On a transport problem and monoids of non-negative integers
aa r X i v : . [ m a t h . G R ] M a r On a transport problem and monoids ofnon-negative integers
Aureliano M. Robles-P´erez ∗† and Jos´e Carlos Rosales ∗‡ Abstract
A problem about how to transport profitably a group of cars leadsus to study the set T formed by the integers n such that the system ofinequalities, with non-negative integer coefficients, a x + · · · + a p x p + α ≤ n ≤ b x + · · · + b p x p − β has at least one solution in N p . We will see that T ∪ { } is a submonoidof ( N , +). Moreover, we show algorithmic processes to compute T . Keywords:
Transport problem, Diophantine inequalities, monoids.
A transport company is dedicated to carry cars from the factory to the autho-rised dealer. For that, the company uses small and large trucks with a capacityof three and six cars, respectively. Moreover, we have the following determi-nants. • The use of those trucks represents, for the company, a cost of 1200 and1500 euros, respectively. • The company charges to the dealer 300 euros for each transported car. • Without charge for the customer, an additional car is loaded facing pos-sible eventualities. ∗ Both authors are supported by the project MTM2014-55367-P, which is funded by Mi-nisterio de Econom´ıa y Competitividad and Fondo Europeo de Desarrollo Regional FEDER,and by the Junta de Andaluc´ıa Grant Number FQM-343. The second author is also partiallysupported by Junta de Andaluc´ıa/Feder Grant Number FQM-5849. † Departamento de Matem´atica Aplicada, Universidad de Granada, 18071-Granada, Spain.E-mail: [email protected] ‡ Departamento de ´Algebra, Universidad de Granada, 18071-Granada, Spain.E-mail: [email protected]
1f the company consider that a transport order is cost-effective when it hasprofits of at least 900 euros, how many cars must be transported in order toachieve that purpose?It is clear that a transport order is profitable if and only if there exist x, y ∈ N (where N is the set of non-negative integers) such that300 n ≥ x + 150 y + 900 n + 1 ≤ x + 6 y ) . (1.1)Simplifying the first inequality of (1.1), we have the equivalent system n ≥ x + 5 y + 3 n ≤ x + 6 y − ) . (1.2)Consequently, the set (cid:8) n ∈ N | (1.2) has a solution in N (cid:9) is formed by the non-negative integers which give us an affirmative answer to the proposed problem.We can generalize the above problem in the following way: if we consider a = ( a , . . . , a p ) , b = ( b , . . . , b p ) ∈ N p and α, β ∈ N , then we want to computethe set S ( a, b, α, β ) = ( n ∈ N a x + · · · + a p x p + α ≤ n ≤ b x + · · · + b p x p − β for some ( x , . . . x p ) ∈ N p ) . (1.3)In this work we will prove that S ( a, b, α, β ) ∪ { } is a submonoid of ( N , +)(that is, a subset of N that is closed under addition and contains the zeroelement) and our main purpose will be to give an algorithm in order to computethe minimal system of generators for such a monoid.To finish this introduction, let us observe that, with the above notation, theproblem studied in [4] is associated to the particular sets S ( a, b, , ( α, β ) = (0 , In this section will be suppose that a = ( a , . . . , a p ), b = ( b , . . . , b p ) belong to N p and that α, β ∈ N . Lemma 2.1. If m, n ∈ S ( a, b, α, β ) , then m + n ∈ S ( a, b, α, β ) .Proof. If m, n ∈ S ( a, b, α, β ), then there exist ( x , . . . , x p ) , ( y , . . . , y p ) ∈ N p such that a x + · · · + a p x p + α ≤ m ≤ b x + · · · + b p x p − β and a y + · · · + a p y p + α ≤ n ≤ b y + · · · + b p y p − β. Therefore, a ( x + y )+ · · · + a p ( x p + y p )+ 2 α ≤ m + n ≤ b ( x + y )+ · · · + b p ( x p + y p ) − β. a ( x + y ) + · · · + a p ( x p + y p ) + α ≤ m + n ≤ b ( x + y ) + · · · + b p ( x p + y p ) − β, and, consequently, m + n ∈ S ( a, b, α, β ).As an immediate consequence of the above result we have the followingproposition. Proposition 2.2. S ( a, b, α, β ) ∪ { } is a submonoid of ( N , +) . Let X be a non-empty subset of N k . We will denote by h X i the submonoidof ( N k , +) generated by X , that is, h X i = { λ x + · · · + λ n x n | n ∈ N \ { } , x , . . . , x n ∈ X, λ , . . . , λ n ∈ N } . If M is a submonoid of ( N k , +) and M = h X i , then we will say that X is a system of generators of (for) M or, equivalently, that M is generated by X . Inaddition, if no proper subset of X generates M , then we will say that X is a minimal system of generators of (for) M .The next result is [6, Corollary 2.8]. Proposition 2.3.
Every submonoid of ( N , +) admit a unique minimal systemof generators. Moreover, such a system is finite. Our objective in this work is to show an algorithm to compute the minimalsystem of generators of S ( a, b, α, β ) ∪ { } . In this section we will solve theproblem when ( α, β ) = (0 , M of ( N p , +) is finitely generated if there exists a finite set X such that M = h X i . From Proposition 2.3 we know that every submonoid of( N , +) is finitely generated. However, if k ≥
2, then there exist submonoids of( N k , +) which are not finitely generated.Let z = ( z , . . . , z p ) ∈ Z p (where Z is the set of integers) and let A ( z ) = { ( x , . . . , x p ) ∈ N p | z x + · · · + z p x p ≥ } . It is well known that A ( z ) is a fini-tely generated submonoid of ( N p , +) and, moreover, in [1] it is shown an algo-rithm to compute a finite system of generators of A ( z ).If s, t ∈ Z (with s ≤ t ), then we will denote by [ s, t ] = { x ∈ Z | s ≤ x ≤ t } . Theorem 2.4.
Let { m , . . . , , m q } be a system of generators of A ( b − a ) , where m i = ( m i , . . . , m ip ) for all i ∈ { , . . . , q } . Then S ( a, b, , ∪ { } is the sub-monoid of ( N , +) generated by L = S qi =1 [ a m i + · · · + a p m ip , b m i + · · · + b p m ip ] .Proof. It is clear that S ( a, b, ,
0) = [ ( x ,...,x p ) ∈ A ( b − a ) [ a x + · · · + a p x p , b x + · · · + b p x p ] . Therefore, S ( a, b, , ∪ { } is a submonoid of ( N , +) which contains L . In orderto finish the proof, we will see that, if T is a submonoid of ( N , +) which contains3, then S ( a, b, , ⊆ T . For that we will prove that, if ( x , . . . , x p ) ∈ A ( b − a ),then [ a x + · · · + a p x p , b x + · · · + b p x p ] ⊆ T .If ( x , . . . , x p ) ∈ A ( b − a ), then we have ( x , . . . , x p ) = λ m + · · · + λ q m q for some λ , . . . , λ q ∈ N . By induction over λ + · · · + λ q , we will show that[ a x + · · · + a p x p , b x + · · · + b p x p ] ⊆ T . Thus, if λ + · · · + λ q = 0 the resultis obvious. Let us suppose that λ + · · · + λ q ≥ i ∈ { , . . . , q } suchthat λ i = 0. If we take ( y , . . . , y p ) = λ m + · · · + ( λ i − m i + · · · + λ q m q ,then ( x , . . . , x p ) = ( y , . . . , y p ) + m i and, by the induction hypothesis, we getthat [ a y + · · · + a p y p , b y + · · · + b p y p ] ⊆ T . Since T is a monoid, then[ a y + · · · + a p y p , b y + · · · + b p y p ]+[ a m i + · · · + a p m ip , b m i + · · · + b p m ip ] ⊆ T and, consequently, [ a x + · · · + a p x p , b x + · · · + b p x p ] ⊆ T .Now, we are going to describe an algorithmic process, given in [4], in orderto compute a system of generators for A( z ). Thereby, we get a self-containedpaper and, in addition, we will be able to describe examples without necessityof referencing to [1].Let B( z ) = { ( x , . . . , x p , x p +1 ) ∈ N p +1 | z x + · · · + z p x p − x p +1 = 0 } . It iswell known (see [5]) that B( z ) is a finitely generated submonoid of ( N p +1 , +) and,in addition, its set of minimal generators coincide with the minimal elements(with the usual order in N p +1 ) of the set B( z ) \ { (0 , . . . , } . Moreover, weknow that, if ( x , . . . , x p , x p +1 ) is a minimal element of B( z ) \ { (0 , . . . , } , then x + · · · + x p +1 ≤ | z | + · · · + | z p | +2 (Pottier’s bound, [3]). Finally, it is easy to seethat, if { b , . . . , b q } is a system of generators of B( z ), then { π ( b ) , . . . , π ( b q ) } is a system of generators for A( z ) (where π ( x , . . . , x p , x p +1 ) = ( x , . . . , x p )).Therefore, we have an algorithm to compute a system of generators for A( z ). Example 2.5.
We are going to compute the minimal system of generators for S = S (cid:0) (4 , , (3 , , , (cid:1) ∪ { } . We begin computing a system of generators for A ( − ,
1) = { ( x, y ) ∈ N | − x + y ≥ } . For that, we compute the minimal ele-ments of B ( − , \{ (0 , , } , where B ( − ,
1) = { ( x, y, z ) ∈ N | − x + y − z = 0 } .By applying the Pottier’s bound, we have that (1 , , , (0 , ,
1) are such mini-mal elements and, in consequence, { (1 , , (0 , } is a system of generators for A ( − , , ∪ [5 ,
6] = { , , } is a system of generators for S . Thus, S = { , , , , , , , , →} (where the symbol → means that every number greater than 14 belongs to S ). ( α, β ) = (0 , Let A = { ( x , . . . , x p ) ∈ N p | ( b − a ) x + · · · + ( b p − a p ) x p ≥ α + β } . Then itis clear that S ( a, b, α, β ) = [ ( x ,...,x p ) ∈A [ a x + · · · + a p x p + α, b x + · · · + b p x p − β ] . Let us define over A the binary relation ≤ A ( b − a ) as follow: x ≤ A ( b − a ) y if y − x ∈ A ( b − a ) .
4n [7] it is shown that ≤ A ( b − a ) is an order relation. Moreover, from the resultsin such a work, we also know that, if D is a system of generators of A ( b − a ) and C = minimals ≤ A ( b − a ) A , then A = C + hDi . Furthermore, in [7] it is also givenan algorithm to compute C and D .Let B ⊆ N . We will say that M is a B -monoid if M is a submonoid of( N , +) fulfilling that ( M \ { } ) + B ⊆ M .It is clear that N is a B -monoid and that the intersection of B -monoids isanother B -monoid. Therefore, we can give the notion of smallest B -monoidwhich contains a given set A of non-negative integers . Theorem 3.1.
With the above notation, let
C = [ ( c ,...,c p ) ∈C [ a c + · · · + a p c p + α, b c + · · · + b p c p − β ] and let D = [ ( d ,...,d p ) ∈D [ a d + · · · + a p d p , b d + · · · + b p d p ] . Then S ( a, b, α, β ) ∪ { } is the smallest D -monoid which contains C .Proof. It is clear that S ( a, b, α, β ) ∪ { } is a D-monoid which contains C. Now,let us see that, if T is a D-monoid containing C, then S ( a, b, α, β ) ∪ { } ⊆ T .For that, we will prove that, if ( x , . . . , x p ) ∈ A , then[ a x + · · · + a p x p + α, b x + · · · + b p x p − β ] ⊆ T. (3.1)Let us suppose that D = { d , . . . , d q } , where d i = ( d i , . . . , d ip ) for all i ∈{ , . . . , q } . Since ( x , . . . , x p ) ∈ A , then there exist c ∈ C and λ , . . . , λ q ∈ N such that ( x , . . . , x p ) = c + λ d + · · · + λ q d q . Now, in order to show (3.1), we useinduction over λ + · · · + λ q following the reasoning exposed in Theorem 2.4.At this moment, we propose to give an algorithm that allows us to computethe smallest B -monoid containing a given set A . Proposition 3.2.
Let M be a submonoid of ( N , +) and let A = { a , . . . , a n } ⊆ N \ { } be a system of generators of M . Then M is a B -monoid if and only if A + B ⊆ M .Proof. The necessary condition is trivial. So, let us see the sufficient one.If m ∈ M \ { } , then there exists ( λ , . . . , λ n ) ∈ N n \ { (0 , . . . , } such that m = λ a + · · · + λ n a n . Now, by induction over λ + · · · + λ n , we will prove that m + b ∈ M for all b ∈ B . Firstly, the result is trivially true for λ + · · · + λ n = 1.Let us suppose that λ + · · · + λ n ≥ i ∈ { , . . . , n } such that λ i = 0. Bythe induction hypothesis, we easily deduce that ( m − a i ) + b ∈ M . Therefore,( m − a i ) + b + a i ∈ M and, consequently, m + b ∈ M .If M is a submonoid of ( N , +), we will denote by msg( M ) the minimal systemof generators of M . 5 lgorithm 3.3. INPUT: A finite set A of positive integers.OUTPUT: The minimal system of generators for the smallest B -monoidcontaining A .(1) X = msg( h A i ).(2) Y = X ∪ ( X + B ).(3) If msg( h Y i ) = X , then return X .(4) Set X = msg( h Y i ) and go to (2).By using Propositon 3.2, it is easy to see that the above algorithm oper-ates suitably. On the other hand, observe that the most complex process inAlgorithm 3.3 is to compute msg( h Y i ), that is, compute the minimal system ofgenerators for a monoid starting from any system of generators of it. For thispurpose, we can use the GAP package numericalsgps (see [2]).
Example 3.4.
By using Algorithm 3.3, we are going to compute the smallest { , } -monoid containing { , } . • X = { , } . • Y = { , , , , } . • msg( h Y i ) = { , , , } . • X = { , , , } . • Y = { , , , , , , } . • msg( h Y i ) = { , , , , } . • X = { , , , , } . • Y = { , , , , , , , , } . • msg( h Y i ) = { , , , , } = X .Therefore, h , , , , i = { , , , →} is the smallest { , } -monoid containing { , } .As a consequence of the previous results, we have an algorithm which allowsus to compute the minimal system of generators of the monoid S ( a, b, α, β ) ∪{ } . Algorithm 3.5.
INPUT: a, b ∈ N p and α, β ∈ N such that ( α, β ) = (0 , S ( a, b, α, β ) ∪ { } .(1) Compute a finite system of generators D for A ( b − a ).(2) Compute C = minimals ≤ A ( b − a ) A , where A = { ( x , . . . , x p ) ∈ N p | ( b − a ) x + · · · + ( b p − a p ) x p ≥ α + β } . (3) C = S ( c ,...,c p ) ∈C [ a c + · · · + a p c p + α, b c + · · · + b p c p − β ] andD = S ( d ,...,d p ) ∈D [ a d + · · · + a p d p , b d + · · · + b p d p ].(4) Return the minimal system of generators of the smallest D-monoid con-taining C. 6n Section 2, we have described an algorithmic process to compute D . Atthe beginning of this section, we also have commented that, from the results of[7], we have an algorithm to compute C . In order to give a self-contained paperand to show examples without necessity of referencing to [7], we are going tomention briefly how to build C . For that, we need to consider the followingequations and inequalities: ( b − a ) x + · · · + ( b p − a p ) x p ≥ α + β, (3.2)( b − a ) x + · · · + ( b p − a p ) x p − x p +1 − ( α + β ) x p +2 = 0 , (3.3)( b − a ) x + · · · + ( b p − a p ) x p ≥ . (3.4)The next result is a direct consequence of the results from [7, Section 2]. Proposition 3.6.
Let A = { α , . . . , α t } , with α i = ( α i , . . . , α i p +2 ) , be a systemof generators of the monoid formed by the set of non-negative solutions of (3.3) .Assume that α , . . . , α d are the elements in A with the last coordinate equal tozero and α d +1 , . . . , α g are those elements in A with the last coordinate equalto one. Let π : N p +2 → N p be the projection onto the first p coordinates.Then the set of non-negative integer solutions of (3.2) is { π ( α d +1 ) , . . . , π ( α g ) } + h π ( α ) , . . . , π ( α d ) i . Moreover, A ( b − a ) = h π ( α ) , . . . , π ( α d ) i . An immediate consequence of the above proposition is the next result.
Corollary 3.7. D = { π ( α ) , . . . , π ( α d ) } .2. C = minimals ≤ A ( b − a ) { π ( α d +1 ) , . . . , π ( α g ) } .Remark . Let us observe that it is really easy to determine C using (3.4). Ineffect, let F = { ( x , . . . , x p +2 ) ∈ N p +2 | ( b − a ) x + · · · + ( b p − a p ) x p − x p +1 − ( α + β ) x p +2 = 0 } . It is well known (see [5]) that F is a finitely generated monoidof ( N p +2 , +) and, in addition, its set of minimal generators coincide with the setof minimal elements (with respect the usual order in N p +2 ) of F \ { (0 , . . . , } .Moreover, from [3], we have that, if { ( x , . . . , x p +2 ) is a minimal element of F \ { (0 , . . . , } , then x + · · · + x p +2 ≤ | b − a | + · · · + | b p − a p | + α + β + 2. Example 3.9.
We are going to compute the minimal system of generators of S = S (cid:0) (4 , , (3 , , , (cid:1) ∪ { } using Algorithm 3.5.Let F = { ( x , x , x , x ) ∈ N | − x + x − x − x = 0 } . Having in mindRemark 3.8, it is easy to see thatminimals (cid:0) F \ { (0 , , , } (cid:1) = { (1 , , , , (0 , , , , (0 , , , } . Now, by Corollary 3.7, we have that D = { (1 , , (0 , } and C = { (0 , } .Consequently, D = [9 , ∪ [5 ,
6] = { , , } and C = [23 ,
23] = { } . Finally, byapplying Algorithm 3.3, the smallest { , , } -monoid containing { } is S = { , , , , , , , , , →} . By the way, let us observe that S is the set of solutions for the example in theintroduction. 7 A brief remark on numerical semigroups
By Proposition 2.2, we know that S ( a, b, α, β ) ∪ { } is a submonoid of ( N , +).On the other hand, we have that the sets of solutions in Examples 2.5, 3.4, and3.9 are numerical semigroups (that is, a submonoid S of ( N , +) such that N \ S is finite). In the next example the answer is related with a monoid that is nota numerical semigroup. Example 4.1.
Let us calculate S = S (cid:0) (4 , , (3 , , , (cid:1) (observe that 0 ∈ S ).First, we need a system of generators for A ( − ,
0) = { ( x, y ) ∈ N | − x +0 y ≥ } .For that, we compute the minimal elements of B ( − , \ { (0 , , } , where B ( − ,
0) = { ( x, y, z ) ∈ N | − x + 0 y − z = 0 } . It is obvious that (0 , ,
0) is theunique minimal element of B ( − , \ { (0 , , } and, in consequence, { (0 , } isa system of generators for A ( − , ,
5] = { } is a system of generators for S . Thus, S = h i = 5 N .At this point, a question arise in a natural way: when is S ( a, b, α, β ) ∪ { } a numerical semigroup? In order to give an answer, we will study three cases. Case 1. a i > b i for all i ∈ { , . . . , r } and ( α, β ) ∈ N or a i ≥ b i for all i ∈ { , . . . , r } and ( α, β ) ∈ N \ (0 , a x + · · · + a p x p + α > b x + · · · + b p x p − β for all ( x , . . . x p ) ∈ N p . Thus, S ( a, b, α, β ) = ∅ and S ( a, b, α, β ) ∪ { } = { } (that is, the trivial submonoid). Case 2. a i ≥ b i for all i ∈ { , . . . , r } , there exists i ∗ ∈ { , . . . , r } such that a i ∗ = b i ∗ , and ( α, β ) = (0 , E = (cid:8) i ∈ { , . . . , r } | a i = b i (cid:9) and I = (cid:8) i ∈ { , . . . , r } | a i > b i (cid:9) , then itis clear that E ∪ I = { , . . . , r } and E ∩ I = ∅ . On the other hand, a x + · · · + a p x p ≤ b x + · · · + b p x p ⇔ ≤ X i ∈ I ( b i − a i ) x i . Since b i − a i < i ∈ I and ( x , . . . , x p ) ∈ N p , we have that the lastinequality is true if and only if x i = 0 for all i ∈ I . Therefore, S ( a, b, ,
0) = ( n ∈ N (cid:12)(cid:12)(cid:12) X i ∈ E a i x i ≤ n ≤ X i ∈ E b i x i for some ( x i ) i ∈ E ∈ N r ) , where r is the cardinality of E . Now, since P i ∈ E a i x i = P i ∈ E b i x i for all( x i ) i ∈ E ∈ N r , we conclude that S ( a, b, ,
0) is the submonoid generated by theset A = { a i | i ∈ E } . In addition, S ( a, b, ,
0) will be a numerical semigruoup ifand only if gcd( A ) = 1 (see [6, Lemma 2.1]). Case 3.
There exists j ∈ { , . . . , r } such that a j < b j .It is clear that ( b j − a j ) x j − β − α > x j . Therefore,the numerical semigroup generated by { a j x j + α, a j x j + α + 1 } is a subset of S ( a, b, α, β ) ∪ { } which, in consequence, is also a numerical semigroup.8 Conclusion
Starting from a real world situation, the aim of this work has been to computethe set S ( a, b, α, β ) = ( n ∈ N a x + · · · + a p x p + α ≤ n ≤ b x + · · · + b p x p − β for some ( x , . . . x p ) ∈ N p ) , where a = ( a , . . . , a p ) , b = ( b , . . . , b p ) ∈ N p and α, β ∈ N .We have achieved our purpose in two steps. Firstly, we have studied thecase ( α, β ) = (0 , α, β ) = (0 , S ( a, b, α, β ) ∪ { } is a submonoidof ( N , +) and, in some cases, a numerical semigroup. References [1] F. Ajili, E. Contejean, Avoiding slack variables in the solving of linear dio-phantine equations and inequations,
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