Commutative Ideal Extensions of Abelian Groups - Springer Link

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Ideal extensions of semigroups were introduced by Clifford in [1] and since then they have been widely studied (see for instance [2]). Our aim here is to.
Semigroup Forum Vol. 62 (2001) 311–316

c 2001 Springer-Verlag New York Inc. °

DOI: 10.1007/s002330010057

RESEARCH ARTICLE

Commutative Ideal Extensions of Abelian Groups J. C. Rosales, P. A. Garc´ıa-S´ anchez, and J. I. Garc´ıa-Garc´ıa Communicated by John Fountain

Introduction Ideal extensions of semigroups were introduced by Clifford in [1] and since then they have been widely studied (see for instance [2]). Our aim here is to characterize commutative ideal extensions of Abelian groups. We show that they are those commutative semigroups with an idempotent Archimedean element, or equivalently, those commutative semigroups E such that E/R is an Abelian group, where R is the least congruence that makes E/R cancellative. These characterizations give rise to an algorithm for deciding from a presentation of a finitely generated commutative monoid whether it is an ideal extension of an Abelian group. Finally we present a procedure that enables us to compute the set of idempotents of a finitely generated commutative monoid. All semigroups appearing in this paper are commutative and for this reason in the sequel we sometimes omit the adjective commutative whenever we refer to a commutative semigroup (the same holds for monoids). The authors would like to thank the referee for his/her comments and suggestions. 1. Commutative ideal extensions of Abelian groups Let S be a semigroup. An ideal extension of S is a semigroup E fulfilling that S is one of its ideals. In this section we characterize semigroups that are ideal extensions of Abelian groups. An element x in a semigroup S is an idempotent if 2x = x. The element x ∈ S is Archimedean if for every y ∈ S there exist k ∈ N \ {0} and z ∈ S such that kx = y + z (where N denotes the set of nonnegative integers). Theorem 1.1. Let E be a commutative semigroup. Then E is an ideal extension of an Abelian group if and only if E has an element that is Archimedean and idempotent.

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Proof. Necessity. Assume that E is an ideal extension of the Abelian group G . Let u be the identity element of G . Clearly u is an idempotent of E . Take x ∈ E . Since G is an ideal, we have that u + x ∈ G . It follows that there exists y ∈ G such that u + x + y = u , whence 1u = x + (u + y) . Thus u is an Archimedean element of E . Sufficiency. If u is an idempotent Archimedean element of E , then u belongs to every ideal of E and u + E is the minimal ideal of E , which turns out to be a group. Theorem 1.2. Let E be a commutative semigroup and R be the least congruence on E such that E/R is cancellative (that is, x R y if x + z = y + z for some z ∈ E ). Then E is an ideal extension of an Abelian group if and only if E/R is an Abelian group and E has at least one Archimedean element. Proof. Necessity. Assume that E is an ideal extension of the Abelian group G . Let u be the identity element of G , which in view of the proof of Theorem 1.1 is Archimedean. In addition, the element [u]R is an idempotent in a cancellative semigroup, so it is the only idempotent and an identity by exercise 2.6.1 from [3]. That E/R is a group follows from the fact that u is Archimedean. Sufficiency. By Theorem 1.1, it suffices to show that E has one idempotent Archimedean element. By hypothesis E has an Archimedean element m. Let [u]R be the identity element of E/R . Then there exists x ∈ E such that [m]R + [x]R = [u]R , which means that m + x + y = u + y for some y ∈ E . In addition [u]R + [u]R = [u]R and thus 2u + z = u + z for some z ∈ E . It follows that m + x + y + z + u = 2u + y + z = u + y + z. Set s = u + y + z , whence m + x + s = s. Since m is Archimedean, so is m + x. Hence there exist k ∈ N \ {0} and v ∈ E such that k(m + x) = s + v . From the equality m + x + s = s, we obtain m + x + s + v = s + v , and substituting s + v by k(m + x) , we deduce that (k + 1)(m + x) = k(m + x) . Using induction on n , it can be shown that (k + n)(x + m) = k(x + m) ; in particular 2(k(x + m)) = k(x + m) , which implies that k(x + m) is an idempotent. The fact that k(x + m) is Archimedean follows from the fact that m is Archimedean. Corollary 1.3. Let E be a finitely generated commutative semigroup. Then E is an ideal extension of an Abelian group if and only if E/R is a group (where R is defined as in Theorem 1.2 ). Proof. If s1 , . . . , sn are the generators of E , then s1 + · · · + sn is an Archimedean element of E . In view of Theorem 1.2, we obtain the desired result.

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2. Idempotents of a finitely generated commutative monoid Let x = (x1 , . . . , xp ) ∈ Zp (where Z denotes the set of integers). The support of x is the set supp(x) = {i ∈ {1, . . . , p} | xi 6= 0}. We say that x is positive if x ∈ Np and supp(x) = {1, . . . , p}. Let σ be a congruence on Np and ρ = {(a1 , b1 ), . . . , (at , bt )} be one of its systems of generators. We give an algorithm for deciding from ρ whether Np /σ has idempotents. As an easy consequence of the procedure presented here we will be able to compute the set of idempotents of Np /σ . The first step to accomplish is to give a method for deciding whether Np /σ has an idempotent Archimedean element, which by Theorem 1.1 is equivalent to deciding whether Np /σ is an ideal extension of an Abelian group. Associated to σ , define Mσ = {a − b ∈ Zp | (a, b) ∈ σ}. The reader can check that Mσ is a subgroup of Zp and that it is generated by the set {a1 − b1 , . . . , at − bt } . Conversely, given a subgroup M of Zp , define ∼M = {(a, b) ∈ Np | a − b ∈ M }. In [4] the following result, showing the relationship between σ and ∼Mσ is presented. Lemma 2.1.

Let σ be a congruence on Np .

1. σ ⊆ ∼Mσ . 2. For every (a, b) ∈ ∼Mσ , there exists c ∈ Np such that (a + c, b + c) ∈ σ . Proposition 2.2. Let σ be a congruence on Np . The monoid Np /σ has an idempotent Archimedean element if and only if Mσ has a positive element. Proof. Necessity. Let [m]σ be an idempotent Archimedean element of Np /σ . Then there exists k ∈ N \ {0} and c ∈ Np such that k[m]σ = [m]σ = [c + e1 + · · · + ep ]σ (ei denotes the element in Np having all its coordinates equal to zero except the ith which is equal to one). Hence [c + e1 + · · · + ep ]σ is an idempotent of Np /σ , which implies that 2(c + e1 + · · · + ep ) σ (c + e1 + · · · + ep ) and thus (c + e1 + · · · + ep ) is a positive element of Mσ .

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Sufficiency. Take x to be an element in Mσ ∩Np fulfilling that supp(x) = {1, . . . , p}. Then (x, 0) ∈ ∼Mσ , which owing to Lemma 2.1 leads to (x+c, c) ∈ σ for some c ∈ Np . Since supp(x) = {1, . . . , p} , there exists k ∈ N and d ∈ Np such that kx = c + d (note that [x]σ is an Archimedean element of Np /σ ). Thus [(k + 1)x]σ = [x + c + d]σ = [c + d]σ = [kx]σ . Using induction on n , it can be easily shown that [(k + n)x]σ = [kx]σ for all n ∈ N , which, in particular, implies that [2(kx)]σ = [kx]σ . Clearly [kx]σ is Archimedean. As a consequence of Theorem 1.1 and Proposition 2.2, we obtain a method for deciding from ρ whether Np /σ is an ideal extension of an Abelian group. It suffices to determine whether Mσ has a positive element, and as we know a system of generators of Mσ , this can be performed using MCP algorithm appearing in [5] (an alternative way for deciding whether Mσ has a positive element is to compute the set {m1 , . . . , mt } of minimal elements of Mσ ∩ Np as explained in [6] and then check whether m1 + · · · + mt is positive). The rest of the section is devoted to the computation of the set of idempotents of Np /σ . A semigroup is Archimedean if all its elements are Archimedean. Given a semigroup S , define on S the equivalence relation N by a N b if there exists k, l ∈ N \ {0} and c, d ∈ S such that ka = b + c and lb = a + d . An Archimedean component of S is an element of S/N . Every Archimedean component of a semigroup is also a subsemigroup. The following are known results that can be found in [2]. Proposition 2.3. one idempotent.

Every Archimedean commutative semigroup has at most

Proposition 2.4. Let S be a commutative semigroup generated by p elements. Then S has at most 2p Archimedean components. Since Archimedean components of Np /σ are Archimedean subsemigroups of N /σ , we deduce that Np /σ has at most 2p idempotents. For a given Archimedean component C of Np /σ , define [ supp(x), supp(C) = p

[x]∈C

N = {(x1 , . . . , xp ) ∈ N | xi = 0 for all i 6∈ supp(C)}. C

p

The set NC is a subsemigroup of Np that is isomorphic to Nr , where r is the cardinality of supp(C) . If [x]σ ∈ C , then x ∈ NC . Denote by σNC the restriction of σ to NC , that is, for a, b ∈ NC , a σNC b if and only if a σ b .

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Lemma 2.5. Let σ be a congruence on Np and C be an Archimedean component of Np /σ . Then C is an Archimedean component of NC /σNC . Proof. Since [x]σ ∈ C implies that x ∈ NC , it makes sense to think of C as a subset of NC /σNC . The proof follows from the fact that C is an Archimedean component of Np /σ . Lemma 2.6. Let σ be a congruence on Np and C be an Archimedean p component of N /σ . Then C contains an Archimedean element of NC /σNC . Proof. Let {i1 , . . . , ir } = supp(C) . From the definition of supp(C) and NC , for every ij ∈ supp(C) , there exists [xj ]σ ∈ C such that its ij th coordinate is nonzero. The element x = x1 + · · · + xr fulfills that [x]σ ∈ C and supp(x) = supp(C). The reader can check that [x]σ is an Archimedean element of NC /σNC . If a is an Archimedean element of a semigroup S and b N a, then b is also an Archimedean element of S . Thus we have the following consequence of Lemmas 2.5 and 2.6. Corollary 2.7. Let σ be a congruence on Np and C be an Archimedean p component of N /σ . Then C = {[x]σ | [x]σ is an Archimedean element of NC /σNC }. As we have pointed out before, if a is an Archimedean element of a semigroup S , then so is a + c for all c ∈ S . Corollary 2.8. Let σ be a congruence on Np and C be an Archimedean p component of N /σ . Then C is an ideal of NC /σNC . Another consequence of the last results is the following statement. Corollary 2.9. Let σ be a congruence on Np and C be an Archimedean p component of N /σ . Then C has an idempotent if and only if NC /σNC has an idempotent Archimedean element. The set supp(C) can be computed from ρ as explained in [7] and a system of generators ρC of σNC can be obtained from ρ by eliminating the coordinates not belonging to supp(C) (see [7] for more details). The concept of positive element of Mσ C translates to an element in Mσ C ∩ NC such N N that its support coincides with supp(C) . In this way, using Proposition 2.2

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and the remark after it, we can decide whether NC /σNC has an idempotent Archimedean element and therefore whether C has an idempotent. Provided that such an element exists, for computing it, owing to the proof of Proposition 2.2, it suffices to find a positive element x in Mσ C (in this case an element N x ∈ NC ∩ Mσ C such that supp(x) = supp(C) ) and then find k ∈ N \ {0} for N which (k +1)x = kx. Since we can compute the set of Archimedean components of Np /σ once we are given ρ (see [7] for details), we can compute the set of all idempotents of Np /σ . References [1] Clifford, A. H., Extensions of semigroups, Trans. Amer. Math. Soc., 68 (1950), 165–173. [2] Grillet, P. A., “Semigroups. An Introduction to the Structure Theory”, Marcel Dekker, New York, 1995. [3] Howie, J. M., “Fundamentals of Semigroup Theory”, The Clarendon Press, Oxford University Press, New York, 1995. [4] R´edei, L., “The Theory of Finitely Generated Commutative Semigroups”, Pergamon, Oxford-Edinburgh-New York, 1965. [5] Rosales, J. C., On finitely generated submonoids of Nk , Semigroup Forum, 50 (1995), 251–262. [6] Rosales, J. C. and P. A. Garc´ıa-S´ anchez, On normal affine semigroups, Linear Algebra Appl., 286 (1999), 175–186. [7] Rosales, J. C. and P. A. Garc´ıa-S´ anchez, “Finitely Generated Commutative Monoids”, Nova Science Publ., New York, 1999.

´ Departamento de Algebra Universidad de Granada Granada E-18071 Spain [email protected] [email protected], [email protected]

Received May 26, 1999 and in final form September 21, 1999