 or k=7, k=3(mod4), and 4,k,k t,k t 2 for k > 6 or k=6, k=2(mod 4), respectively. We will prove that Saturated numerical semigroups such that multiplicity four. Also, we will give formulas Frobenius number F S( ) , Pseudo Frobenius number PF S( ) , gaps H S( ) and genus g S( ) of these numerical semigroups.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 Arf numerical semigroups with multiplicity eight(Yıldız Teknik Üniversitesi, 2017-05) Süer, Meral; İlhan, Sedat; Karakaş, İbrahimÖğe Some extension of a class of pseudo symmetric numerical semigroups(Selçuk Üniversitesi, 2011-07) Süer, Meral; İlhan, SedatÖğe 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 Betti numbers of some telescopic numerical semigroups(IKSAD Publishing House, 2018-05-01) Süer, Meral; Sezgin, Mehmet ŞirinLet be the set of nonnegative integers. A numerical semigroup is a nonempty subset M of that is closed under addition, contains the zero element, and whose complement in is finite. In this study, we will examine the Betti numbers of some telescopic numerical semigroup families with generated triply. And we will try to express in terms of generators of these numerical semigroup families. So we will find a formula for the Betti numbers of these numerical semigroup families.
