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2 Journal of Algebra ) Contents lists available at SciVerse ScienceDirect Journal of Algebra Gröbner Shirshov bases for semirings L.A. Bokut a,b,1, Yuqun Chen a,, Qiuhui Mo a,2 a School of Mathematical Sciences, South China Normal University, Guangzhou , PR China b Sobolev Institute of Mathematics, Novosibirsk , Russia article info abstract Article history: Received 19 August 2012 Available online Communicated by Louis Rowen In memory of Jean-Louis Loday MSC: 16Y60 16S15 13P10 In the paper we derive a Gröbner Shirshov algorithm for semirings and commutative semirings. As applications, we obtain Gröbner Shirshov bases and A. Blass s 1995) and M. Fiore and T. Leinster s 2004) normal forms of the semirings N[]/ = ) and N[]/ = ), correspondingly Elsevier Inc. All rights reserved. Keywords: Gröbner Shirshov basis Semiring Congruence Normal form 1. Introduction Gröbner bases and Gröbner Shirshov bases were invented independently by A.I. Shirshov for ideals of free commutative, anti-commutative) non-associative algebras [38,40], free Lie algebras [39,40] and implicitly free associative algebras [39,40] see also [2,5]), by H. Hironaka [31] for ideals of the power Supported by the NNSF of China ), the Research Fund for the Doctoral Program of Higher Education of China ), the NSF of Guangdong Province S ) and the Program on International Cooperation and Innovation, Department of Education, Guangdong Province 2012gjhz0007). * Corresponding author. addresses: bokut@math.nsc.ru L.A. Bokut), yqchen@scnu.edu.cn Y. Chen), scnuhuashimomo@126.com Q. Mo). 1 Supported by RFBR , LSS and SB RAS Integration grant No Russia) and Federal Target Grant Scientific and educational personnel of innovation Russia for government contract No ). 2 Supported by the Innovation Grant of the Postgraduate of South China Normal University 2012kyjj112) /$ see front matter 2013 Elsevier Inc. All rights reserved.

3 48 L.A. Bokut et al. / Journal of Algebra ) series algebras both formal and convergent), and by B. Buchberger [21] for ideals of the polynomial algebras. Gröbner bases and Gröbner Shirshov bases theories have been proved to be very useful in different branches of mathematics, including commutative algebra and combinatorial algebra, see, for eample, the books [1,20,22,23,28,29], the papers [2,4,5], and the surveys [6,7,15,17 19]. Up to now, different versions of Composition-Diamond lemma are known for the following classes of algebras apart those mentioned above: color) Lie super-algebras [33 35], Lie p-algebras [34], associative conformal algebras [16], modules[27,32] see also [25]), right-symmetric algebras [12], dialgebras [10], associative algebras with multiple operators [14], Rota Bater algebras [11], and so on. It is well-known Shirshov s result [37,40] that every finitely or countably generated Lie algebra over a field k can be embedded into a two-generated Lie algebra over k. Actually, from the technical point of view, it was a beginning of the Gröbner Shirshov bases theory for Lie algebras and associative algebras as well). Another proof of the result using eplicitly Gröbner Shirshov bases theory is refereed to L.A. Bokut, Yuqun Chen and Qiuhui Mo [13]. A.A. Mikhalev and A.A. Zolotykh [36] prove the Composition-Diamond lemma for a tensor product of a free algebra and a polynomial algebra, i.e. they establish Gröbner Shirshov bases theory for associative algebras over a commutative algebra. L.A. Bokut, Yuqun Chen and Yongshan Chen [8] prove the Composition-Diamond lemma for a tensor product of two free algebras. Yuqun Chen, Jing Li and Mingjun Zeng [26] prove the Composition-Diamond lemma for a tensor product of a non-associative algebra and a polynomial algebra. L.A. Bokut, Yuqun Chen and Yongshan Chen [9] establish the Composition-Diamond lemma for Lie algebras over a polynomial algebra, i.e. for double free Lie algebras. It provides a Gröbner Shirshov bases theory for Lie algebras over a commutative algebra. Yuqun Chen and Yongshan Chen [24] establish the Composition-Diamond lemma for matabelian Lie algebras. In this paper, we establish Gröbner Shirshov bases for semirings and commutative semirings. We show that for a given monomial ordering on the free commutative) semiring, each ideal of the free commutative) semiring algebra has the unique reduced Gröbner Shirshov basis. In 2004, M. Fiore and T. Leinster [30] find a strongly normalizing reduction system and a normal form of the semiring N[]/ = ) where N is the set of natural numbers which is regarded as a semiring and = ) is the congruence on the semiring N[] generated by = In 1995, A. Blass [3] finds a normal form of the semiring N[]/ = ).Now,weusethe Composition-Diamond lemma for commutative semirings to find Gröbner Shirshov bases and Fiore Leinster s and Blass s normal forms for the above semirings respectively. Also we show that each congruence of the semiring N is generated by one element and that the commutative semiring N[] is not Noetherian. 2. Free semiring Let A be an Ω-algebra, i.e. universal algebra with signature Ω, Ω = n=1 Ω n where Ω n is the set of n-ary operations, for eample, aryδ) = n if δ Ω n.wewillcalla an Ω-groupoid and each ω wouldbecalledaproductofa. Letk be a field and ka the groupoid algebra over k, i.e. the linear space over k with a basis A and multi-linear products ω Ω that are etended by linearity from A to ka.such ka is called a linear Ω-algebra. Now let A,,,θ,1) be a semiring, i.e. A,,θ) is a commutative monoid, A,, 1) is a monoid, θ a = a θ = θ for any a A, and is distributive relative to from left and right. Some people call rig instead of semiring. The class of semirings is a variety. So a free semiring Rig X generated by a set X is defined as usual as for any variety of universal systems. Let X,, 1) bethefreemonoidgeneratedbyx. If one fies some linear ordering < on X, then any element of Rig X has a unique form w = u 1 u 2 u n, where each u i X, u 1 u 2 u n, n 0 and w = θ if n = 0. For any u 1, u 2,...,u n X we have u 1 u 2 u n Rig X and u 1 u 2 u n = u i1 u i2 u in, where i 1 i 2...i n is a permutation of 1, 2,...,n and u i1 u i2 u in.

4 L.A. Bokut et al. / Journal of Algebra ) The free semiring Rig X generated by X is just the semiring of non-commutative polynomials on X with natural numbers coefficients, i.e. Rig X =N X. Rig X =X,, ) is an Ω-groupoid, where Ω ={, }. From now on, we assume that Ω ={, }. Let k be a field. We call the groupoid algebra krig X a semiring algebra. Any element in Rig X is called a monomial and any element in krig X is called a polynomial. For any u 1 u 2 u n Rig X, u i X,wedenote u = n. For any u u u u where u u u u = n, u X,wewilldenoteitbyu n. For any u = w 1 w 2 w m u m+1 u n, v = w 1 w 2 w m v m+1 v t, where w l, u i, v j X, such that u i v j for any i = m + 1,...,n, j = m + 1,...,t, we denote lcm u, v) = w 1 w 2 w m u m+1 u n v m+1 v t the least common multiple of u and v in Rig X with respect to. Throughout this paper, we denote N the set of natural numbers, Rig X S the semiring with generators X and defining relations S, i.e.rig X S :=Rig X /ρs), where ρs) is the congruence of Rig X generated by S. Let us remind that θ and 0 are different elements of krig X since θ Rig X and 0 / Rig X. Let A be a semiring. If ρ is a congruence relation on A generated by pairs a i, b i ), i I, then, as linear Ω-algebras, ka/ρ) ka/ IdS), αi u i ρ) α i u i + IdS) is an isomorphism, where S ={a i b i, i I} and IdS) is the Ω-ideal of ka generated by S a set B is an Ω-ideal of ka if B is a subspace of ka such that for any δ = δ n Ω, δa 1,...,a n ) B whenever at least one a i B). It means that any monomial linear basis of ka/ IdS), i.e. a basis that consists of elements of A, is a set of normal forms of the semiring A/ρ, where a set T A) of normal forms of the semiring A/ρ means that for any aρ A/ρ there uniquely eists a t T such that tρ = aρ. We want to create Gröbner Shirshov bases theory for the semiring Rig X. From the above and the Shirshov algorithm see the last paragraph of the net section), as for semigroups or groups, it follows that it is enough to create Gröbner Shirshov bases theory for the semiring algebra krig X. 3. Composition-Diamond lemma for semirings Let / X. Bya -monomial we mean a monomial in Rig X with only one occurrence of. Let u be a -monomial and s krig X. Then we call an s-monomial. For eample, if u s = u s then u = Rig X and s = u s = u s = s 3 = Let > be any monomial ordering on Rig X, i.e.> is a well ordering such that for any v, w Rig X and u a -monomial, w > v u w > u v.

5 50 L.A. Bokut et al. / Journal of Algebra ) For every polynomial f krig X, f has the leading monomial f. If the coefficient of f is 1, then we call f to be monic. For any set S krig X, wesays monic if any s S is monic. Definition 3.1. Let > be a monomial ordering on Rig X. Let krig X. f, g be two monic polynomials in I) If there eist a, b X such that lcm fa, bḡ) < fa + bḡ then we call f, g) w = fa u bg v the intersection composition of f and g with respect to w, where w = lcm fa, bḡ) = fa u = bḡ v. II) If there eist a, b X such that lcm f, aḡb) < f + aḡb then we call f, g) w = f u agb v the inclusion composition of f and g with respect to w, where w = lcm f, aḡb) = f u = aḡb v. In the above definition, w is called the ambiguity of the composition. Clearly, f, g) w Id f, g) and f, g) w < w, where Id f, g) is the ideal of krig X generated by the set { f, g}. Remark. We regard krig X as a linear Ω-algebra. In Definition 3.1, theidealid f, g) means Ω-ideal. In this paper, the ideal of krig X will be Ω-ideal. Let f, g be polynomials and g monic with f = aḡb u for some a, b X, u Rig X. Then the transformation f f αaḡb u is called the elimination of the leading term ELT) of f by g, where α is the coefficient of the leading term of f. Definition 3.2. Suppose that w is a monomial, S a set of monic polynomials in krig X and h a polynomial. Then h is trivial modulo S, w), denoted by h 0 mods, w), if h = i α ia i s i b i u i, where each α i k, a i, b i X, u i Rig X, s i S and a i s i b i u i < w. The set S is called a Gröbner Shirshov basis in krig X if any composition in S is trivial modulo S and corresponding w. AsetS is called a minimal Gröbner Shirshov basis in krig X if S is a Gröbner Shirshov basis in krig X and for any f, g S with f g, a, b X, u Rig X, s.t., f = aḡb u. Denote IrrS) = { w Rig X w a sb u for any a, b X, u Rig X, s S }. A Gröbner Shirshov basis S in krig X is reduced if for any s S, supps) IrrS \{s}), where supps) ={u 1, u 2,...,u n } if s = n α i=1 iu i,0 α i k, u i Rig X. If the set S is a Gröbner Shirshov basis in krig X, then we call also S is a Gröbner Shirshov basis for the ideal IdS) or the algebra krig X S :=krig X / IdS). Let I be an ideal of krig X. Then there eists uniquely the reduced Gröbner Shirshov basis S for I, see Theorem 3.5. If a subset S of krig X is not a Gröbner Shirshov basis for IdS) then one can add to S anontrivial composition f, g) w of f, g S and continue this process repeatedly actually using the transfinite induction) in order to obtain a set S comp of generators of IdS) such that S comp is a Gröbner Shirshov basis in krig X. Such a process is called Shirshov algorithm.

6 L.A. Bokut et al. / Journal of Algebra ) Suppose that S ={u i v i i I} where for any i, u i, v i Rig X. In this case we call u i v i a semiring relation. It is clear that if S is a set of semiring relations then so is S comp.inorderto find a normal form of the semiring Rig X S =Rig X /ρs), where ρs) is the congruence of Rig X generated by the set {u i, v i ) i I}, it is enough to find a monomial k-linear basis of the semiring algebra krig X S. Lemma 3.3. Let S be a Gröbner Shirshov basis in k Rig X and s 1, s 2 S. If w = as 1 b u = cs 2 d vforsome a, b, c, d X,u, v Rig X,then as 1 b u cs 2 d v mods, w). Proof. There are two cases to consider. I) lcm as 1 b, cs 2 d) = as 1 b cs 2 d which means there eists u 1 Rig X such that u = u 1 cs 2 d, v = as 1 b u 1. Then as 1 b u cs 2 d v = as 1 b u 1 cs 2 d cs 2 d as 1 b u 1 = as 1 b cs 2 d u 1 as 1 b cs 2 d u 1 + as 1 b cs 2 d u 1 as 1 b cs 2 d u 1 = as 1 b cs 2 s 2 )d u 1 + as 1 s 1 )b cs 2 d u 1. Since s 2 s 2 < s 2 and s 1 s 1 < s 1,wehave as 1 b cs 2 s 2 )d u 1 < as 1 b cs 2 d u 1 = w and as 1 s 1 )b cs 2 d u 1 < as 1 b cs 2 d u 1 = w. It follows that as 1 b u cs 2 d v mods, w). II) lcm as 1 b, cs 2 d) < as 1 b + cs 2 d,i.e.s 1 = u 1 u 2 u m u m+1 u n, s 2 = v 1 v 2 v m v m+1 v t such that au 1 b = cv 1 d, au 2 b = cv 2 d,..., au m b = cv m d and au i b cv j d for any i = m + 1,...,n, j = m + 1,...,t, where u i, v j X. In this case, there eists u Rig X such that u = cv m+1 d cv m+2 d cv t d u, v = au m+1 b au m+2 b au n b u. There are four subcases to consider.

7 52 L.A. Bokut et al. / Journal of Algebra ) ) u 1 b 1 = c 1 v 1 for some b 1, c 1 X.Inthissubcase,c = ac 1, b = b 1 d.then as 1 b u cs 2 d v = as 1 b 1 d ac 1 v m+1 d ac 1 v m+2 d ac 1 v t d u ac 1 s 2 d au m+1 b 1 d au m+2 b 1 d au n b 1 d u = as 1 b 1 c 1 v m+1 c 1 v m+2 c 1 v t c 1 s 2 u m+1 b 1 u m+2 b 1 u n b 1 )d u = ) a s 1, s 2 ) w d u where w = s 1 b 1 c 1 v m+1 c 1 v m+2 c 1 v t = c 1 s 2 u m+1 b 1 u m+2 b 1 u n b 1. Since S is a Gröbner Shirshov basis in krig X, wehave Then s 1, s 2 ) w 0 mod S, w ). as 1 b u cs 2 d v = ) a s 1, s 2 ) w d u 0 mod S, aw d u ) 0 mods, w). 2) a 1 u 1 = v 1 d 1 for some a 1, d 1 X. This subcase is similar to subcase 1). We omit the proof. 3) u 1 = c 1 v 1 d 1 for some c 1, d 1 X.Thenc = ac 1, d = d 1 b and as 1 b u cs 2 d v = as 1 b ac 1 v m+1 d 1 b ac 1 v m+2 d 1 b ac 1 v t d 1 b u ac 1 s 2 d 1 b au m+1 b au m+2 b au n b u = as 1 c 1 v m+1 d 1 c 1 v m+2 d 1 c 1 v t d 1 c 1 s 2 d 1 u m+1 u m+2 u n )b u = ) a s 1, s 2 ) w b u where w = s 1 c 1 v m+1 d 1 c 1 v m+2 d 1 c 1 v t d 1 = c 1 s 2 d 1 u m+1 u m+2 u n. Since S is a Gröbner Shirshov basis in krig X, wehave Similar to the subcase 1), we have s 1, s 2 ) w 0 mod S, w ). as 1 b u cs 2 d v 0 mods, w). 4) a 1 u 1 b 1 = v 1 for some a 1, b 1 X. This subcase is similar to the subcase 3). We omit the proof. The lemma is proved. Theorem 3.4 Composition-Diamond lemma for semirings). Let S be a set of monic polynomials in k Rig X, > a monomial ordering on Rig X and IdS) the Ω-ideal of k Rig X generated by S. Then the following statements are equivalent. 1) S is a Gröbner Shirshov basis in k Rig X.

8 L.A. Bokut et al. / Journal of Algebra ) ) f IdS) f = a sb u for some a, b X,u Rig X and s S. 2 ) f IdS) f = α 1 a 1 s 1 b 1 u 1 + α 2 a 2 s 2 b 2 u 2 + +α n a n s n b n u n,wherea 1 s 1 b 1 u 1 > a 2 s 2 b 2 u 2 > > a n s n b n u n, 0 α i k, a i, b i X,u i Rig X, s i S. 3) IrrS) ={w Rig X w a sb u for any a, b X, u Rig X, s S} is a k-linear basis of krig X S =krig X / IdS). Proof. 1) 2) Let 0 f IdS). Then f = n α i a i s i b i u i where each α i k, a i, b i X, u i Rig X, s i S. Let w i = a i s i b i u i and we arrange this leading terms in non-increasing ordering by Now we prove the result by induction on m. If m = 1, then f = a 1 s 1 b 1 u 1. Now we assume that m 2. Then i=1 w 1 = w 2 = = w m > w m+1 w n. a 1 s 1 b 1 u 1 = w 1 = w 2 = a 2 s 2 b 2 u 2. Since S is a Gröbner Shirshov basis in krig X, by Lemma 3.3, wehave a 2 s 2 b 2 u 2 a 1 s 1 b 1 u 1 = β j c j s j d j v j where each β j k, c j, d j X, v j Rig X, s j S, and c j s j d j v j < w 1. Therefore, since α 1 a 1 s 1 b 1 u 1 + α 2 a 2 s 2 b 2 u 2 = α 1 + α 2 )a 1 s 1 b 1 u 1 + α 2 a 2 s 2 b 2 u 2 a 1 s 1 b 1 u 1 ), we have f = α 1 + α 2 )a 1 s 1 b 1 u 1 + α 2 β j c j s j d j v j + n α i a i s i b i u i. If either m > 2 orα 1 + α 2 0, then the result follows from the induction on m. Ifm = 2 and α 1 + α 2 = 0, then the result follows from the induction on w 1. 2) 2 )isclear. 2) 3) For any f krig X, by the ELTs, we can obtain that f + IdS) can be epressed as a linear combination of elements of IrrS). Now suppose α 1 u 1 + α 2 u 2 + +α n u n = 0inkRig X S with u i IrrS), u 1 > u 2 > > u n and each α i 0. Then, in krig X, g = α 1 u 1 + α 2 u 2 + +α n u n IdS). By 2), we have u 1 = ḡ / IrrS), a contradiction. So IrrS) is k-linearly independent. This shows that IrrS) is a k-basis of krig X S. 3) 2) Let 0 f IdS). Suppose that f IrrS). Then i=3 f + IdS) = α f + IdS) ) + αi ui + IdS) ),

9 54 L.A. Bokut et al. / Journal of Algebra ) where α, α i k, u i IrrS) and f > u i. Therefore, f + IdS) 0, a contradiction. So f = a sb u for some a, b X, u Rig X, s S. 2) 1) By the definition of the composition, we have f, g) w IdS). If f, g) w 0, then by 2), f, g) w = a 1 s 1 b 1 u 1 for some a 1, b 1 X, u 1 Rig X, s 1 S. Let h = f, g) w α 1 a 1 s 1 b 1 u 1, where α 1 is the coefficient of f, g) w.then h < f, g) w and h IdS). By induction, we can get the result. Suppose that > is a monomial ordering on Rig X and I an ideal of krig X. Then there eists a Gröbner Shirshov basis S krig X for the ideal I = IdS), for eample, we may take S = I. By Theorem 3.4, we may assume that the leading terms of the elements of S are different with each other. For any g S, denote and S 1 = S \ g S g. g = { f S f g and f = aḡb u for some a, b X, u Rig X } For any f IdS) we show that there eists an s 1 S 1 such that f = as 1 b u for some a, b X, u Rig X. In fact, by Theorem 3.4, f = a hb u for some a, b X, u Rig X and h S. Suppose that h S \ S 1. Then we have h g S g, say, h g, i.e. h g and h = aḡb u for some a, b X, u Rig X. We claim that h > ḡ. Otherwise, h < ḡ. It follows that h = aḡb u > a hb u and so we have an infinite descending chain h > a hb u > a 2 hb 2 aub u > a 3 hb 3 a 2 ub 2 aub u > which contradicts that > is well ordered. Suppose that g / S 1. Then by the above proof, there eists a g 1 S such that g g1 and ḡ > g 1. Since > is well ordered, there must eist an s 1 S 1 such that f = a 1 s 1 b 1 u 1 for some a 1, b 1 X, u 1 Rig X. Let f 1 = f α 1 a 1 s 1 b 1 u 1, where α 1 is the coefficient of the leading term of f.then f 1 IdS) and f > f 1. By induction on f, we know that f IdS 1 ) and hence I = IdS 1 ). Moreover, by Theorem 3.4, S 1 is clearly a minimal Gröbner Shirshov basis for the ideal IdS). Assume that S is a minimal Gröbner Shirshov basis for the ideal I. For any s S, wehaves = s + s, where supps ) IrrS \{s}), s IdS \{s}). SinceS is a minimal Gröbner Shirshov basis, we have s = s for any s S. Then S 2 ={s s S} is the reduced Gröbner Shirshov basis for the ideal I. In fact, it is clear that S 2 IdS) = I. Forany f IdS), bytheorem3.4, f = a1 s 1 b 1 u 1 = a 1 s 1 b 1 u 1 for some a 1, b 1 X, u 1 Rig X. Suppose that S, R are two reduced Gröbner Shirshov bases for the ideal I. For any s S, by Theorem 3.4, s = a rb u, r = cs 1 d v for some a, b, c, d X, u, v Rig X and hence s = acs 1 db avb u. Since s supps) IrrS \{s}), we have s = s 1. It follows that a = b = c = d = 1 and u = v = θ and so s = r. If s r then 0 s r I = IdS) = IdR). ByTheorem3.4, s r = a 1 r 1 b 1 u 1 = c 1 s 2 d 1 v 1 for some a 1, b 1, c 1, d 1 X, u 1, v 1 Rig X with r 1, s 2 < s = r. This means that s 2 S \{s} and r 1 R \{r}. Noting that s r supps) suppr), wehaveeithers r supps) or s r suppr). Ifs r supps) then s r IrrS \{s}) which contradicts s r = c 1 s 2 d 1 v 1 ;ifs r suppr) then s r IrrR \{r}) which contradicts s r = a 1 r 1 b 1 u 1. This shows that s = r and then S R. Similarly, R S.

10 L.A. Bokut et al. / Journal of Algebra ) Therefore, we have proved the following theorem. Theorem 3.5. Let I be an ideal of k Rig X and > a monomial ordering on Rig X. Then there eists uniquely the reduced Gröbner Shirshov basis S for I. 4. Composition-Diamond lemma for commutative semirings In this section, we will give Gröbner Shirshov bases theory for commutative semirings which is almost the same as the case of semirings. The compositions of commutative semiring are simpler. A semiring A,,,θ,1) is commutative if A,, 1) is a commutative monoid. The class of commutative semirings is a variety. A free commutative semiring Rig[X] generated by a set X is defined as usual. Let [X],, 1) be the free commutative monoid generated by X. If one fies some linear ordering < on the set [X], then any element of Rig[X] has a unique form θ, orw = u 1 u 2 u n, where u i [X], u 1 u 2 u n, n 1. Note that Rig[X] =N[X], the semiring of commutative polynomials on X with natural numbers coefficients. For any u, v [X], wedenotelcm u, v) the least common multiple of u and v in [X]. Then there eist uniquely a, b [X] such that lcm u, v) = au = bv. Definition 4.1. Let < be a monomial ordering on Rig[X]. Let f, g be two monic polynomials in krig[x] and f = u 1 u 2 u n, ḡ = v 1 v 2 v m where each u i, v j [X]. For any pair a, b) {a ij, b ij ) 1 i n, 1 j m} where a ij, b ij [X] such that lcm u i, v j ) = a ij u i = b ij v j, we call f, g) w = af u bg v the composition of f and g with respect to w where w = lcm a f, bḡ) = a f u = bḡ v. Definition 4.2. Suppose that w is a monomial in Rig[X], S a set of monic polynomials in krig[x] and h a polynomial. Then h is trivial modulo S, w), denotedbyh 0modS, w), ifh = i α ia i s i u i, where each α i k, a i [X], u i Rig[X], s i S and a i s i u i < w. The set S is called a Gröbner Shirshov basis in krig[x] if any composition in S is trivial modulo S and corresponding w. Remark. For any given monic polynomials f, g krig[x], there are finitely many compositions f, g) w. Therefore we may use computer to realize Shirshov s algorithm to find a Gröbner Shirshov basis S comp for a finite set S in krig[x]. However, the reduced Gröbner Shirshov basis of IdS) is generally infinite even if both S and X are finite, see Eample 5.9. The following theorems can be similarly proved to Theorems 3.4 and 3.5 respectively. We omit the detail. Theorem 4.3 Composition-Diamond lemma for commutative semirings). Let S be a set of monic polynomials in k Rig[X] and > a monomial ordering on Rig[X]. Then the following statements are equivalent. 1) S is a Gröbner Shirshov basis in k Rig[X]. 2) f IdS) f = a s u for some a [X], u Rig[X] and s S. 2 ) f IdS) f = α 1 a 1 s 1 u 1 + α 2 a 2 s 2 u 2 + +α n a n s n u n,wherea 1 s 1 u 1 > a 2 s 2 u 2 > > a n s n u n, α i k, a i [X], u i Rig[X], s i S. 3) IrrS) ={w Rig[X] w a s u for any a [X], u Rig[X], s S} is a k-linear basis of k Rig[X S]= krig[x]/ IdS). Theorem 4.4. Let I be an ideal of k Rig[X] and > a monomial ordering on Rig[X]. Then there eists uniquely the reduced Gröbner Shirshov basis S for I.

11 56 L.A. Bokut et al. / Journal of Algebra ) Applications In 2004, M. Fiore and T. Leinster [30] find a strongly normalizing reduction system and a normal form of the semiring N[]/ = ). ActuallyN[]/ = ) = Rig[ = 1 2 ]. Now, we use the Composition-Diamond lemma for commutative semirings, i.e. Theorem 4.3, to find a Gröbner Shirshov basis and a normal form of this semiring. We define a monomial ordering on Rig[] first. We order [] by degree ordering : n m n m. For any u Rig[], u can be uniquely epressed as u = u 1 u 2 u n, where u 1, u 2,...,u n [], and u 1 u 2 u n.denote wtu) = u n, u n 1,...,u 1 ). We order Rig[] as follows: for any u, v Rig[], if one of the sequences is not a prefi of other, then u < v wtu)<wtv) leicographically; if the sequence of u is a prefi of the sequence of v, thenu < v. Then, it is clear that < on Rig[] is a monomial ordering. Theorem 5.1. Let the ordering on Rig[] be as above. Then k Rig[ = 1 2 ]=krig[ S] and S is a Gröbner Shirshov basis in k Rig[], where S consists of the following relations: 1. 4 = 1 1 2, 2. 3 = 1 2, n = n 1 n 3). Proof. We denote i j the composition of the type i and type j. Let us check all the possible compositions. For 1 1, there is no composition. For 1 2, the ambiguities w of all possible compositions are: 1) 4 6,2) 2 4. For 1 3, the ambiguities w of all possible compositions are: 3) 4 6 n+4 4) 2 4 n+2 5) 4 n 6 n 4 where 1 n 3. For 2 2, the ambiguity w of all possible composition is: 6) 3 5. For 2 3, the ambiguities w of all possible compositions are: 7) 3 5 n+3 8) 3 5 n 3 n 9) n 10) n+2 n ) 3 n+1 where 1 n 3. For 3 3, the ambiguities w of all possible compositions are: 12) 1 2 n n+2 n+m 13) m 14) 1 2 p p 2 p+m 2 15) 1 2 t 2+t n t n 16) 1 n 2 4 m+2 17) ) ) n 1 2 m where 1 n,m, t 3, 2 p 3 and t n.

12 L.A. Bokut et al. / Journal of Algebra ) We have to check that all these compositions are trivial mods, w). Here, for eample, we just check 11) and 17). Others are similarly proved. For 17), let f = 1 2, g = 1 2. Thenw = and f, g) w = 1 2 ) ) 1 = From this it follows that we have the relation 2. For 11), there are three cases to consider. Case 1. n = 1, f = 1 2, g = Thenw = 3 2 and f, g) w = 1 2 ) 3 1 2) 2 = Case 2. n = 2, f = , g = Thenw = 3 3 and f, g) w = ) 3 1 2) 3 = By Case 1 and Case 2 we have the relations 3. Case 3. n = 3, f = , g = Thenw = 3 4 and f, g) w = ) 3 1 2) 4 = By Case 3 we have the relation 1. So S is a Gröbner Shirshov basis in krig[]. The above proof implies that krig[ = 1 2 ]=krig[ S]. By Theorems 4.3 and 5.1, we have the following corollary. Corollary 5.2. See [30].) A normal form of the semiring Rig[ = 1 2 ] is the set { 1 m+1) 2, 1 m n, 1 m 3) n, m 2) n, 2 ) m 3 ) n n,m 0 }. In 1995, A. Blass [3] finds a normal form of the semiring N[]/ = ). Clearly,N[]/ = ) = Rig[ = 1 2 ].WeuseTheorem4.3 to find a Gröbner Shirshov basis and a normal form of this semiring which is different from [3].

13 58 L.A. Bokut et al. / Journal of Algebra ) Theorem 5.3. Let the ordering on Rig[] be as in Theorem 5.1.ThenkRig[ = 1 2 ]=krig[ S] and S is a Gröbner Shirshov basis in k Rig[], where S consists of the following relations: =, 2. 4 = 1 3, 3. 5 = 1 4, n = n 3 n 4). Proof. Let us check all the possible compositions. For 1 1, the ambiguity w of all possible composition is: 1) For 1 2, the ambiguities w of all possible compositions are: 2) ) 2 4 4) 3 4 5) 6 4. For 1 3, the ambiguities w of all possible compositions are: 6) 3 5,7) 7 5. For 1 4, the ambiguities w of all possible compositions are: 8) n+2 9) 3 1 n 10) n 2 n ) n 12) n 13) n+2 n 1 3. where 3 n 4. For 2 2, the ambiguity w of all possible composition is: 14) 4 7. For 2 3, the ambiguities w of all possible compositions are: 15) 2 5,16) 8 5. For 2 4, the ambiguities w of all possible compositions are: 17) 4 7 n+4 18) 4 4 n 7 n 19) n 20) n+3 n ) 4 n+1 where 3 n 4. For 3 3, there is no composition. For 3 4, the ambiguities w of all possible compositions are: 22) 8 n ) 2 n ) 5 n 8 n 5 where 3 n 4. For 4 4, the ambiguities w of all possible compositions are: 25) n+m m+3 m ) m 3 m+n 3 m ) ) 6 n m 29) where 3 n,m 4. We have to check that all these compositions are trivial mods, w). Here, for eample, we just check 1), 4) and 21). Others are similarly proved. For 1), let f = 1 2, g = 1 2. Thenw = 1 2 4, and f, g) w = 1 2 ) ) 2 1 = It follows that we have the relation 2.

14 L.A. Bokut et al. / Journal of Algebra ) For 4), let f = 1 2, g = Thenw = 3 4, and f, g) w = 1 2 ) ) 3 = Then we have the relation 4 for n = 3. For 21), there are two cases to consider. Case 1. n = 3, let f = 4 1 3, g = Thenw = 4 4,and f, g) w = 4 1 3) ) = Then we have the relation 4 for n = 4. Case 2. n = 4, let f = 4 1 3, g = Thenw = 4 5, and f, g) w = 4 1 3) ) = Then we have the relation 3. Therefore S is a Gröbner Shirshov basis in krig[]. The above proof implies that krig[ = 1 2 ]=krig[ S]. We complete the proof. By Theorems 4.3 and 5.3, we have the following corollary. Corollary 5.4. A normal form of the semiring Rig[ = 1 2 ] is the set { 1 n m) t, 1 n 3, 1 n 4) m n,m 0, 0 t 3 }. In order to compare an another normal form of the semiring Rig[ = 1 2 ] { 1 n 2 4, 1 n 2) m, 2 ) m 4 ) t, 1 n 4) t n,m, t 0 } given by A. Blass [3], we need the following lemma. Lemma 5.5. Suppose that Γ,Σ are two subsets of the semiring Rig X and ρ is a congruence on Rig X. Suppose that Γ is a normal form of Rig X ρ. If f : Γ Σ is a bijective mapping such that for any u Γ, f u)ρ = uρ,thenσ is also a normal form of Rig X ρ.

15 60 L.A. Bokut et al. / Journal of Algebra ) Proof. For any u Rig X, sinceγ is a normal form of the semiring Rig X ρ, there is uniquely v Γ, such that uρ = vρ. Henceuρ = f v)ρ, where f v) Σ. For any two different u, v Σ, ifuρ = vρ, then f 1 u)ρ = f 1 v)ρ and hence f 1 u) f 1 v), a contradiction. This shows that Σ is a normal form of the semiring Rig X ρ. Corollary 5.6. See [3].) A normal form of the semiring Rig[ = 1 2 ] is the set { 1 n 2 4, 1 n 2) m, 2 ) m 4 ) t, 1 n 4) t n,m, t 0 }. Proof. We denote Γ = { 1 n m) t, 1 n 3, 1 n 4) m n,m 0, 0 t 3 }, Σ = { 1 n 2 4, 1 n 2) m, 2 ) m 4 ) t, 1 n 4) t n,m, t 0 } and ρ the congruence on Rig[] generated by { = 1 2 }. Define f : Γ Σ, 1 n m 1 n+m) 2) m, 1 n m) 1 n 2) n+m), 1 n m) 2 2) n+m) 4 ) m, 1 n m) 3 2) n 4 ) n+m), 1 n 3 1 n 2 4, 1 n 4) m 1 n 4) m. Then f is a bijective mapping and for any u Γ, f u)ρ = uρ since f u) is obtained by u replacing, 3 for 1 2, 2 4 respectively. Now the result follows from Corollary 5.4 and Lemma 5.5. Let X beawellorderedset,z the integer ring and Z X the semigroup ring over Z.Itiseasyto see that Z X,, ) is a semiring with the operations f g := f + g, f g := f g, where f, g are polynomials in Z X. Now, we represent the semiring Z X by generators and defining relations. Let X 1 ={ 1 X}. We define a monomial ordering on Rig X X first. For any, y X, wedefine 1 > > y 1 > y if > y and > 1 1 > 1. Then we define the inverse deg-le ordering on {X X }. For any u Rig X X 1 1 1, u can be uniquely epressed as u = u 1 u 2 u n, where u 1, u 2,...,u n {X X } and u 1 u 2 u n.denote wtu) = degu),n, u 1, u 2,...,u n ). We order Rig X X as follows: for any u, v Rig X X 1 1 1, u > v wtu)>wtv) leicographically. Then, it is clear that > on Rig X X is a monomial ordering.

16 L.A. Bokut et al. / Journal of Algebra ) Theorem 5.7. Let the ordering be as above. Then Z X = Rig X X S as semirings and a Gröbner Shirshov basis S in k Rig X X consists of the following relations: 1. 1 = θ, = θ, 3. 1 y 1 = y, 4. y 1 = 1 y, 5. ɛ 1 1 = ɛ, ɛ = ɛ, where, y X,ɛ =±1. As a result, a normal form of the semiring Rig X X S is the set IrrS) = { ɛ n1 ɛ n2 ɛ m m1 m2 mnm ij X, m 0, ɛ i =±1, n i 0, i = 1,...,m }, where ɛ 1 i1 i2 ini = 1 ɛ 1 if n i = 0. Proof. It is easy to see that σ : Z X Rig X X S, ɛ i1 i2 it ɛ i1 i2 it, 0 θ is a semiring isomorphism, where ɛ =±1. Since IrrS) = σ Z X ), IrrS) is a k-basis of krig X X S. Therefore, by using Theorem 3.4, S is a Gröbner Shirshov basis in krig X X Let N,, ) be the natural numbers semiring, where for any n,m N, n m := n +m, n m := n m. Then N,, ) = Rig[ = 1]. For any congruence ρ on N, wehaven/ρ = Rig[ = 1, ρ]. Letthe ordering on Rig[] be defined as in Theorem 5.1. By Shirshov algorithm, we are able to find a Gröbner Shirshov basis { = 1, ρ} comp for the set { = 1, ρ}. Suppose { = 1} S ={ = 1, ρ} comp. Then by Theorem 4.4, we may assume that { = 1} S is the reduced Gröbner Shirshov basis. Since { = 1} S is minimal, each element in S has the form 1 n = 1 m, n > m N and S contains only one element, say, 1 n = 1 m, n > m N. It follows that the congruence ρ on N is generated by one element n,m). Thus, we have the following corollary. Corollary 5.8. Each congruence on the semiring N is generated by one element. In particular, N is Noetherian. For a commutative algebra k[x S] with X <, it is well known that a reduced Gröbner Shirshov basis of k[x S] must be finite. It is also well known that if the ring R is Noetherian then so is the polynomial ring R[X] if X <. However, it is not the case for the semiring N[]. Eample 5.9. Considering the semiring N[]/ + 1 = ) = Rig[ 1 = ], it is easy to have that krig[ 1 = ]=krig[ S] where S ={ n 1 = n n 1} is the reduced Gröbner Shirshov basis in krig[] with the ordering in Theorem 5.1. Now, we construct an ascending chain of ideals in krig[] as follows. I 1 I 2 I n where I n = Id 1, 2 1,..., n 1).

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