ON PRESENTATIONS OF COMMUTATIVE MONOIDS

Transcription

1 ON PRESENTATIONS OF COMMUTATIVE MONOIDS J C ROSALES, P A GARCÍA-SÁNCHEZ and J M URBANO-BLANCO Departamento de Algebra, Universidad de Granada Granada, España s: {jrosales, pedro, jurbano}@goliatugres Communicated by D Klarner Received April AMS Mathematics Subject Classification: 20M05, 20M14, 20M30 In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids As a consequence, we give a method to obtain a presentation for any commutative monoid 0 Introduction In this paper, all the monoids considered are commutative If S is a monoid generated by {m 1,,m n },thensis isomorphic to a quotient monoid of N n by the kernel congruence σ of the map ϕ : N n S, ϕ(k 1,,k n )= n i=1 k im i Under this setting, a finite presentation for S is a finite subset ρ of N n N n such that the congruence generated by ρ is equal to σ Rédei proves in [5] that every congruence on N n is finitely generated, and so that every finitely generated monoid is finitely presented Of all the subsets which generate σ, we are interested in those which have a minimal cardinality and therefore, we will call them presentations of minimal cardinality A congruence σ on N n is said to be reduced if the quotient monoid N n /σ is reduced (ie the only unit is the zero element) If M is a subgroup of Z n, then we define the set M = {(a, b) N n N n ; a b M}, which is in fact a congruence on N n Conversely, for a given congruence σ N n N n, we define the following subgroup of Z n : M σ = {z Z n : z = a b for some pair (a, b) σ} It is well-known that if σ is a congruence on N n and N n / Mσ is reduced, then N n /σ is reduced According to this, we say that a congruence σ is strongly reduced if N n / Mσ is reduced 539 International Journal of Algebra and Computation Vol 9, No 5 (1999) c World Scientific Publishing Company

2 540 J C ROSALES, P A GARCÍA-SÁNCHEZ and J M URBANO-BLANCO The purpose of this paper is to characterize the minimal presentations of strongly reduced monoids It will permit us to give a method to compute a presentation for any finitely generated monoid The main part of this paper consists of the extension of the results given in [6], where an algorithm to compute presentations for numerical monoids is given As an interesting consequence, it is derived that the concepts of minimal presentation and minimal cardinality presentation are equivalent for the class of strongly reduced monoids This is presented in the first section As a consequence of the results obtained in the first section, we will be able to compute minimal presentations for finitely generated monoids which are reduced, cancellative (a + c = b + c implies a = b for all a, b, c in the monoid) and torsion-free (ka = kb implies a = b for every a, b in the monoid and every positive integer k) Next, we will extend this procedure to cancellative and reduced monoids, and finally we will deal with the general case The idea to solve the general case, consists in associating a strongly reduced monoid to any given monoid in such a way that a presentation for the initial monoid be easily reconstructed from a presentation for the strongly reduced one Note that these results have a direct application in Ring Theory In fact, if R is a ring with unit and R[S] denotes the semigroup-ring of S on R, thenr[s] = R[X 1,,X n ]/I S,whereI S is the ideal of R[X 1,,X n ] which is generated by the set {X a X b : a b} More precisely, if {(s i,t i ):i=1,,k} is a presentation for S, then{x si X ti :i=1,,k} is a generating system for I S Many algebraic properties of R[S] can be deduced from the ideal I S and specially from its minimal generating systems (see [2, 3]) 1 Characterization of Minimal Presentations of Strongly Reduced Monoids Our main goal of this section is to prove Theorem 7, which gives a necessary and sufficient condition for a subset of a congruence σ to generate the whole congruence We will use in our proofs, the following well-known construction for the congruence generated by a subset of it (see [1]) For a subset ρ of N n N n, there always exists the smallest congruence ρ containing ρ, which can be described in three steps: (1) Put ρ (0) = ρ ρ 1 τ, whereρ 1 ={(v, w) such that (w, v) ρ} and τ = {(v, v) such that v N n } (2) Put ρ (1) = {(v + u, w + u) such that (v, w) ρ (0) and u N n } (3) (Transitive closure) Define (v, w) N n N n to be an element of ρ if there exist v 0,v 1,,v k N n,v 0 = v, v k = w with (v i,v i+1 ) ρ 1 for all i =0,1,,k 1 We will also refer to ρ as the congruence generated by ρ Definition 1 For any x, y N n, we will write xr σ y when either x = y =0 or there exist z 1,,z k N n such that x = z 1, y = z k and for every i {1,,k 1}, there exist a pair (a i,b i ) σ and an element c i 0suchthat(z i,z i+1 )= (a i +c i,b i +c i )

3 ON PRESENTATIONS OF COMMUTATIVE MONOIDS 541 Clearly, R σ is an equivalence relation on N n and R σ σ Thus, for every [a] σ N n /σ, we can contruct the quotient set [a] σ /R σ The elements of [a] σ /R σ are called the R σ -classes of [a] σ Definition 2 Let X be a set, P = {B 1,,B t } a partition on X and R a binary relation on X We define the graph G R =(V,E) associated to R with respect to the partition P as follows: V = P and the edge B i B j is in E, with i j, ifthere exist x B i and y B j such that (x, y) R R 1 Proposition 3 Suppose that [a] σ N n /σ and let X 1,,X r be the list of all the R σ -classes contained in [a] σ If β is a binary relation on N n which generates σ, and β a = β ([a] σ [a] σ ), then the graph G βa associated to β a with respect to the partition P = {X 1,,X r } of [a] σ is a connected graph Proof Suppose that t, s {1,,r}, with s t We have to prove that there exists a path in G βa from X t to X s Wepickupk X t and h X s Then, clearly, kσh Since we are supposing that β = σ, itimpliesthatthere exist v 1,,v l N n such that k = v 1, h = v l and (v i,v i+1 ) β (1) for every i =1,,l 1 But (v i,v i+1 ) β (1) means that (v i,v i+1 )=(a i,b i )+(c i,c i )for some pair (a i,b i ) β β 1 and c i N n (since s t, we may suppress the reflexive steps in β (1) ) At this point, two cases are possible: If c i 0,thenv i and v i+1 are in the same R σ -class Otherwise (v i,v i+1 )=(a i,b i ) β a βa 1 We conclude that those elements v i for which c i = 0 determine a path in G βa from X t to X s If we write e i to represent the n-tuple with 1 at its ith position and 0 s elsewhere, we will always assume that e i M σ Now, we concentrate on the proof of the converse to Proposition 3 for the case of σ been a strongly reduced congruence We need the following lemmas: Lemma 4 Let σ be a congruence on N n The following statements are equivalent: (1) If (x + c, x) σ, then c =0 (2) σ is a strongly reduced congruence Proof Suppose that N n / Mσ is not reduced Then, there exists a nonzero element c N n such that c Mσ 0 This implies that there exists a N n such that (a + c)σ(a + 0) Conversely, suppose that N n / Mσ is reduced and (a + c)σa Then c M σ and so c Mσ 0 Since we are assuming that no generator is in the same class of 0, the only possibility is c =0 In the rest of this section, σ will represent a strongly reduced congruence on N n Lemma 5 The binary relation defined on N n /σ by [a] σ [b] σ [c] σ such that [a] σ +[c] σ =[b] σ is an order relation

4 542 J C ROSALES, P A GARCÍA-SÁNCHEZ and J M URBANO-BLANCO The proof is very easy and we only note that the strongly reduced condition is used to prove the antisymmetry property Theorem 6 Suppose that β is a binary relation on N n which is contained in σ and such that the graph G βa is connected, for every a N n Then σ = β Proof The inclusion β σis trivial For the other one, note that it is equivalent to show that [a] σ [a] σ β, for every [a] σ N n /σ Suppose for a contradiction, that the element [a] σ is minimal (with respect to the order considered in Lemma 5) amongst all those elements for which the condition [a] σ [a] σ β does not hold This implies that there exists (x, y) [a] σ [a] σ such that (x, y) β Two cases are possible: Case 1 If (x, y) R σ, then there exist x 1,,x q N n such that x 1 = x, x q = y and for every i =1,,q 1, there exist (a i,b i ) σ and c i 0suchthat (x i,x i+1 )=(a i,b i )+(c i,c i ) In particular, this implies that [a i ] σ +[c i ] σ =[a] σ and so that [a i ] σ < [a] σ Thus, by the minimality of [a] σ, we get that (a i,b i ) β It means that (x i,x i+1 )=(a i,b i )+(c i,c i ) β So, we obtain (x, y) β which is a contradiction Case 2 If (x, y) R σ,then#[a] σ /R σ 2 Since we are assuming that G βa is a connected graph, there exists a path in G βa from the R σ -class containing x, say X, tother σ -class containing y, sayy Suppose that such path is given by X = X 1,X 2,,X l =YSinceX i X i+1 is an edge of G βa from the definitions, we get that there exists (a i,b i ) β a βa 1 with a i X i and b i X i+1 But now, by Case 1, we have that (x, a 1 ), (b l 1,y),(b i,a i+1 ) β So (x, y) β,whichisa contradiction The following theorem sums up the results obtained above: Theorem 7 If β is a binary relation which is contained in σ, then σ = β if and only if the graph G βa is connected for every a N n The following two corollaries are a direct consequence of the previous theorems Corollary 8 If β is a binary relation contained in σ, then β is a minimal cardinality presentation for σ if and only if the graph G βa is a tree, for every a N n Corollary 9 The concepts of minimal presentation (with respect to the inclusion order of sets) and of minimal cardinality presentation coincide in the class of strongly reduced semigroups 2 Computing Presentations for Affine Semigroups In this section, we study a special type of strongly reduced monoids, the so-called affine semigroups An affine semigroup is a finitely generated monoid isomorphic to a subsemigroup of N r for some positive integer r A finitely generated semigroup is an affine semigroup if and only if it is cancellative, reduced and torsion free (see [8] for more details)

5 ON PRESENTATIONS OF COMMUTATIVE MONOIDS 543 Let S be a subgroup of N r with minimal set of generators A = {n 1,,n r, n r+1,,n r+m } We can always assume that the rank of the abelian group generated by S is r because of the following observation: assume that the mentioned rank is d<r Let (a i,1,,a i,r )=n i and let σ be the kernel of the semigroup homomorphism ϕ : N r+m S which is defined by ϕ(k 1,,k r+m )=k 1 n 1 + +k r+m n r+m Then z =(z 1,,z r+m ) M σ if and only if it verifies the system of linear equations: a 1,1 x a r+m,1 x r+m =0 a 1,r x a r+m,r x r+m =0 Since N n /σ is a cancellative monoid, we have that σ = Mσ Moreover, since the rank of the columns of the previous system of equations is d, it follows that M σ can be defined by a subset of d equations Suppose that these equations are: a 1,i1 x a r+m,i1 x r+m =0 a 1,id x a r+m,id x r+m =0 Then S = N n / Mσ = (a1,i1,,a 1,id ),,(a r+m,i1,,a r+m,id ) N d Note also that since in this case σ = Mσ, the concepts of reduced and strongly reduced coincide Hence, we are looking for a set β such that the graph G βa is connected for all a N r+m In order to get such a set, we define the following graphs, n, for every 0 n S, whose vertices are: and whose edges are: V( n )={n i :n n i S}, E( n )={n i n j :n (n i +n j ) S} Next result gives us an idea on how the set of relators must be Proposition 10 Let 0 n S and a N r+m such that ϕ(a) =n There is a bijective map between the set of connected components of n and the set [a] σ /R σ Proof Let us denote by C n the set of connected components of n Let us define a bijective map f : C n [a] σ /R σ GivenC C n,taken i a vertex in C (note that V( n ) because n 0) Hence, n n i S and therefore, there must be an element a i in [a] σ verifying that its ith coordinate is nonzero Define f(c) =[a i ] Rσ Let us show that f is well-defined, that is to say, it neither depends on the choice of n i nor on the choice of a i Take n i n j C There must be an element in [a] σ such that its jth coordinate is nontrivial, say a j Since n i and n j are both in C, theremust be a path connecting them Let n i = n i0,,n it = n j be this path Since

6 544 J C ROSALES, P A GARCÍA-SÁNCHEZ and J M URBANO-BLANCO n (n ik + n ik+1 ) S, there must be elements b k [a] σ such that b k has its coordinates i k and i k+1 nonzero, for k {0,,t 1} Notice that (b k e ik+1,b k+1 e ik+1 ) σ for all k, (b 0 e i,a i e i )and(b t 1 e j,a j e j ) σ This leads to (a i,a j ) R σ Thus, they belong to the same R σ -class Now, take a i a i fulfilling that its ith coordinate is not equal to zero Then, (a i e i,a i e i) σ and therefore (a i,a i ) R σ Finally, let us show that the map f is bijective Injective: If f(c) =f(c ), and we choose a i with the ith coordinate not equal to zero by the definition of f(c), then n i V(C) andn i V(C ) (because we have just shown that the definition of f(c) does not depend on the election of n i ), and this leads to C = C Surjective: Take [b] Rσ [a] σ /R σ Since a 0, then by the definition of R σ, b 0 Thus, we can choose a coordinate of b not being zero Let us assume that it is the ith coordinate Then, n i V( n ) If C is the connected component of n containing n i, then clearly f(c) =[b] Rσ Using the proof of this lemma and Theorem 7, it is straightforward to show the following result Corollary 11 For any n S, define ρ n as follows: (1) If n is connected, then ρ n = (2) If n is not connected and 1 n,, t n are the connected components of n, then choose a vertex n ji V( i n ) and an element αi n =(ai 1,,ai r+m ) N r+m such that ϕ(α i n )=nand ai j i 0;define (3) Take ρ = n S ρ n ρ n = {(α 2 n,α 1 n),,(α t n,α 1 n)} Then G ρa is connected for every a N r+m Furthermore, G ρa is a tree for all a N r+m and therefore ρ is a minimal presentation for S The problem is, of course, that the set {n S : n is not connected} is not known (it is known to be finite, because for any finitely generated affine semigroup, the cardinal of a minimal system of generators for σ is finite) In the next section, we give a bound for the elements that can be in this set For subsemigroups of N, the first author found a bound in [6] and for Cohen-Macaulay simplicial affine semigroups, the first and the second authors found in [9] an optimal bound for the number of elements in this set 21 A bound for the cardinal of a minimal system of generators for σ The number of elements of a minimal system of generators is finite and a bound for its cardinal, can be constructed as Sturmfels shows in [11] In this section, we sketch this construction We also give a bound for the elements in S such that its associated graph is not connected, which is essential if we want to give an effective algorithm to compute a minimal presentation of an affine semigroup

7 ON PRESENTATIONS OF COMMUTATIVE MONOIDS 545 Throughout this section, we denote by <, the usual partial ordering on N r, ie (a 1,,a r )<(b 1,,b r ) if and only if a i b i for all i {1,,r}, and there exists j {1,,r} such that a j <b j and by the lexicographical ordering on N r, ie (a 1,,a r ) (b 1,,b r ) if and only if there exists i {1,,r} such that a j = b j for all j {1,,i 1} and a i <b i Givenanelementu=(u 1,,u r+m ) Z r+m,theset{i {1,,r+m}:u i 0} is denoted by supp(u) The element u can be written uniquely as u = u + u, where u +,u N r+m and supp(u + ) supp(u )= The element u is a circuit, if (u +,u ) σ, supp(u) is minimal with respect to inclusion and the coordinates of u are relatively prime The concept of circuit is crucial to develop the rest of this section In the following lemma, we recall some properties of circuits (see [11] for more details) Lemma 12 If u =(u 1,,u r+m ) is a circuit, then (1) #supp(u) r +1, (2) u i det(n 1,,n r ) Given u, v Z r+m, the element u is conformal to v if supp(u + ) supp(v + )and supp(u ) supp(v ) Next lemma shows the importance of the concept of circuit: Lemma 13 For every (a, b) σ, a b can be written as a nonnegative rational linear combination of m circuits, where each of them is conformal to a b The elements of a minimal system of generators are irreducible in the following sense: Lemma 14 Let ρ be a set generating σ and (a, b) ρ If there exists (a 1,b 1 ) and (a 2,b 2 ) in σ such that (a, b) =(a 1,b 1 )+(a 2,b 2 ), then ρ is not a minimal set of generators for σ Theorem 15 Let ρ be a minimal system of generators for σ For all (a, b) ρ, where (u 1,,u r+m ) 1 = r+m i=1 u i a 1, b 1 (r +1)mdet(n 1,,n r ), Proof This result is a reformulation of Theorem 47 of [11] Note that every element in ρ is primitive in the sense of Sturmfels by Lemma 14 This theorem gives a bound for the elements appearing in a minimal system of generators of σ If we want to compute this minimal system of generators in an effective way, we need to know for which elements n in S, the graph n is not connected It is possible to derive a bound for these elements of S as the next theorem shows

8 546 J C ROSALES, P A GARCÍA-SÁNCHEZ and J M URBANO-BLANCO Theorem 16 Let n S If n is not connected, then n m det(n 1,,n r )(n 1 + +n r+m ) Proof If n is not connected, then ρ n, constructed as above, is not empty and therefore, there exists (a, b) ρ n ρ, with ϕ(a) = ϕ(b) = n Thus, by Lemma 13, we know that there are c 1,,c m circuits, which are conformal to a b, and λ 1,,λ m Q +, such that: which means that: a b = (a, b) = m λ i c i, i=1 m λ i (c + i,c i ) i=1 (Note that (a b) + = a since (a, b) ρ, and the elements c i are conformal to a b) Hence, a = m i=1 λ ic + i and therefore, n = ϕ(a) = m i=1 λ iϕ(c + i ) Furthermore, by Lemma 14, λ i < 1 for all i {1,,m}Thus,n m i=1 ϕ(c+ i ) Besides, each coordinate of c + i is less or equal than det(n 1,,n r ) and this implies that ϕ(c + i ) det(n 1,,n r )(n 1 + +n r+m ) Putting all together, we are done This theorem enables us to compute all the elements in S having whose graph are nonconnected, because it shows that these elements are always contained in a certain box, which can be easily scanned 22 An algorithm to compute ρ Let (b 1,,b r )=mdet(n 1,,n r )(n 1 + +n r+m ) Observe that: (1) All the elements n S such that n is not connected are in the box Γ=([0,b 1 ] [0,b r ]) N r, and they can be scanned using the lexicographical ordering (2) Once we have solved the membership problem to S for all the elements in this box, which are smaller (wrt lexicographical order) than a certain n, the question of whether n belongs or not to S, can be answered in the following way: if n S then, there must exist i {1,,r+m} such that n n i S Notethatn n i nand therefore, we already know whether n n i belongs to S or not (3) Additionally, using the same idea exposed in the previous remark, we get the elements of V( n ) because we can check if n n i S, for any i {1,,r+m} (4) Assuming that we know the vertices of the graph associated to all the elements of S which are smaller than n S (wrt the lexicographical order), it is easy to compute the graph n and its connected components by taking into account the following facts:

9 ON PRESENTATIONS OF COMMUTATIVE MONOIDS 547 (a) If n n i S, thenv( n ni ) V( n ) Furthermore, V( n ni )is contained in a connected component of n (the connected component where lies n i ) (b) V( n )= n n (V( i S n n i ) {n i }) (c) If V( n ni ) V( n nj ), thenv( n ni )andv( n nj ) are contained in the same connected component (d) If V( n )={n i1,,n ik }, we define Bj 0 =V( n n ij ) {n ij },and Pn={B 0 1,,B 0 k 0} We construct P n l in the following way: if Pn l 1 = {B1 l 1,,B l 1 k l 1 } and there exist i, j such that B l 1 i,b l 1 j are not disjoint, then we set Pn l = {Bl 1 h : h i, j} {B l 1 i B l 1 j } This process ends when Pn l is a partition of V( n) It is easy to see that, in this case, the elements in Pn l are the sets of vertices of the connected components of n (5) Once we know the connected components of the graph associated to each element n S Γ, we must construct ρ n For this purpose, we have to find the tuples α i n described in the definition of ρ n Assume that 1 n,, tn n are the connected components of n Taken i1 V( i n), then n n i1 S Two cases are possible: n n i1 = 0 and therefore, α i n = e i1 ;orn n i1 0 In the latter case, there exists n i2 such that (n n i1 ) n i2 S (hence n i2 V( i n)) This process must stop and when it stops, we have that n k j=1 n i j =0 Thus,α i n = k j=1 e i j 221 Description of the algorithm It is clear that following the steps given above, one can construct ρ Let us define the appropriate procedures in order to give the algorithm to compute ρ First of all, we need to compute the sets S Γ = S Γ,T = {n : n S Γand n not connected} and P = {P n : n T }, wherep n is the partition whose elements are the sets of vertices of the connected components of n We need the following procedures: (1) succ, which, given an element n Γ, returns the element l =min {k Γ: n k}and if this number does not exist, it returns 0 (2) vertices computes the set V( n )foragivenn Γ It returns if n S (3) partition returns a partition of the set of vertices of n for a given n S Γ, such that the elements in this partition are the sets of vertices of the connected components of n function succ Input: n =(k 1,,k r ) Γ Output: 0,ifn=(b 1,,b r ); min {k Γ:n k}, otherwise i =0; while k r i = b r i do i = i +1; if i = r then

10 548 J C ROSALES, P A GARCÍA-SÁNCHEZ and J M URBANO-BLANCO return (0,,0); else return (k 1,,k r i +1,0,,0); function vertices Input: n Γ, the set SΓ n = {k S Γ:k n} Output:, ifn S; V( n ), otherwise V = ; for i =1to r + m do if n n i SΓ n then V = V {n i }; return V ; function partition Input: n Γ, the set V( n ) Output: a partition which elements are the set of vertices of the connected components of n P = ; for each n i V( n ) do P = P {vertices(n n i ) {n i }}; partition = false; while partition = false do if there exist A, B P such that A B then P =(P\{A, B}) {A B}; else partition = true; return P ; Now, we are ready to describe the function compute ρ which computes the sets T and P as described above function compute ρ Input: ThesetAverifying the conditions imposed to S Output: ρ S Γ =0;ρ= ; T ={0};P= ; n=succ(0); while n 0do V = vertices(n, S Γ ); if V then P n = partition(n, V ); S Γ = S Γ {n}; if #P n > 1 then P = P {P n }; T =T {n}; n=succ(n);

11 ON PRESENTATIONS OF COMMUTATIVE MONOIDS 549 for each n T do for i =1to #P n do α i n =0; while n ϕ(α i n) do for each n j in the ith element of P n do if n ϕ(α i n) n j S Γ then α i n = α i n + e j ; ρ = ; for each n T do for i =2to #P n do ρ = ρ {(α i n,α1 n )}; return ρ; 222 Complexity of the algorithm The number of iterations of the body of the while statement in the algorithm is r i=1 b i In the body of the while statement, the procedures succ, vertices and partition are executed The complexity of the first one is linear, the complexity of the second is bounded by r i=1 b i, and the complexity of the third is polynomial in the number of generators of S Thus, the leading complexity in the main body of the while statement is the one given by the execution of vertices The other loops appearing in the algorithm have less or equal complexity than the while part of the algorithm The amount r i=1 b i is a polinomial of the form m r p,wherepis a polynomialin the coordinates of the generatorsof S with degree O(r)+O(m) Thus, the complexity of the algorithm is polinomial in the coordinates of the generators of the semigroup and the codimension of S (ie m), and it is exponential in the dimension (r) and codimension of S Example 17 Let S = (2, 0), (0, 1), (1, 2), (3, 1) N 2 We have that the greatest determinant of two generators of S is det((3, 1), (1, 2)) = 5 Hence, the elements n in S such that n may be nonconnected verify that n 2 5 ((2, 0) + (0, 1) + (1, 2) + (3, 1)) = (60, 40) Thus, Γ = {(n, m) N 2 : n 60,m 40} Ifwe scan this box looking for elements in S, wegetthats Γ=S Γ =Γ\({(2n +1,0) : 0 n 29} {(1, 1)}) The nonconnected graphs are: Graph Connected components Relators (3,2) {(2, 0), (1, 2)}, {(0, 1), (3, 1)} e 1 + e 3 = e 2 + e 4 (6,2) {(2, 0), (0, 1)}, {(3, 1)} 3e 1 +2e 2 =2e 4 (4,3) {(2, 0), (0, 1)}, {(1, 2), (3, 1)} 2e 1 +3e 2 =e 3 +e 4 (2,4) {(2, 0), (0, 1)}, {(1, 2)} e 1 +4e 2 =2e 3 And therefore ρ = {((1, 0, 1, 0), (0, 1, 0, 1)), ((3, 2, 0, 0), (0, 0, 0, 2)), ((2, 3, 0, 0), (0, 0, 1, 1)), ((1, 4, 0, 0), (0, 0, 2, 0))}

12 550 J C ROSALES, P A GARCÍA-SÁNCHEZ and J M URBANO-BLANCO 3 Computing Presentations for Cancellative and Reduced Monoids If S is a cancellative and reduced monoid, then S is isomorphic to a subsemigroup of Z d1 Z dr Z k for some positive integers d 1,,d r,k(see [8] for more details) Note that, under this settings, σ = Mσ and therefore, the concepts of strongly reduced semigroup and reduced semigroup coincide Next theorem, appearing in [7], will allow us to use the results and procedures exposed in the previous section to achieve an algorithm to compute minimal presentations for finitely generated, cancellative and reduced monoids Theorem 18 Let M be the subgroup of Z n given by the system of equations a 1,1 x 1 + a 2,1 x a n,1 x n 0(mod d 1 ) a 1,2 x 1 + a 2,2 x a n,2 x n 0(mod d 2 ) a 1,r x 1 + a 2,r x a n,r x n 0(mod d r ) a 1,r+1 x 1 + a 2,r+1 x a n,r+1 x n = 0 a 1,r+k x 1 + a 2,r+k x 2 + +a n,r+k x n = 0 and let M be the subgroup of Z n+r given by the system of equations a 1,1 x a n,1 x n + d 1 x n x n x n+r = 0 a 1,2 x a n,2 x n + 0 x n+1 + d 2 x n x n+r = 0 a 1,r x a n,r x n + 0 x n x n d r x n+r = 0 a 1,r+1 x a n,r+1 x n + 0 x n x n x n+r = 0 a 1,r+k x a n,r+k x n + 0 x n x n x n+r = 0 Suppose that ρ = {(α 1, β 1 ),,(α k,β 1 )} is a generating system for the congruence M, where α i =(α i,1,,α i,n,α i,n+1,,α i,n+r ) and β i =(β i,1,,β i,n, β i,n+1,,β i,n+r ) If we define α i =(α i,1,,α i,n ) and β i =(β i,1,,β i,n ), then ρ = {(α 1,β 1 ),,(α k,β k )} is a generating system for M If S is a reduced cancellative submonoid of Z d1 Z dr Z k minimally generated by {(a 1,1,,a 1,s ),,(a n,1,,a n,s )} and σ represents its associated kernel congruence then, M σ is the subgroup of Z n which can be defined by the following set of linear equations: a 1,1 x a n,1 x n 0(mod d 1 ) a 1,r x a n,r x n 0(mod d r ) a 1,r+1 x a n,r+1 x n = 0 a 1,s x a n,s x n = 0

13 ON PRESENTATIONS OF COMMUTATIVE MONOIDS 551 Besides, it can be shown that S is reduced if and only if the only element having all its coordinates greater or equal than zero is the zero element, and this implies that there exists a strongly positive element (a 1,,a n ) N n (that is to say a i > 0 for every i) such that a 1 x 1 + +a n x n =0forall(x 1,,x n ) M σ (see [8], where it is also given an algorithm to find such an element) Thus, we can add this new equation to our system of equations defining M σ By using this new equation, we can transform the system of equations into a new system such that all coefficients are nonnegative Hence, we may assume that {(a 1,1,,a 1,s ),,(a n,1,,a n,s )} N s Let S be the submonoid of N s generated by the set {(a 1,1,,a 1,s ),,(a n,1,,a n,s ), (d 1, 0,,0), (0,d 2,0,,0),,(0,,0,d r,0,,0)}, and let σ be its associated kernel congruence Using the algorithm given in the previous section, we can compute a minimal presentation for S, and using the previous theorem we can get a generating system for Mσ = σ We close this section with an example illustrating the procedure outlined above Example 19 Suppose that S is the submonoid of Z 2 Z 2 N which is generated by {(1, 1, 1), (0, 1, 1), (1, 0, 1)} Then, S = (1, 1, 1), (0, 1, 1), (1, 0, 1), (2, 0, 0), (0, 2, 0) Using the algorithm for affine semigroups one gets and therefore, ρ = {((0, 2, 0, 1, 0), (2, 0, 0, 0, 0)), ((0, 0, 2, 0, 1), (2, 0, 0, 0, 0))}, which is a minimal presentation for S ρ = {((0, 2, 0), (2, 0, 0)), ((0, 0, 2), (2, 0, 0))}, 4 A Procedure to Compute a Presentation for any Monoid Now we show the existing connection between arbitrary congruences and strongly reduced congruences More precisely, we will associate to a given congruence σ a strongly reduced one σ, in such a way that for a presentation for σ, we will derive a presentation for σ Definition 20 Let σ and σ be two congruences on N n and N n+1 respectively Then, we say that σ is a projection of σ when for every (x 1,,x n ),(y 1,,y n ) N n, it holds that (x 1,,x n )σ(y 1,,y n ) if and only if there exists x n+1,y n+1 N such that: (x 1,,x n,x n+1 )σ(y 1,,y n,y n+1 ) Given an element ((x 1,,x n,x n+1 ), (y 1,,y n,y n+1 )) of N n+1 N n+1, we will refertoitsprojection as the element ((x 1,,x n ),(y 1,,y n )) of N n N n

14 552 J C ROSALES, P A GARCÍA-SÁNCHEZ and J M URBANO-BLANCO The following Theorem is a direct consequence of the three steps process for generating a congruence from a given subset Theorem 21 Let σ and σ be two congruences on N n and N n+1 respectively The following assertions are equivalents: (1) σ is a projection of σ (2) The projection of a generating system for σ is a generating system for σ The following proposition proves that every congruence σ is a projection of a strongly reduced congruence σ, whose classes are all finite The proof is very easy and it relies only on definitions It establishes the final link necessary to derive an effective implementation for the general case Proposition 22 Let σ be a congruence on N n and let σ be the binary relation defined by (x 1,,x n,x n+1 )σ(y 1,,y n,y n+1 ) if and only if (x 1,,x n ) σ(y 1,,y n ) and Then: (1) σ is a congruence on N n+1 (2) σ is a projection of σ (3) σ is strongly reduced (4) Every σ-class is finite x 1 + +x n +x n+1 = y 1 + +y n +y n+1 Before ending, we would like to point out the following facts The first one is the importance of strongly reduced monoids, which classify, up to projections, the whole class of monoids Secondly, in the case of cancellative monoids (that is to say, submonoids of Z n1 Z nr Z k ), there exist explicit methods to compute a presentation for a congruence, which can be considered as a particular case of that given in this paper (see [7]) Finally, as we have mentioned before, we must impose to a finitely generated semigroup the condition of being strongly reduced if we want the concepts of minimal presentation with respect to inclusion and with respect to cardinality to coincide Next example, which can be found in [4], shows that if unfortunately the given semigroup is not strongly reduced, then these two concepts may be different Example 23 Let S = (0, 0, 1), (2, 1, 1), (1, 0, 3), ( 2, 1, 3) Z Z 2 Z 4 Then, and {((2, 0, 0, 0), (0, 0, 4, 2)), ((0, 1, 0, 0), (0, 0, 8, 3)), ((0, 0, 0, 0), (0, 0, 8, 4))} are minimal presentations for S {((0, 0, 0, 0), (0, 1, 0, 1)), ((0, 0, 0, 0), (4, 0, 0, 0)), ((8, 4, 0, 0), (0, 0, 8, 0)), ((10, 6, 0, 0), (0, 0, 12, 0))}

15 ON PRESENTATIONS OF COMMUTATIVE MONOIDS 553 References 1 A H Clifford and G B Preston, The Algebraic Theory of Semigroups, American Mathematics Society Math Surveys, No 7, R Gilmer, Commutative Semigroup Rings, The University of Chicago Press, J Herzog, Generators and Relations of Abelian Semigroups and Semigroup Rings, Manuscripta Mathematica 3 (1970), P Pisón Casares and A Vignerón Tenorio, Ideales de semigrupos con torsión: Cálculos mediante Maple V, Proc EACA 96 September, 1996, Sevilla (Spain) 5 L Rédei, The Theory of Finitely Generated Commutative Semigroups, Pergamon, J C Rosales, An algorithmic method to compute a minimal relation for any numerical semigroup, Int J Algebra Comput 6(4) (1996), J C Rosales and J M Urbano-Blanco, A deterministic algorithm to decide if a finitely presented abelian monoid is cancellative, Commun Algebra 24(13) (1996), J C Rosales, On finitely generated submonoids of N k, Semigroup Forum 50 (1995), J C Rosales and P A García-Sánchez, On relations of Cohen Macaulay simplicial affine semigroups, Proc Edinburg Mathematical Society 41 (1998), B Sturmfels, Gröbner bases of toric varieties, Tôhoku Math J 43 (1991), B Sturmfels, Gröbner bases and convex polytopes, University Lecture Series 8 American Mathematics Society, Providence, 1996