Rose-Hulman Undergraduate Mathematics Journal Volume 14 Issue 1 Article 5 Trees of Irreducible Numerical Semigroups Taryn M. Laird Northern Arizona University, [email protected] Jose E. Martinez Northern Arizona University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Laird, Taryn M. and Martinez, Jose E. (2013) "Trees of Irreducible Numerical Semigroups," Rose-Hulman Undergraduate Mathematics Journal: Vol. 14 : Iss. 1 , Article 5. Available at: https://scholar.rose-hulman.edu/rhumj/vol14/iss1/5 Rose- Hulman Undergraduate Mathematics Journal Trees of Irreducible Numerical Semigroups Taryn M. Laird and Jos´eE. Martinez a Volume 14, No. 1, Spring 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Northern Arizona University Rose-Hulman Undergraduate Mathematics Journal Volume 14, No. 1, Spring 2013 Trees of Irreducible Numerical Semigroups Taryn M. Laird and Jos´eE. Martinez Abstract.A 2011 paper by Blanco and Rosales describes an algorithm for construct- ing a directed tree graph of irreducible numerical semigroups of fixed Frobenius numbers. This paper will provide an overview of irreducible numerical semigroups and the directed tree graphs. We will also present new findings and conjectures concerning the structure of these trees. Acknowledgements: We would like to give a special thanks to our mentor, Jeff Rushall, for his help and guidance, and to Ian Douglas for sharing his expertise in programming. Page 58 RHIT Undergrad. Math. J., Vol. 14, No. 1 1 Introduction There are many problems in which we unknowingly encounter numerical semigroups in mathematics. Two of the most well known examples of these problems are the coin problem and the postage stamp problem. Here is one of many such puzzles. Example 1.1. Suppose we are given an infinite number of 4 cent, 6 cent, and 9 cent coins. We would like to determine the largest amount of exact change that we cannot make using combinations of these coins. In order to find this maximum number, we must consider all possible combinations of the coins. That is, fall possible amounts of changeg = f4k1 + 6k2 + 9k3jk1; k2; k3 2 Ng = f0; 4; 6; 8; 9; 10; 12; 13; 14; 15; 16; :::g Since we have made 4 consecutive integer amounts (12 cents, 13 cents, 14 cents and 15 cents) we know that all integer amounts greater than 15 cents can be made by adding more 4 cent coins to these consecutive amounts. This means that the largest amount of exact change we cannot make is 11 cents. The next problem is similar to Example 1.1, but it has an important difference. Example 1.2. Suppose we are given an infinite number of 4 cent and 14 cent coins and we again wish to find the largest amount of change that we cannot make with these coins. We again consider all possible combinations of the coins: fall possible amounts of changeg = f4k1 + 14k2jk1; k2 2 Ng = f0; 4; 8; 12; 14; 16; 18; :::g Note that we can never make an odd amount of change with our coins. This means that there cannot be a largest un-makeable amount. The infinitely large set of change amounts in the first example is called a numerical semi- group, and the largest uncreatable number is called the Frobenius number of the numerical semigroup. We now ask if there are other sets of coin denominations that have a largest un-makeable amount of 11 cents. It turns out that other such sets of denominations do share the same largest unmakeable amount and a special subset of the corresponding numerical semigroups can be recursively constructed in a directed tree graph. In the following sections of the paper we will first define numerical semigroups and certain properties that are important to both the numerical semigroups themselves and the directed tree graphs. We will then briefly examine how numerical semigroups are connected to com- munative ring theory. In section 4 we will present an algorithm that creates the directed tree graphs and construct a few examples. To conclude the paper, we will present some new results that help describe the structure of these directed tree graphs. RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 59 2 Numerical Semigroups We first need a solid grasp of what a numerical semigroup is, and more precisely, what it means to be an irreducible numerical semigroup. Definition 2.1. A numerical semigroup is a subset S of the nonnegative integers N that satisfies the following: 1. S contains 0. 2. S is closed under addition. 3. NnS is finite. We say fa1; a2; :::; ang is a generating set for S if S =fk1a1+k2a2+:::+knanjk1; k2; :::; kn 2 Ng, and we call each ai a generator of S. If no proper subset is a generating set for S, we say fa1; a2; :::; ang is the minimal generating set and we write S = ha1; a2; :::; ani; 0 < a1 < a2 < ::: < an. Example 2.1. Consider the numerical semigroup generated by 5, 6 and 13. Then S = h5; 6; 13i = f5k1 + 6k2 + 13k3jk1; k2; k3 2 Ng = f0; 5; 6; 10; 11; 12; 13; 15; 16; 17; 18; 19; 20; 21; 22; 23; :::g = f0; 5; 6; 10; 11; 12; 13; 15; !g The arrow indicates that all integers greater than 15 are in S. Associated with every numerical semigroup are two important integers which are impor- tant for our studies. Definition 2.2. The Frobenius number of a numerical semigroup S, denoted F (S), is the largest integer not in S. The multiplicity of S, denoted m(S), is the smallest positive element of S. Example 2.2. Consider once again the numerical semigroup generated by 5, 6, and 13. That is, S = h5; 6; 13i = f0; 5; 6; 10; 11; 12; 13; 15; !g Here F (S) = 14 as it is the largest integer not in S, and m(S) = 5, as 5 is the smallest positive element in S. Next we define the two special types of numerical semigroups that are the focus of our research. Page 60 RHIT Undergrad. Math. J., Vol. 14, No. 1 Definition 2.3. A numerical semigroup S is said to be symmetric if and only if F (S) is odd and if x 2 Z 2= S, then F (S) − x 2 S: Example 2.3. Consider, S = h6; 7; 10; 11i = f0; 6; 7; 10; 11; 12; 13; 14; 16; !g. Here F (S) = 15, which is odd, as required. In addition, for every integer x2 = S we have F (S) − x 2 S as depicted in Figure 1 below. For example, if x = 8, we have 15 − 8 = 7 2 S, and the same can be verified for all elements in Z 2= S. Thus S is symmetric. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Figure 1 Definition 2.4. A numerical semigroup S is said to be pseudosymmetric if and only if F (S) F (S) is even and if x 2 Z 2= S, then either x = 2 or F (S) − x 2 S. Example 2.4. Again consider S = h5; 6; 13i = f0; 5; 6; 10; 11; 12; 13; 15; !g. As previously discussed, F (S) = 14 is even as required. As indicated in Figure 2, for every integer x2 = S F (S) we have either x = 2 or F (S) − x 2 S, and so S is pseudosymmetric. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Figure 2 The reason for our emphasis on symmetric and pseudosymmetric numerical semigroups will be made clear in the remaining sections of the paper. RHIT Undergrad. Math. J., Vol. 14, No. 1 Page 61 2.1 Irreducible Numerical Semigroups Definition 2.5. A numerical semigroup S is irreducible if it cannot be expressed as an intersection of two numerical semigroups that properly contain S. For a given Frobenius number F , the set of all irreducible numerical semigroups with Frobenius number F is denoted I(F ). In [2], Branco and Rosales showed that a numerical semigroup is irreducible if and only if it is symmetric or pseudosymmetric. The following theorem [1] defines a certain numerical semigroup unique to every Frobenius number. Theorem 2.1. For every positive integer F , there exists a unique irreducible numerical F semigroup C(F ) whose generators are all larger than 2 . Moreover, ( F +1 f0; 2 ; !gnfF g if F is odd, C(F ) = F f0; 2 + 1; !gnfF g if F is even. ( F +1 F +3 h 2 ; 2 ; :::; F − 1i if F is odd, = F F h 2 + 1; 2 + 2; :::; F − 1;F + 1i if F is even. Example 2.5. Using Theorem 2.1 we will now construct C(15). 15 + 1 C(15) = f0; ; !gnf15g 2 = f0; 8; 9; 10; 11; 12; 13; 14; 15; 16; !gnf15g = f0; 8; 9; 10; 11; 12; 13; 14; 16; !g = h8; 9; 10; 11; 12; 13; 14i Note that when F is odd, C(F ) is generated by an interval of consecutive integers of N. This type of numerical semigroup, first investigated by Garc´ıa-S´anchez and Rosales in [3], led to a new characterization of symmetry for interval-generated numerical semigroups. The interested reader can consult [3] for more information. 3 Graph Theory Necessities The remainder of this paper will focus on graphs of families of specific numerical semi- groups that share a common Frobenius number. Therefore, a brief review of basic graph theory terminology is appropriate.