††footnotetext: 2020 _Mathematics Subject Classification._ 13P10, 16S15,
16P90, 17A30, 17A61.

# Gelfand-Kirillov dimension of bicommutative algebras

Yuxiu Bai  Y.B., School of Mathematical Sciences, South China Normal
University Guangzhou 510631, P. R. China<EMAIL_ADDRESS>,  Yuqun Chen
Y.C., School of Mathematical Sciences, South China Normal University,
Guangzhou 510631, P. R. China<EMAIL_ADDRESS>and  Zerui Zhang♯ Z.Z.,
School of Mathematical Sciences, South China Normal University, Guangzhou
510631, P. R. China<EMAIL_ADDRESS>
###### Abstract.

We first offer a fast method for calculating the Gelfand-Kirillov dimension of
a finitely presented commutative algebra by investigating certain finite set.
Then we establish a Gröbner–Shirshov bases theory for bicommutative algebras,
and show that every finitely generated bicommutative algebra has a finite
Gröbner–Shirshov basis. As an application, we show that the Gelfand-Kirillov
dimension of a finitely generated bicommutative algebra is a nonnegative
integer.

###### Key words and phrases:

commutative algebra; bicommutative algebra; Gelfand-Kirillov dimension;
Gröbner basis; Gröbner–Shirshov basis

♯ Corresponding author

##  1\. Introduction

The Gelfand-Kirillov dimension measures the asymptotic rate of growth of
algebras. And it has become an important tool in the study of algebras. There
are many well-known results on the Gelfand-Kirillov dimensions of algebras.
For instance, the Gelfand-Kirillov dimension of a finitely generated
commutative algebra $\mathcal{A}$ over a field is the classical Krull
dimension of $\mathcal{A}$ [4, Theorem 4.5], in particular, it is a
nonnegative integer.

There exist a lot of important results concerning the Gelfand-Kirillov
dimensions of algebras. Bergman’s gap Theorem [5] shows that there is no
algebra with Gelfand-Kirillov dimension strictly between one and two. Martinez
and Zelmanov[18] studied finitely generated linear Jordan algebras of Gelfand-
Kirillov dimension one and showed that the gap theorem holds for Jordan
algebras. Petrogradsky and Shestakov [19] showed that the Gelfand-Kirillov
dimension of a Jordan superalgebra can be an arbitrary number in
$\\{0\\}\cup[1,+\infty]$. The Gelfand-Kirillov dimension of a Leavitt path
algebra was investigated by Alahmadi et al. in [2]. Qi et al. [21] proved that
no finitely generated locally finite nonsymmetric operad has Gelfand-Kirillov
dimension strictly between 1 and 2. And Bao et al. [3] classified $k$-linear
symmetric operads of low Gelfand-Kirillov dimension. Petrogradsky, Shestakov,
Zelmanov [20] and Shestakov, Zelmanov [22] investigated the Gelfand-Kirillov
dimensions of a family of infinite-dimensional nil Lie algebras. Drensky and
Papistas [11] showed that, under some natural restrictions, the Gelfand-
Kirillov dimension of the group of tame automorphisms of relatively free
algebras $F_{n}(V)$ is equal to the Gelfand-Kirillov dimension of the algebra
$F_{n}(V)$. Centrone [9] proved a strict relation between the Gelfand-Kirillov
dimension of the relatively free (graded) algebra of a PI-algebra and its
(graded) exponent. Zhao and Zhang [26] offered an algorithm for computing the
Gelfand-Kirillov dimension of certain type of differential difference modules.

Recall that an algebra $\mathcal{B}$ over a field $k$ is called bicommutative
if it satisfies the identities

$\displaystyle x(yz)$ $\displaystyle=$ $\displaystyle y(xz)\ \ \ \ \mbox{(left
commutativity)}$ $\displaystyle(xy)z$ $\displaystyle=$ $\displaystyle(xz)y\ \
\ \ \mbox{(right commutativity)}$

for all $x,y,z\in\mathcal{B}$. We denote by $\mathsf{BC}(X)$ the free
bicommutative algebra generated by $X$. In [14], Dzhumadil’daev, Ismailov and
Tulenbaev constructed a linear basis for $\mathsf{BC}(X)$ when $X$ is finite,
which can be easily generalized to the case when $X$ is an arbitrary linearly
ordered set, see also [12]. Drensky and Zhakhayev [12] proved that every
finitely generated bicommutative algebra is weakly noetherian and offered a
positive solution to the finite basis problem for varieties of bicommutative
algebras over an arbitrary field of any characteristic. Classifications on low
dimensional bicommutative algebras also attracts peoples’ attraction, and it
is proved that there are only three pairwise nonisomorphic two-dimensional
bicommutative algebras over an algebraically closed field by Drensky [10].
Kaygorodov, Páez-Guillán and Voronin [16] offered the algebraic and geometric
classification of 4-dimensional nilpotent bicommutative algebras.
Dzhumadil’daev and Ismailov [13] established criterions for elements of a free
bicommutative algebra to be Lie or Jordan.

We continue to study bicommutative algebras and show that the Gelfand-Kirillov
dimension of an arbitrary finitely generated bicommutative algebra is a
nonnegative integer. Moreover, we show that, in order to calculate the
Gelfand-Kirillov dimension of a finitely generated bicommutative algebra, it
suffices to investigate a finite specific set instead of constructing a
possibly infinite linear basis for the bicommutative algebra under
consideration. The method we applied for the proof consists of using Gröbner
bases and Gröbner–Shirshov bases, as introduced by Buchberger [7, 8] and
Shirshov [24, 25]. For more history on Gröbner bases and Gröbner–Shirshov
bases theory, we refer to [6].

The paper is organized as follows. In Section 2, we recall the Gröbner bases
theory for commutative algebras and offer a fast method for calculating the
Gelfand-Kirillov dimension of a finitely generated commutative algebra. In
Section 3, we establish a Gröbner–Shirshov bases theory for bicommutative
algebras and show that every finitely generated bicommutative algebra has a
finite Gröbner–Shirshov basis. In Section 4, we apply the method of
Gröbner–Shirshov bases to prove that the Gelfand-Kirillov dimension of a
finitely generated bicommutative algebra is a nonnegative integer. Moreover,
in Theorem 4.2 we offer a fast method for calculating the Gelfand-Kirillov
dimension of a finitely presented bicommutative algebra $\mathcal{B}$ by
investigating certain finite set in $\mathcal{B}$.

##  2\. Gelfand-Kirillov dimension of commutative algebras

Our aim in this section is to offer a new proof for the known fact that the
Gelfand-Kirillov dimension of a finitely generated commutative algebra is a
nonnegative integer by using the Gröbner bases theory for commutative
algebras.

Let $(X,\prec)$ be a fixed well-ordered set and let $\mathbb{N}$ (resp.
$\mathbb{N^{+}}$) be the set of all nonnegative integers (resp. all positive
integers). Furthermore, we denote by $[X]$ (resp. $[X]^{+}$) the free
commutative monoid (resp. free commutative semigroup) generated by $X$, that
is

$[X]=\\{x_{i_{1}}^{n_{i_{1}}}\cdots x_{i_{t}}^{n_{i_{t}}}\mid
x_{i_{1}},\dots,x_{i_{t}}\in X,\ x_{i_{1}}\prec\cdots\prec
x_{i_{t}},t,n_{i_{1}},...\leavevmode\hbox{}\hskip
0.56905pt,n_{i_{t}}\in\mathbb{N}\\}$

and $[X]^{+}=[X]\setminus\\{\varepsilon\\}$, where $x^{0}=\varepsilon$ for
every ${x\in X}$ and $\varepsilon$ is the unit of the monoid $[X]$. When we
write a monomial $u=x_{i_{1}}^{n_{i_{1}}}\cdots x_{i_{t}}^{n_{i_{t}}}$, we
always assume that all the letters $x_{i_{1}},...\leavevmode\hbox{}\hskip
0.56905pt,x_{i_{t}}$ are pairwise different unless otherwise specified.
Finally, we denote by $k[X]$ the free commutative algebra generated by $X$,
and for an arbitrary set $Y$, we denote by $\\#Y$ the cardinality of $Y$.

Now we recall the Gröbner bases theory for commutative algebras, for more
details, see [7, 1]. For every monomial $u=x_{i_{1}}^{n_{i_{1}}}\cdots
x_{i_{t}}^{n_{i_{t}}}\in[X]$, we define the _length_ of $u$, denoted by
$\ell(u)$, to be $\sum_{j=1}^{t}n_{i_{j}}$. Let $<$ be a monomial order on
$[X]$, that is, $<$ is a well order and for all monomials $u,v,w\in[X]$, we
have $u<v\Rightarrow wu<wv$. For every nonzero polynomial
$f=\sum\alpha_{i}u_{i}$ in $k[X]$, where each $0\neq\alpha_{i}$ lies in $k$,
each $u_{i}$ lies in $[X]$ and $u_{1}>u_{2}>\cdots$, we call $u_{1}$ the
_leading monomial_ of $f$ and denote it by $\bar{f}$. We call $f$ a _monic
polynomial_ if $\alpha_{1}=1$. And a set $\SS\subseteq k[X]$ is called a
_monic set_ if every polynomial in $\SS$ is monic. By convention, the empty
set $S$ is a monic set but $\\{0\\}$ is not. We denote by $\mathsf{Id}(\SS)$
the ideal of $k[X]$ generated by $\SS$ and define
$\overline{S}=\\{\overline{f}\mid f\in S\\}$.

Let $I$ be an ideal of $k[X]$. A monic set $\SS$ is called a _Gröbner basis_
(in $k[X]$) for the quotient algebra $k[X]/I$ if $\mathsf{Id}(\SS)=I$ and for
every nonzero polynomial $f\in I$, we have ${\overline{f}=\overline{sv}}$ for
some $s\in\SS$, $v\in[X]$. By the Gröbner bases theory for commuative algebras
[7, 1], if $S$ is a Gröbner basis for the algebra
$k[X|\SS]:=k[X]/\mathsf{Id}(\SS)$, then the set

$\mathsf{Irr}(\SS)=[X]\setminus\\{w\in[X]\mid w=\overline{sv}\ \mbox{for
some}\ s\in\SS,v\in[X]\\}$

forms a linear basis for the algebra $k[X|\SS]$. In particular, if $S$ is
empty, then $S$ is a Gröbner basis for $k[X|\SS]=k[X]$ and we have
$\mathsf{Irr}(\SS)=[X]$.

Now we recall the definition of the Gelfand-Kirillov dimensions for
associative algebras. Let $V$ be a finite dimensional subspace of an
associative algebra $\mathcal{A}$ with a linear basis
$\\{a_{1},...\leavevmode\hbox{}\hskip 0.56905pt,a_{r}\\}$. Define $V^{1}=V$,
and for $n\geq 1$, denote by $V^{n}$ the subspace of $\mathcal{A}$ spanned by
all monomials of the form $a_{i_{1}}\cdots a_{i_{n}}$, where
$a_{i_{1}},...\leavevmode\hbox{}\hskip
0.56905pt,a_{i_{n}}\in\\{a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{r}\\}$. Finally, define $V^{\leq n}=V^{1}+\cdots+V^{n}$.

######  Definition 2.1.

[17] Let $\mathcal{A}$ be an associative algebra over a field $k$. Then the
Gelfand-Kirillov dimension of $\mathcal{A}$ is defined to be

$\mathsf{GKdim}(\mathcal{A})=\sup\limits_{V}\overline{\lim\limits_{n\to\infty}}\log_{n}\mathsf{dim}(V^{\leq
n}),$

where the supremum is taken over all finite dimensional subspaces of
$\mathcal{A}$.

It is well-known that if $\mathcal{A}$ is an associative algebra generated by
a finite set $X$, then we have

$\mathsf{GKdim}(\mathcal{A})=\overline{{\lim\limits_{n\to\infty}}}\log_{n}\mathsf{dim}((kX)^{\leq
n}).$

In particular, for a commutative algebra $\mathcal{A}=k[X|\SS]$, where $X$ is
a finite set and $\SS$ is a Gröbner basis for $\mathcal{A}$, we have

$\mathsf{GKdim}(\mathcal{A})=\overline{{\lim\limits_{n\to\infty}}}\log_{n}\\#\\{u\in\mathsf{Irr}(\SS)\mid\ell(u)\leq
n\\}.$

Obviously, for all commutative algebras, their Gelfand-Kirillov dimensions
have very close relations with their Gröbner bases. Now we have a close look
at the case of finitely generated commutative algebras. It is well-known that
every finitely generated commutative algebra has a finite Gröbner basis by
using Hilbert’s Theorem, see also [1, Sect. 1.6].

We recall that for all monomials $u,v\in[X]$, the monomial $u$ is called a
_multiple_ of $v$, denoted by $v|u$, if $u=vw$ in $[X]$ for some monomial
$w\in[X]$. Let $X=\\{x_{1},\dots,x_{r}\\}$ be a nonempty finite set and let
$\SS$ be a finite Gröbner basis for $k[X|S]$. Define

$\displaystyle C_{\SS}=\begin{cases}0,&\mbox{if }\SS=\emptyset\\\
\min\\{n\in\mathbb{N}\mid\bar{s}|x_{1}^{n}\cdots
x_{r}^{n},s\in\SS\\},&\mbox{if }\SS\neq\emptyset.\end{cases}$

In particular, if $S=\\{\varepsilon\\}$, then we have $C_{\SS}=0$. For every
$u=x_{1}^{n_{1}}\cdots x_{r}^{n_{r}}\in[X]$, define

$\mathsf{Sdeg}(u)=\\#\\{i\mid n_{i}>C_{S},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}.$

It follows immediately that we have

$\mathsf{Sdeg}(\varepsilon)=0.$

Finally, for an arbitrary nonempty subset $B\subseteq\mathsf{Irr}(\SS)$, we
define

$\mathsf{Sdeg}(B)=\max\\{\mathsf{Sdeg}(\mu)\mid\mu\in B\\}$

and define

$\mathsf{Sdeg}(\emptyset)=0.$

We shall soon see that $\mathsf{Sdeg}(B)$ is always finite if $X$ is a finite
set.

Now we offer a characterization for the Gelfand-Kirillov dimension of a
finitely generated commutative algebra. In particular, as far as the authors
know, it offers a new alternative proof for the fact that the Gelfand-Kirillov
dimension of a finitely generated commutative algebra is a nonnegative
integer.

######  Lemma 2.2.

Let $X$ be a nonempty finite set and let $S$ be a finite Gröbner basis for
$k[X|\SS]$. Then we have
$\mathsf{GKdim}(k[X|\SS])=\mathsf{Sdeg}(\mathsf{Irr}(\SS))$.

###### Proof.

Assume that $X=\\{x_{1},\dots,x_{r}\\}$. Define
$m=\mathsf{Sdeg}(\mathsf{Irr}(\SS))$. If $m=0$, then we have
$\mathsf{Irr}(\SS)\subseteq\\{x_{1}^{n_{1}}\cdots x_{r}^{n_{r}}\mid n_{i}\leq
C_{S},i\in\\{1,...\leavevmode\hbox{}\hskip 0.56905pt,r\\}\\}$. It follows that
$\mathsf{GKdim}(k[X|S])=0$.

Now we suppose $m\geq 1$. Since $m=\mathsf{Sdeg}(\mathsf{Irr}(\SS))$, for
every $u\in\mathsf{Irr}(\SS)\subseteq[X]$, we obtain that
$\mathsf{Sdeg}(u)\leq m$. Therefore, we have

$\mathsf{Irr}(\SS)\subseteq\\{u\in[X]\mid\mathsf{Sdeg}(u)\leq m\\}.$

Moreover, there exists $u\in\mathsf{Irr}(\SS)$ such that $\mathsf{Sdeg}(u)=m$.
Without loss of generality, we assume $u=x_{1}^{n_{1}}\cdots x_{r}^{n_{r}}$,
where each $n_{i}\in\mathbb{N}$ and $n_{1},\dots,n_{m}>C_{\SS}$. For every
$w\in\mathsf{Irr}(\SS)$ and for every monomial $v\in[X]$ such that $v|w$, it
is clear that $v$ also lies in $\mathsf{Irr}(\SS)$. So we deduce that

$x_{1}^{C_{\SS}+1}\cdots x_{m}^{C_{\SS}+1}\in\mathsf{Irr}(\SS).$

Now we show that

$\mathsf{Irr}(\SS)\supseteq\\{x_{1}^{p_{1}}\cdots x_{m}^{p_{m}}\mid
p_{1},\dots,p_{m}\in\mathbb{N}\\}\setminus\\{\varepsilon\\}.$

It suffices to show that $x_{1}^{n}\cdots x_{m}^{n}$ belongs to
$\mathsf{Irr}(\SS)$ for every $n>C_{\SS}$. Suppose for the contrary that
${x_{1}^{n}\cdots x_{m}^{n}\notin\mathsf{Irr}(\SS)}$ for some integer
$n>C_{\SS}+1$. Then we define

$n_{0}=\mathsf{min}\\{n\mid x_{1}^{n}\cdots
x_{m}^{n}\notin\mathsf{Irr}(\SS)\\}.$

Clearly, we have

$n_{0}>C_{\SS}+1$

and $x_{1}^{n_{0}-1}\cdots x_{m}^{n_{0}-1}\in\mathsf{Irr}(\SS)$. Therefore,
there exists some element $s\in S$ such that we have
$\bar{s}|x_{1}^{n_{0}}\cdots x_{m}^{n_{0}}$ while $\bar{s}\nmid
x_{1}^{n_{0}-1}\cdots x_{m}^{n_{0}-1}$. By the definition of $C_{S}$, we
deduce $C_{S}\geq n_{0}$, which contradicts with the fact that
$n_{0}>C_{\SS}+1$.

Therefore we have

$\\{x_{1}^{p_{1}}\cdots x_{m}^{p_{m}}\mid
p_{1},\dots,p_{m}\in\mathbb{N}\\}\setminus\\{\varepsilon\\}\subseteq\mathsf{Irr}(\SS)\subseteq\\{u\in[X]\mid\mathsf{Sdeg}(u)\leq
m\\}.$

By straightforward calculations, we obtain $\mathsf{GKdim}(k[X|S])=m$
immediately. ∎

######  Theorem 2.3.

Let $X=\\{x_{1},\dots,x_{r}\\}$ be a nonempty finite set and $S$ a finite
Gröbner basis for $k[X|\SS]$. Then we have

$\mathsf{GKdim}(k[X|\SS])=\mathsf{Sdeg}(\\{u\mid u=x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}}\in\mathsf{Irr}(\SS),n_{i}\in\\{0,C_{S}+1\\},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}).$

###### Proof.

Define

$m^{\prime}=\mathsf{Sdeg}(\\{u\mid u=x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}}\in\mathsf{Irr}(\SS),n_{i}\in\\{0,C_{S}+1\\},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\})$

and

$m=\mathsf{Sdeg}(\mathsf{Irr}(\SS)).$

Clearly we have $m^{\prime}\leq m$. By Lemma 2.2, it suffices to show $m\leq
m^{\prime}$.

If $m^{\prime}=0$, then $\\{u\mid u=x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}}\in\mathsf{Irr}(\SS)\setminus\\{\varepsilon\\},n_{i}\in\\{0,C_{S}+1\\},1\leq
i\leq r\\}$ is an empty set. It follows that we have
$\mathsf{Irr}(\SS)\subseteq\\{x_{1}^{n_{1}}\cdots x_{r}^{n_{r}}\mid n_{i}\leq
C_{S},1\leq i\leq r\\}$. Thus we have $m=0=m^{\prime}$.

Suppose $0<m^{\prime}$. Then we have $m\geq m^{\prime}>0$. So there exists a
monomial $u\in\mathsf{Irr}(\SS)$ with $\mathsf{Sdeg}(u)=m>0$. Thus we may
assume

$u=y_{1}^{t_{1}}\cdots y_{m}^{t_{m}}u^{\prime},$

where $u^{\prime}\in[X]$, $\mathsf{Sdeg}(u^{\prime})=0$,
$t_{1},...\leavevmode\hbox{}\hskip 0.56905pt,t_{m}\geq C_{\SS}+1$,
$y_{1},...\leavevmode\hbox{}\hskip 0.56905pt,y_{m}\in X$ and
$y_{1}<\cdots<y_{m}$. Then $y_{1}^{C_{\SS}+1}\cdots y_{m}^{C_{\SS}+1}$ also
lies in $\mathsf{Irr}(\SS)$, in other words, we have
$m^{\prime}\geq\mathsf{Sdeg}(u)=m$.

Therefore, by Lemma 2.2, we obtain

$\mathsf{GKdim}(k[X|S])=m=\mathsf{Sdeg}(\\{u\mid u=x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}}\in\mathsf{Irr}(\SS),n_{i}\in\\{0,C_{S}+1\\},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}).$

The proof is completed. ∎

Theorem 2.3 offers a fast method for calculating the Gelfand-Kirillov
dimension of a finitely presented commutative algebra by investigating the
finite set

$\\{u\in\mathsf{Irr}(\SS)\mid u=x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}},n_{i}\in\\{0,C_{S}+1\\},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}$

rather than $\mathsf{Irr}(\SS)$. As a special case, if
$X=\\{x_{1},\dots,x_{r}\\}$ and $S=\emptyset$, then we have $C_{S}=0$,
$x_{1}^{1}\cdots x_{r}^{1}\in\mathsf{Irr}(S)$, and thus
$\mathsf{GKdim}(k[X])=r$.

######  Remark 2.4.

Theorem 2.3 offers an algorithm for computing the Gelfand-Kirillov dimension
(or equivalently, Krull dimension [4, Proposition 5.1]) of a finitely
presented commutative algebra. More precisely, let $X=\\{x_{1},\dots,x_{r}\\}$
be a nonempty finite set and let $T$ be a finite subset of $k[X]$. Then by
applying Buchberger’s algorithm [1, Algotithm 1.7.1], we can find a finite
Gröbner basis $\SS$ for the algebra $\mathcal{A}:=k[X|T]$. Then we obtain the
nonnegative integer $C_{\SS}$ and the set $B:=\\{u\mid u=x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}}\in\mathsf{Irr}(\SS),n_{i}\in\\{0,C_{S}+1\\},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}$. Since $B$ and $S$ are finite sets, there exists an
algorithm for calculating $\mathsf{Sdeg}(B)$, which is exactly
$\mathsf{GKdim}(\mathcal{A})$.

######  Example 2.5.

Let $X=\\{x_{1},...\leavevmode\hbox{}\hskip 0.56905pt,x_{6}\\}$ and
$\SS=\\{x_{1}^{2}x_{2}^{3},x_{3}x_{4}\\}$. Then $C_{\SS}=3$. For every subset
$\\{y_{1},...\leavevmode\hbox{}\hskip 0.56905pt,y_{5}\\}\subseteq X$, we have
$y_{1}^{4}\cdots y_{5}^{4}\notin\mathsf{Irr}(\SS)$. Since
$x_{2}^{4}x_{3}^{4}x_{5}^{4}x_{6}^{4}$ lies in $\mathsf{Irr}(\SS)$. Thus we
obtain $\mathsf{GKdim}(k[X|\SS])=4$.

######  Corollary 2.6.

Let $\SS$ be a nonempty subset of $[X]^{+}$ and $u\in[X]^{+}$. Suppose that
$\mathsf{GKdim}(k[X|\SS])=p\geq 1$. If there exists a subset
$\\{y_{1},...\leavevmode\hbox{}\hskip 0.56905pt,y_{p}\\}\subseteq X$ such that
some monomial in $\mathsf{Irr}(S)$ of the form $y_{1}^{C_{S}+1}\cdots
y_{p}^{C_{S}+1}$ lies in $\mathsf{Irr}(\SS\cup\\{u\\})$, where $y_{i}\neq
y_{j}$ for all distinct $i,j\leq p$, then we have

$\mathsf{GKdim}(k[X|\SS\cup\\{u\\}])=\mathsf{GKdim}(k[X|\SS]);$

Otherwise, we have

$\mathsf{GKdim}(k[X|\SS\cup\\{u\\}])=\mathsf{GKdim}(k[X|\SS])-1.$

###### Proof.

Obviously, every nonempty subset of $[X]^{+}$ is a Gröbner basis in $k[X]$,
and thus $S$ is a Gröbner basis for $k[X|\SS]$. By assumption, we have

$\mathsf{GKdim}(k[X|\SS])=\mathsf{Sdeg}(\\{x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}}\in\mathsf{Irr}(\SS)\mid n_{i}\in\\{0,C_{S}+1\\},1\leq i\leq
r\\})=p,\mbox{ where }p\geq 1.$

If there exists a subset $\\{y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{p}\\}\subseteq X$, such that some monomial in $\mathsf{Irr}(S)$
of the form $y_{1}^{C_{S}+1}\cdots y_{p}^{C_{S}+1}$ lies in
$\mathsf{Irr}(\SS\cup\\{u\\})$, where $y_{i}\neq y_{j}$ for all distinct
$i,j\leq p$, then we have

$\mathsf{GKdim}(k[X|\SS\cup\\{u\\}])=p=\mathsf{GKdim}(k[X|\SS]).$

Now we assume that for every subset $\\{y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{p}\\}\subseteq X$, every monomial in $\mathsf{Irr}(S)$ of the
form $y_{1}^{C_{S}+1}\cdots y_{p}^{C_{S}+1}$ does not belong to the set
$\mathsf{Irr}(\SS\cup\\{u\\})$, then we have $u|y_{1}^{C_{S}+1}\cdots
y_{p}^{C_{S}+1}$. If $p=1$, then $\\{x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}}\in\mathsf{Irr}(\SS\cup\\{u\\})\mid
n_{i}\in\\{0,C_{S}+1\\},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}=\\{\varepsilon\\}$, and thus

$\mathsf{GKdim}(k[X|\SS\cup\\{u\\}])=0=p-1=\mathsf{GKdim}(k[X|\SS])-1.$

If $p>1$, then without loss of generality, we may assume
$x_{i_{1}}^{C_{S}+1}\cdots x_{i_{p}}^{C_{S}+1}\in\mathsf{Irr}(\SS)$ and
$x_{i_{1}}|u$. It follows that $x_{i_{2}}^{C_{S}+1}\cdots x_{i_{p}}^{C_{S}+1}$
lies in $\mathsf{Irr}(\SS\cup\\{u\\})$. Therefore we obtain that

$\mathsf{GKdim}(k[X|\SS\cup\\{u\\}])=p-1=\mathsf{GKdim}(k[X|\SS])-1.$

The proof is completed. ∎

The following proposition offers a rough estimation of the Gelfand-Kirillov
dimension of a finitely presented commutative algebra.

######  Proposition 2.7.

Let $X=\\{x_{1},\dots,x_{r}\\}$ be a nonempty finite set and let
$S=\\{f_{1},...\leavevmode\hbox{}\hskip 0.56905pt,f_{t}\\}$ be a finite
Gröbner basis for $k[X|\SS]$. Then the following hold:

(i) If $1\leq t\leq r$ and $\varepsilon\notin\SS$, then we have

$\mathsf{GKdim}(k[X|\SS])\in\\{r-t,...\leavevmode\hbox{}\hskip
0.56905pt,r-1\\};$

(ii) If $t>r$, then we have

$\mathsf{GKdim}(k[X|\SS])\in\\{0,...\leavevmode\hbox{}\hskip
0.56905pt,r-1\\}.$

Moreover, if $r>1$ and $t<r$, then
$\mathsf{GKdim}(k[X|\SS])=0\Leftrightarrow\varepsilon\in\SS$.

###### Proof.

(i) We use induction on $t$. If $t=1$, then we may assume that
$\bar{f}_{1}=x_{1}^{n_{1}}\cdots x_{r}^{n_{r}}$, where $n_{i}\in\mathbb{N}$
and $n_{1}>0$. Thus we have $C_{S}=\max\\{n_{1},...\leavevmode\hbox{}\hskip
0.56905pt,n_{r}\\}$. Clearly we have

$x_{2}^{C_{S}+1}\cdots x_{r}^{C_{S}+1}\in\mathsf{Irr}(S)\mbox{ and
}x_{1}^{C_{S}+1}x_{2}^{C_{S}+1}\cdots x_{r}^{C_{S}+1}\notin\mathsf{Irr}(S).$

Therefore by Theorem 2.3, we obtain

$\mathsf{GKdim}(k[X|\SS])=\mathsf{Sdeg}(\\{u\mid u=x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}}\in\mathsf{Irr}(\SS),n_{i}\in\\{0,C_{S}+1\\},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\})=r-1.$

Now we assume $t\geq 2$. Obviously,
$\\{\overline{f_{1}},...\leavevmode\hbox{}\hskip
0.56905pt,\overline{f_{t-1}}\\}$ is a Gröbner basis in $k[X]$ and
$\mathsf{Irr}(\SS)=\mathsf{Irr}(\overline{\SS})$. By induction hypothesis, we
have $r-t+1\geq 1$ and

$p:=\mathsf{GKdim}(k[X|\\{\overline{f_{1}},...\leavevmode\hbox{}\hskip
0.56905pt,\overline{f_{t-1}}\\}])\in\\{r-t+1,...\leavevmode\hbox{}\hskip
0.56905pt,r-1\\}.$

By Corollary 2.6, we have

$\mathsf{GKdim}(k[X|\overline{\SS}])=\mathsf{GKdim}(k[X|\SS])\in\\{p,p-1\\}\subseteq\\{r-t,...\leavevmode\hbox{}\hskip
0.56905pt,r-1\\}.$

(ii) Since $x_{1}^{C_{S}+1}x_{2}^{C_{S}+1}\cdots
x_{r}^{C_{S}+1}\notin\mathsf{Irr}(S)$, by Theorem 2.3, the result follows. ∎

######  Example 2.8.

Let $X=\\{x_{1},\dots,x_{r}\\}$ be a nonempty finite set. Given a positive
integer $t$, for every integer $p$ such that $1\leq p\leq\min\\{t,r\\}$, we
define $m=t-p$ and define

$\SS=\\{x_{i}^{4m+1}\mid 1\leq i\leq p\\}\cup\\{x_{1}^{j}x_{r}^{3m-j}\mid
m\leq j\leq 2m-1\\}.$

Note that if $t=p$, then we have $\SS=\\{x_{i}\mid 1\leq i\leq t\\}$. Clearly
$\SS$ is a Gröbner basis for $k[X|\SS]$ with $\\#\SS=t$ and we have
$C_{S}=4m+1$. Let

$B:=\\{u\in[X]\mid u=x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}},n_{i}\in\\{0,C_{S}+1\\},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}.$

If $p<r$, then we have

$x_{p+1}^{C_{S}+1}\cdots x_{r}^{C_{S}+1}\in B\cap\mathsf{Irr}(\SS),$

and for every monomial $u\in B$ with $\mathsf{Sdeg}(\mu)=r-p+1$, $u$ does not
lie in $\mathsf{Irr}(\SS)$. Thus we obtain

$\mathsf{GKdim}(k[X|\SS])=r-p\in\\{r-t,...\leavevmode\hbox{}\hskip
0.56905pt,r-1\\}.$

If $p=r$, then we have $B\cap\mathsf{Irr}(\SS)=\\{\varepsilon\\}.$ Thus
$\mathsf{GKdim}(k[X|\SS])=r-p=0.$

It follows that if $S$ is a Gröbner basis consisting of $t\geq 1$ elements and
$\varepsilon\notin\SS$, then the possible Gelfand-Kirillov dimension of the
algebra $k[X|S]$ can be an arbitrary integer between $\max\\{r-t,0\\}$ and
$r-1$.

######  Remark 2.9.

In Proposition 2.7, a Gröbner basis $S$ with smaller cardinality might improve
the estimation. So in Proposition 2.7, we may assume $S$ to be a minimal
Gröbner basis. Recall that a Gröbner basis $S$ is called _minimal_ if for all
$f,g\in S$, $\bar{f}$ is not a multiple of $\bar{g}$. Moreover, when the order
is fixed, the cardinality of a minimal Gröbner basis is also minimum among all
the possible cardinalities of Gröbner bases for $k[X|S]$. If we use another
monomial order $\prec$ on $[X]$, then we can find a minimal Gröbner basis $T$
for $k[X|S]$ with respect to the order $\prec$ and probably we have
$\\#T\neq\\#S$.

##  3\. Gröbner–Shirshov bases theory for bicommutative algebras

Our aim in this section is to establish a Gröbner–Shirshov bases theory for
bicommutative algebras and show that every finitely generated bicommutative
algebra has a finite Gröbner–Shirshov basis.

###  3.1. Composition-Diamond lemma for bicommutative algebras

We first recall that _terms_ over $X$ are “words with brackets”; they are
defined by the following induction:

(i) Every element of $X$ is a term over $X$;

(ii) If $\mu,\nu$ are terms over $X$, then so is $(\mu\nu)$.

We denote by $X^{(*)}$ the set of all terms over $X$.

Let $kX^{(*)}$ be the free nonassociative algebra generated by $X$. Then
clearly $X^{(*)}$ forms a linear basis for $kX^{(*)}$. For every $\mu$ in
$X^{(*)}$, we define the _content_ $\mathsf{cont}(\mu)$ of $\mu$ to be the set
of all letters that appear in $\mu$ and define the _length_ $\ell(\mu)$ of
$\mu$ to be the number of letters (with repetition) that appear in $\mu$. For
instance, for $\mu=((x_{1}x_{1})x_{2})\in X^{(\ast)}$, we have
$\mathsf{cont}(\mu)=\\{x_{1},x_{2}\\}$ and $\ell(\mu)=3$.

For all $a_{1},...\leavevmode\hbox{}\hskip 0.56905pt,a_{n}$ ($n\geq 1$) in a
bicommutative algebra $\mathcal{B}$, we define

$[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{n}]{{}_{{}_{\mathsf{L}}}}=(\cdots(((a_{1}a_{2})a_{3})\cdots
a_{n-1})a_{n})\ \ \mbox{(left-normed bracketing),}$

and define

$[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{n}]{{}_{{}_{\mathsf{R}}}}=(a_{1}(a_{2}\cdots(a_{n-2}(a_{n-1}a_{n}))\cdots))\
\ \mbox{(right-normed bracketing).}$

In particular, we have
$[a_{1}]_{\mathsf{{}_{L}}}=[a_{1}]_{\mathsf{{}_{R}}}=a_{1}$ and
$[a_{1},a_{2}]_{\mathsf{{}_{L}}}=[a_{1},a_{2}]_{\mathsf{{}_{R}}}=(a_{1}a_{2})$.
For all elements $a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{p},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{q}\in\mathcal{B}$, where $p,q\in\mathbb{N}^{+}$, we define

$[a_{1}\cdots a_{p};b_{1}\cdots b_{q}]=[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{p-1},[a_{p},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}.$

Now we recall some known identities in bicommutative algebras established in
[15, 14], see also [23].

######  Lemma 3.1.

_[15, 14]_ For all $a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{p},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{q},c_{1},...\leavevmode\hbox{}\hskip
0.56905pt,c_{m},d_{1},...\leavevmode\hbox{}\hskip
0.56905pt,d_{n}\in\mathcal{B}$, for all permutations $\sigma\in
S_{p},\delta\in\SS_{q}$, where $p,q,m,n\in\mathbb{N}^{+}$, and
$\SS_{p},\SS_{q}$ are symmetric groups of orders $p,q$ respectively, we have

(i) $((a_{1}(a_{2}a_{3}))a_{4})=(a_{1}((a_{2}a_{3})a_{4}))$,

(ii) $[a_{1}\cdots a_{p};b_{1}\cdots b_{q}]=[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{p-1},[a_{p},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}=[{[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{p},b_{1}]{{}_{{}_{\mathsf{R}}}},b_{2},...\leavevmode\hbox{}\hskip
0.56905pt,b_{q}}]{{}_{{}_{\mathsf{L}}}}$. In particular, we have $[a_{1}\cdots
a_{p};b_{1}\cdots b_{q}]=[a_{\sigma(1)}\cdots
a_{\sigma(p)};b_{\delta(1)}\cdots b_{\delta(q)}]$.

(iii) $([a_{1}\cdots a_{p};b_{1}\cdots b_{q}][c_{1}\cdots c_{m};d_{1}\cdots
d_{n}])=[a_{1}\cdots a_{p}c_{1}\cdots c_{m};b_{1}\cdots b_{q}d_{1}\cdots
d_{n}]$.

Let $\mathsf{BC}(X)$ be the free bicommutative algebra generated by a well-
ordered set $X$. Note that by Lemma 3.1, the ordering of $a_{i}$ and that of
$b_{j}$ do not make difference in the monomial $[a_{1}\cdots a_{p};b_{1}\cdots
b_{q}]$, and what is important is whether they are on the “left hand side” or
on the “right hand side”. In particular, for all $p,q\in\mathbb{N}^{+}$, for
all letters $z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},y_{1},...\leavevmode\hbox{}\hskip 0.56905pt,y_{q}\in X$, when
we write an element of the form $[z_{1}\cdots z_{p};y_{1}\cdots
y_{q}]\in\mathsf{BC}(X)$, it is not necessary to assume $z_{1}\leq\cdots\leq
z_{p}$ or $y_{1}\leq\cdots\leq y_{q}$.

Moreover, for all $u=z_{1}\cdots z_{p},v=y_{1}\cdots y_{q}\in[X]^{+}$, we
define

$[u;v]=[z_{1}\cdots z_{p};y_{1}\cdots y_{q}],$

and

$\mathsf{NF}(X)=\\{[u;v]\mid u,v\in[X]^{+}\\}\cup X.$

Then by [14], $\mathsf{NF}(X)$ forms a linear basis for $\mathsf{BC}(X)$. For
the convenience of the readers, we list the multiplication table below. For
all $z,y\in X$ and $[u;v],[u^{\prime};v^{\prime}]\in\mathsf{NF}(X)\setminus
X$, we have

(3.1) $(zy)=[z;y];\ (z[u;v])=[zu;v];\ ([u;v]z)=[u;vz];\
([u;v][u^{\prime};v^{\prime}])=[uu^{\prime};vv^{\prime}].$

In light of the multiplication table (3.1) above, we can generalize the notion
of multiple in $\mathsf{NF}(X)$. More precisely, for all
$\mu,\nu\in\mathsf{NF}(X)$, we call $\nu$ a _multiple_ of $\mu$, denoted by
$\mu|\nu$, if one of the followings holds: (i) $\mu=x\in X$ and
$x\in\mathsf{cont}(\nu)$; (ii) $\mu=[u;v]$,
$\nu=[u^{\prime};v^{\prime}]\in\mathsf{NF}(X)$ for some
$u,v,u^{\prime},v^{\prime}\in[X]^{+}$, and we have $u|u^{\prime}$,
$v|v^{\prime}$ in $[X]^{+}$.

Let $<$ be a well order on $\mathsf{NF}(X)$. For every nonzero polynomial
$f=\sum\alpha_{i}\mu_{i}$ in $\mathsf{BC}(X)$, where each $0\neq\alpha_{i}$
lies in $k$, each $\mu_{i}$ lies in $\mathsf{NF}(X)$ and
$\mu_{1}>\mu_{2}>\cdots$, we call $\mu_{1}$ the _leading monomial_ of $f$,
denoted by $\bar{f}$; and call $\alpha_{1}$ the _leading coefficient_ of $f$,
denoted by $\mathsf{lc}(f)$. We call $f$ a _monic polynomial_ if
$\mathsf{lc}(f)=1$. Finally, a set $\SS\subseteq\mathsf{BC}(X)$ is called a
_monic set_ if every polynomial in $\SS$ is monic. By convention, the empty
set $S$ is a monic set but $\\{0\\}$ is not.

A well order $<$ on $\mathsf{NF}(X)$ is called a _monomial order_ if for all
monomials $\mu,\nu,\nu^{\prime}\in\mathsf{NF}(X)$, we have
$\mu<\nu\Rightarrow\overline{(\mu\nu^{\prime})}<\overline{(\nu\nu^{\prime})}\mbox{
and }\overline{(\nu^{\prime}\mu)}<\overline{(\nu^{\prime}\nu)}$. Here we offer
two specific examples of monomial order on $\mathsf{NF}(X)$.

######  Example 3.2.

Let $\prec$ be a monomial order on $[X]^{+}$. For every
$\mu\in\mathsf{NF}(X)$, we define

$\mathsf{wt}(\mu)=\begin{cases}(1,1,x),&\text{if}\ \mu=x\in X,\\\
(\ell(u)+\ell(v),\ell(u),u,v)&\text{if}\ \mu=[u;v]\in\mathsf{NF}(X)\setminus
X.\end{cases}$

For all $\mu,\nu\in\mathsf{NF}(X)$, we define

$\mu<\nu\text{ if and only if }\mathsf{wt}(\mu)<\mathsf{wt}(\nu)\text{
lexicographically},$

where the elements in $[X]^{+}$ are compared by $\prec$. Then clearly $<$ is a
monomial order on $\mathsf{NF}(X)$.

######  Example 3.3.

Let $\prec$ be a monomial order on $[X]^{+}$. For every
$\mu\in\mathsf{NF}(X)$, we define

$\mathsf{wt}(\mu)=\begin{cases}(x,x,x),&\text{if}\ \mu=x\in X,\\\
(uv,u,v)&\text{if}\ \mu=[u;v]\in\mathsf{NF}(X)\setminus X,\end{cases}$

where $uv$ is a commutative word in $[X]^{+}$. For all
$\mu,\nu\in\mathsf{NF}(X)$, we define

$\mu<\nu\text{ if and only if }\mathsf{wt}(\mu)<\mathsf{wt}(\nu)\text{
lexicographically},$

where the elements in $[X]^{+}$ are compared by $\prec$. Then clearly $<$ is a
monomial order on $\mathsf{NF}(X)$.

From now on, let $<$ be a fixed monomial order on $\mathsf{NF}(X)$ and let
$\SS$ be a monic subset of $\mathsf{BC}(X)$. Denote by $\mathsf{Id}(\SS)$ the
ideal of $\mathsf{BC}(X)$ generated by $\SS$. For every $s\in S$, we call $s$
an  _$S$ -polynomial_, and for every $S$-polynomial $h$, for every $\mu$ in
$\mathsf{NF}(X)$, both $(\mu h)$ and $(h\mu)$ are called  _$S$ -polynomials_.
Obviously every element in $\mathsf{Id}(\SS)$ can be written as a linear
combination of $S$-polynomials. Among $S$-polynomials, we introduce some
special polynomials in $\mathsf{Id}(\SS)$, which will form a linear generating
set of $\mathsf{Id}(\SS)$ under some conditions.

######  Definition 3.4.

Let $\SS$ be a monic set in $\mathsf{BC}(X)$. For every $s\in\SS$,
(i) the polynomial $s$ is called a _normal $\SS$-polynomial_;
(ii) if $\ell(\bar{s})\geq 2$, then for all $z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},y_{1},...\leavevmode\hbox{}\hskip 0.56905pt,y_{q}\in X$,
$p,q\in\mathbb{N}$, we call $[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[s,y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$ a _normal
$\SS$-polynomial_.

The following lemma shows that the set:

$\mathsf{Irr}(S):=\mathsf{NF}(X)\setminus\\{\mu\in\mathsf{NF}(X)\mid\mu=\overline{h},h\mbox{
is a normal }\SS\mbox{-polynomial}\\}$

is a linear generating set of the bicommutative algebra
$\mathsf{BC}(X|\SS):=\mathsf{BC}(X)/\mathsf{Id}(\SS)$.

######  Lemma 3.5.

For every $f\in\mathsf{BC}(X)$, we have

$f=\sum_{\mu_{i}\leq\bar{f}}\alpha_{i}\mu_{i}+\sum_{\overline{h_{j}}\leq\bar{f}}\beta_{j}h_{j},$

where each $\alpha_{i},\beta_{j}\in k$, $\mu_{i}\in\mathsf{Irr}(\SS)$ and
$h_{j}$ is a normal $\SS$-polynomial. In particular, it follows that
$\mathsf{Irr}(\SS)$ is a set of linear generators of the algebra
$\mathsf{BC}(X|\SS)$.

###### Proof.

The result follows by induction on $\bar{f}$. ∎

######  Definition 3.6.

Let $I$ be an ideal of $\mathsf{BC}(X)$. A monic set $S$ is called a _Gröbner-
Shirshov basis_ (in $\mathsf{BC}(X)$) for $\mathsf{BC}(X)/I$, if
$\mathsf{Id}(\SS)=I$ and for every nonzero polynomial $f\in\mathsf{Id}(\SS)$,
we have $\overline{f}=\overline{h}$ for some normal $\SS$-polynomial $h$.

We shall see that, if $\SS$ is a Gröbner–Shirshov basis for
$\mathsf{BC}(X|\SS)$, then $\mathsf{Irr}(\SS)$ forms a linear basis for
$\mathsf{BC}(X|\SS)$. In particular, if $S$ is an empty set, then $S$ is a
Gröbner–Shirshov basis for $\mathsf{BC}(X|\SS)$ and we have
$\mathsf{Irr}(\SS)=\mathsf{NF}(X)$. But the conditions in Definition 3.7 is
not tractable in practice. Therefore, we shall introduce certain special
polynomials (hereafter called compositions) in $\mathsf{Id}(\SS)$, such that
if these polynomials satisfy some easily managed conditions, then $S$ forms a
Gröbner–Shirshov basis.

Before going there, we introduce the notion of least common multiple in
$\mathsf{NF}(X)\setminus X$. For all $u,v\in[X]$, we denote by
$\mathsf{lcm}(u,v)$ the least common multiple of $u$ and $v$ in $[X]$. For all
$\mu=[u;v],\nu=[u^{\prime};v^{\prime}]\in\mathsf{NF}(X)$, where
$u,v,u^{\prime},v^{\prime}$ lie in $[X]^{+}$, we call
$[\mathsf{lcm}(u,u^{\prime});\mathsf{lcm}(v,v^{\prime})]$ the _least common
multiple_ of $\mu$ and $\nu$, and denote it by $\mathsf{lcm}(\mu,\nu)$.

######  Definition 3.7.

Let $\SS$ be a monic subset of $\mathsf{BC}(X)$. For all $f,g\in\SS$, we
define _compositions_ of $S$ as follows:

(i) If $\ell(\overline{f})=1$, then for every $z\in X$, we call $(zf)$ a _left
multiplication composition_ and $(fz)$ a _right multiplication composition_ ;

(ii) If $\overline{f}=\overline{g}\in X$, then we call
$(f,g)_{\overline{f}}=f-g$ an _inclusion composition_ ;

(iii) If $\ell(\overline{f})\geq 2$ and
$f\notin\mathsf{span}_{k}(\mathsf{NF}(X)\setminus X)$, where
$\mathsf{span}_{k}(\mathsf{NF}(X)\setminus X)$ is the subspace of
$\mathsf{BC}(X)$ spanned by $\mathsf{span}_{k}(\mathsf{NF}(X)\setminus X)$,
then for all $y,z\in X$, we call

$(f,f)_{\overline{(y(fz))}}=(y(fz))-((yf)z)$

a _multiplication composition_ ;

(iv) If
$\bar{f}=[u,v],\bar{g}=[u^{\prime},v^{\prime}]\in\mathsf{NF}(X)\setminus X$,
and $\ell(\mathsf{lcm}(\bar{f},\bar{g}))<\ell(\bar{f})+\ell(\bar{g})$, where
$u,v,u^{\prime},v^{\prime}\in[X]^{+}$,
$\mathsf{lcm}(u,u^{\prime})=ua_{1}\cdots a_{p}=u^{\prime}b_{1}\cdots b_{m}$,
$\mathsf{lcm}(v,v^{\prime})=vc_{1}\cdots c_{q}=v^{\prime}d_{1}\cdots d_{n}$,
and $a_{1},...\leavevmode\hbox{}\hskip 0.56905pt,a_{p}$,
$b_{1},...\leavevmode\hbox{}\hskip 0.56905pt,b_{m}$,
$c_{1},...\leavevmode\hbox{}\hskip 0.56905pt,c_{q}$,
$d_{1},...\leavevmode\hbox{}\hskip 0.56905pt,d_{n}\in X$,
$p,q,m,n\in\mathbb{N}$, then we call

$(f,g)_{\mathsf{lcm}(\bar{f},\bar{g})}=[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{p},[f,c_{1},...\leavevmode\hbox{}\hskip
0.56905pt,c_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}-[b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{m},[g,d_{1},...\leavevmode\hbox{}\hskip
0.56905pt,d_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$

an _intersection composition_.

A composition $h$ in (i) of the form $(zf)$ or $(fz)$ is called _trivial_
(modulo $\SS$) if $h$ can be written as a linear combination of normal
$\SS$-polynomials such that their leading monomials are $\leq\overline{h}$.
And a composition $h$ in (ii)-(iv) of the form $(f_{1},f_{2})_{\mu}$ is called
_trivial_ (modulo $\SS$ with respect to $\mu$) if $h$ can be written as a
linear combination of normal $\SS$-polynomials such that their leading
monomials are $<\mu$. Finally, for every $h\in\mathsf{BC}(X)$ and
$\mu\in\mathsf{NF}(X)$, the polynomial $h$ is said to be _trivial modulo $\SS$
with respect to $\mu$_, denoted by

$h\equiv 0\ \mod(\SS,\mu),$

if $h$ can be written as a linear combination of normal $\SS$-polynomials such
that their leading monomials are $<\mu$;

Now we show that, under certain conditions, we can rewrite some
$S$-polynomials into normal $S$-polynomials.

######  Lemma 3.8.

Let $\SS$ be a monic subset of $\mathsf{BC}(X)$ such that all the left
multiplication compositions, right multiplication compositions and
multiplication compositions of $\SS$ are trivial. Then the followings hold:

(i) Every $\SS$-polynomial $h$ can be written as a linear combination of
normal $S$-polynomials whose leading monomials are $\leq\overline{h}$.

(ii) If $h=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[s,y_{1},y_{2},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$, where
$s\in\SS$, $\ell(\overline{s})\geq 2$, $z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},y_{1},...\leavevmode\hbox{}\hskip 0.56905pt,y_{q}\in X$ and
$p,q\in\mathbb{N}^{+}$, then we have $h-[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},[z_{p},s,y_{1},y_{2},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}\equiv 0\
\mod(\SS,\overline{h})$.

###### Proof.

(i) Since every element in $S$ is a normal $S$-polynomial, it suffices to show
that, for every normal $S$-polynomial $h$, for every $\mu\in\mathsf{NF}(X)$,
the polynomial $(\mu h)$ (resp. $(h\mu)$) can be written as a linear
combination of normal $\SS$-polynomials whose leading monomials are less than
or equal to $\overline{(\mu h)}$ (resp. $\overline{(h\mu)}$).

Assume that $h$ is a normal $\SS$-polynomial. We use induction on $\ell(\mu)$
to prove (i). Suppose that $\ell(\mu)=1$ and $\mu=x\in X$. If
$\ell(\bar{h})=1$, that is, $h=s\in\SS$ and $\ell(\bar{s})=1$, then we have
$(\mu h)=(xs)$ and $(h\mu)=(sx)$. Since all left multiplication compositions
and right multiplication compositions of $\SS$ are trivial, there is nothing
to prove. If $\ell(\bar{h})>1$, then we may assume

$h=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[s,y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}},$

where $z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},y_{1},...\leavevmode\hbox{}\hskip 0.56905pt,y_{q}\in X$ and
$p,q\in\mathbb{N}$. Thus $(\mu h)=(xh)=[x,z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[s,y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$ is already a
normal $\SS$-polynomial. Moreover, since $<$ is a monomial order, it follows
immediately that, for an arbitrary normal $S$-polynomial $f$, for all
$x_{i_{1}},...\leavevmode\hbox{}\hskip 0.56905pt,x_{i_{t}}\in X$, the
polynomial $[x_{i_{1}},...\leavevmode\hbox{}\hskip
0.56905pt,x_{i_{t}},f]{{}_{{}_{\mathsf{R}}}}$ can be written as a linear
combination of normal $\SS$-polynomials whose leading monomials are less than
or equal to $\overline{[x_{i_{1}},...\leavevmode\hbox{}\hskip
0.56905pt,x_{i_{t}},f]{{}_{{}_{\mathsf{R}}}}}$.

If $q=p=0$ or $q>0$, then by Lemma 3.1(ii), we have

$(h\mu)=(sx)\mbox{ or }(h\mu)=(hx)=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[s,y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q},x]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}},$

both of which are already normal $\SS$-polynomials. Note that every
multiplication composition of $\SS$ is trivial. If $q=0$ and $p>0$, then we
may assume

$((z_{p}s)x)-(z_{p}(sx))=\sum_{i}\alpha_{i}h_{i},$

where each $\alpha_{i}$ belongs to $k$ and each $h_{i}$ is a normal
$\SS$-polynomial with $\overline{h_{i}}<\overline{((z_{p}s)x)}$. By Lemma
3.1(ii) again, we obtain

$\displaystyle(h\mu)$ $\displaystyle=([z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},[z_{p},s]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}x)$
$\displaystyle=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},[z_{p},s,x]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},((z_{p}s)x)]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},(z_{p}(sx))]{{}_{{}_{\mathsf{R}}}}+\sum_{i}\alpha_{i}[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},h_{i}]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[s,x]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}+\sum\gamma_{t}h^{\prime\prime}_{t},$

where $[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[s,x]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$ is a
normal $\SS$-polynomial satisfying
$\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[s,x]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}=\overline{(h\mu)}$,
and each $\gamma_{t}$ lies in $k$, each $h^{\prime\prime}_{t}$ is a normal
$\SS$-polynomial satisfying

$\overline{h^{\prime\prime}_{t}}\leq\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},h_{i}]{{}_{{}_{\mathsf{R}}}}}<\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},((z_{p}s)z)]{{}_{{}_{\mathsf{R}}}}}=\overline{(hx)}=\overline{(h\mu)}.$

Now suppose that $\ell(\mu)>1$ and $\mu=(\mu_{1}\mu_{2})$, where
$\mu_{1},\mu_{2}\in\mathsf{NF}(X)$. Since $\ell(\mu_{1})<\ell(\mu)$, by
induction hypothesis, we may assume

$(\mu
h)=((\mu_{1}\mu_{2})h)=((\mu_{1}h)\mu_{2})=\sum_{i}\alpha_{i}(h_{i}\mu_{2}),$

where each $\alpha_{i}$ belongs to $k$, each $h_{i}$ is a normal
$\SS$-polynomial satisfying $\overline{h_{i}}\leq\overline{(\mu_{1}h)}$.
Similarly we have $(h_{i}\mu_{2})=\sum_{j}\beta_{ij}h^{\prime}_{ij}$, where
each $\beta_{ij}\in k$, each $h^{\prime}_{ij}$ is a normal $\SS$-polynomial
with
$\overline{h^{\prime}_{ij}}\leq\overline{(h_{i}\mu_{2})}\leq\overline{(\overline{(\mu_{1}h)}\mu_{2})}=\overline{((\mu_{1}h)\mu_{2})}=\overline{(\mu
h)}$.

Finally, note that $(h\mu)=(h(\mu_{1}\mu_{2}))=(\mu_{1}(h\mu_{2}))$, the claim
follows immediately.

(ii) We use induction on $p+q$. If $p=q=1$, then since every multiplication
composition of $\SS$ is trivial, we have

$[z_{1},[s,y_{1}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}-[z_{1},s,y_{1}]{{}_{{}_{\mathsf{L}}}}\equiv
0\mod(\SS,\overline{[z_{1},[s,y_{1}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}).$

In particular, if $p+q>2$, then we assume

$((z_{p}(sy_{1}))-((z_{p}s)y_{1}))=\sum_{i}\beta_{i}h_{i}^{\prime},$

where each $\beta_{i}$ lies in $k$ and each $h_{i}^{\prime}$ is a normal
$S$-polynomial satisfying
$\overline{h_{i}^{\prime}}\leq\overline{((z_{p}(sy_{1}))}$. By Lemma 3.1, we
have

$\displaystyle h-[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},[((z_{p}s)y_{1}),y_{2},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle=$ $\displaystyle[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},[(z_{p}(sy_{1})),y_{2},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}-[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},[((z_{p}s)y_{1}),y_{2},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle=$ $\displaystyle[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},[((z_{p}(sy_{1}))-((z_{p}s)y_{1})),y_{2},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle=$
$\displaystyle\sum_{i}\beta_{i}[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},[h^{\prime}_{i},y_{2},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}.$

Since $<$ is a monomial order, the claim follows by (i). ∎

The following lemma says that, under certain conditions, two normal
$S$-polynomials with the same leading monomial are not of significant
difference.

######  Lemma 3.9.

Let $\SS$ be a monic subset of $\mathsf{BC}(X)$ such that all compositions of
$S$ are trivial, and let $h_{1}$, $h_{2}$ be two normal $\SS$-polynomials. If
$\overline{h_{1}}=\overline{h_{2}}$, then we have

$h_{1}-h_{2}\equiv 0\mod(\SS,\overline{h_{1}}).$

###### Proof.

If $\ell(\overline{h_{1}})=1=\ell(\overline{h_{2}})$, then we may assume
$h_{1}=s_{1}$ and $h_{2}=s_{2}$, where $s_{1},s_{2}\in\SS$. Since every
inclusion composition of $\SS$ is trivial, we have $h_{1}-h_{2}\equiv
0\mod(\SS,\overline{h_{1}})$. Now we suppose $\ell(\overline{h_{1}})>1$. Then
we may assume

$h_{1}=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{m},[s_{1},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}},$
$h_{2}=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[[a^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a^{\prime}_{m^{\prime}},[s_{2},b^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b^{\prime}_{n^{\prime}}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}},$

where $s_{1},s_{2}\in\SS$, $\ell(\overline{s_{1}}),\ell(\overline{s_{2}})>1$,
$z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q},a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{m},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{n},a^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a^{\prime}_{m^{\prime}},b^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b^{\prime}_{n^{\prime}}$ lie in $X$,
$p,q,n,m,n^{\prime},m^{\prime}\in\mathbb{N}$ and
$\\{a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{m}\\}\cap\\{a^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a^{\prime}_{m^{\prime}}\\}=\\{b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{n}\\}\cap\\{b^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b^{\prime}_{n^{\prime}}\\}=\emptyset$. In particular, we deduce

$\mathsf{lcm}(\overline{s_{1}},\overline{s_{2}})=\overline{[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{m},[s_{1},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}=\overline{[a^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a^{\prime}_{m^{\prime}},[s_{2},b^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b^{\prime}_{n^{\prime}}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}.$

There are two cases to consider. If $n+m<\ell(\overline{s_{2}})$, then we have

$(s_{1},s_{2})_{\mathsf{lcm}(\overline{s_{1}},\overline{s_{2}})}=[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{m},[s_{1},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}-[a^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a^{\prime}_{m^{\prime}},[s_{2},b^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b^{\prime}_{n^{\prime}}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}=\sum_{j}\beta_{j}h^{\prime}_{j},$

where each $\beta_{j}$ lies in $k$, each $h^{\prime}_{j}$ is a normal
$\SS$-polynomial satisfying
$\overline{h^{\prime}_{j}}<\mathsf{lcm}(\overline{s_{1}},\overline{s_{2}})$.
So we obtain

$h_{1}-h_{2}=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[(s_{1},s_{2})_{\mathsf{lcm}(\overline{s_{1}},\overline{s_{2}})},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}=\sum_{j}\beta_{j}[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[h^{\prime}_{j},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}.$

By Lemma 3.8(i), each $[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[h^{\prime}_{j},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$ can be written
as a linear combination of normal $\SS$-polynomials such that their leading
monomials are

$\leq\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[h^{\prime}_{j},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}<\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[\mathsf{lcm}(\overline{s_{1}},\overline{s_{2}}),y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}=\overline{h_{1}}.$

Therefore, we have $h_{1}-h_{2}\equiv 0\mod(\SS,\overline{h_{1}})$.

If $n+m=\ell(\overline{s_{2}})$, then we deduce
$n,m,n^{\prime},m^{\prime}\in\mathbb{N}^{+}$,
$\overline{s_{2}}=[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{m-1},[a_{m},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$ and
$\overline{s_{1}}=[a^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a^{\prime}_{m^{\prime}-1},[a^{\prime}_{m^{\prime}},b^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b^{\prime}_{n^{\prime}}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$.
Thus we have

$(s_{1}\overline{s_{2}})=[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{m},[s_{1},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}},\
(\overline{s_{1}}s_{2})=[a^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a^{\prime}_{m^{\prime}},[s_{2},b^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b^{\prime}_{n^{\prime}}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$

and

$h_{1}=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[s_{1},\overline{s_{2}},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}.$

By Lemma 3.8(ii), we have

$\displaystyle h_{2}$ $\displaystyle=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},a^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a^{\prime}_{m^{\prime}},[s_{2},b^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b^{\prime}_{n^{\prime}},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},a^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a^{\prime}_{m^{\prime}-1},[a^{\prime}_{m^{\prime}},s_{2},b^{\prime}_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b^{\prime}_{n^{\prime}},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}+\sum_{i}\alpha_{i}h^{\prime}_{i}$
$\displaystyle=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[\overline{s_{1}},s_{2},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}+\sum_{i}\alpha_{i}h^{\prime}_{i},$

where each $\alpha_{i}$ lies in $k$, and each $h^{\prime}_{i}$ is a normal
$\SS$-polynomial such that $\overline{h^{\prime}_{i}}<\overline{h_{2}}$.
Finally we obtain

$\displaystyle h_{1}-h_{2}$ $\displaystyle=$
$\displaystyle[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[s_{1},\overline{s_{2}},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}-[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[\overline{s_{1}},s_{2},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}-\sum_{i}\alpha_{i}h^{\prime}_{i}$
$\displaystyle=$ $\displaystyle[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[s_{1},\overline{s_{2}},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}+[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[s_{1},s_{2},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle-[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[s_{1},s_{2},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}-[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[\overline{s_{1}},s_{2},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}-\sum_{i}\alpha_{i}h^{\prime}_{i}$
$\displaystyle=$ $\displaystyle[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[s_{1},(\overline{s_{2}}-s_{2}),y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle+[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[(s_{1}-\overline{s_{1}}),s_{2},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}-\sum_{i}\alpha_{i}h^{\prime}_{i}.$

Since we have

$\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[s_{1},(\overline{s_{2}}-s_{2}),y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}<\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[s_{1},\overline{s_{2}},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}=\overline{h_{1}}$

and

$\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[(s_{1}-\overline{s_{1}}),s_{2},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}<\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p-1},z_{p},[\overline{s_{1}},s_{2},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}=\overline{h_{1}},$

the lemma follows by Lemma 3.8(i). ∎

Now we are ready to prove the Composition-Diamond lemma for bicommutative
algebras.

######  Theorem 3.10.

_(_ Composition-Diamond lemma for bicommutative algebras _)_ Let $\SS$ be a
monic subset of $\mathsf{BC}(X)$. Let $\mathsf{Id}(\SS)$ be the ideal of
$\mathsf{BC}(X)$ generated by $\SS$. Then the followings are equivalent.

(i) All the compositions of $\SS$ are trivial.

(ii) The set $\SS$ is a Gröbner–Shirshov basis for
$\mathsf{BC}(X)/\mathsf{Id}(\SS)$, that is, for every nonzero polynomial
$f\in\mathsf{Id}(\SS)$, we have $\overline{f}=\overline{h}$ for some normal
$\SS$-polynomial $h$.

(iii) The set
$\mathsf{Irr}(\SS)=\mathsf{NF}(X)\setminus\\{\mu\in\mathsf{NF}(X)\mid\mu=\overline{h},h\mbox{
is a normal }\SS\mbox{-polynomial}\\}$ forms a linear basis of the algebra
$\mathsf{BC}(X|\SS):=\mathsf{BC}(X)/\mathsf{Id}(\SS)$.

###### Proof.

(i) $\Rightarrow$ (ii) For every nonzero element $f$ in $\mathsf{Id}(\SS)$, by
Lemma 3.8(i), we may assume that $f=\sum_{i=1}^{n}\alpha_{i}h_{i}$, where each
$\alpha_{i}\in k,\alpha_{i}\neq 0$ and each $h_{i}$ is a normal
$\SS$-polynomial. Define $\mu_{i}=\overline{h_{i}}$ and assume that
$\mu_{1}=\mu_{2}=\cdots=\mu_{{}_{l}}>\mu_{l+1}\geq\cdots.$ Use induction on
$\mu_{1}$. If $\mu_{1}=\overline{f}$, then there is nothing to prove. If
$\mu_{1}>\overline{f}$, then we have $\sum_{i=1}^{l}\alpha_{i}=0$ and

$f=\sum_{i=1}^{l}\alpha_{i}h_{1}-\sum_{i=1}^{l}\alpha_{i}(h_{1}-h_{i})+\sum_{i=l+1}^{n}\alpha_{i}h_{i}=\sum_{j}\beta_{j}h_{j}^{\prime}+\sum_{i=l+1}^{n}\alpha_{i}h_{i},$

where each $\overline{h_{j}^{\prime}}<\mu_{1}$ by Lemma 3.9. Claim (ii)
follows by induction hypothesis.

(ii) $\Rightarrow$ (iii) By Lemma 3.5, $\mathsf{Irr}(\SS)$ generates
$\mathsf{BC}(X|\SS)$ as a vector space. Suppose that
$\sum_{i}\alpha_{i}\mu_{i}=0$ in $\mathsf{BC}(X|\SS)$, where each
$\alpha_{i}\in k\setminus\\{0\\}$ and each $\mu_{i}\in\mathsf{Irr}(\SS)$,
$\mu_{1}>\mu_{2}>\mathrel{\leavevmode\hbox{}\hskip
0.56905pt{\cdot}{\cdot}{\cdot}\leavevmode\hbox{}\hskip 0.56905pt}$. Then we
have $0\neq\sum_{i}\alpha_{i}\mu_{i}\in\mathsf{Id}(\SS)$. Thus
$\overline{\sum_{i}\alpha_{i}\mu_{i}}=\mu_{1}\in\mathsf{Irr}(\SS)$, which
contradicts with (ii).

(iii) $\Rightarrow$ (i) Every composition of $\SS$ is an element in
$\mathsf{Id}(\SS)$. By Lemma 3.5 and (iii), we obtain (i). ∎

###  3.2. Finite Gröbner-Shirshov bases

Our aim in this subsection is to show that every finitely generated
bicommutative algebra has a finite Gröbner–Shirshov basis.

######  Definition 3.11.

Let $\SS$ be a Gröbner–Shirshov basis for $\mathsf{BC}(X|\SS)$. Then we call
$\SS$ a _minimal Gröbner–Shirshov basis_ , if for every $s\in\SS$, we have
$\overline{s}\in\mathsf{Irr}(S\setminus\\{s\\})$.

The following lemma is a general fact for various kind of algebras, but we
still offer a proof for the convenience of the readers.

######  Lemma 3.12.

Let $I$ be an ideal of $\mathsf{BC}(X)$ and $\SS$ a Gröbner–Shirshov basis for
$\mathsf{BC}(X)/I$. For every $T\subseteq S$, if
$\mathsf{Irr}(T)=\mathsf{Irr}(\SS)$ then $T$ is also a Gröbner–Shirshov basis
for $\mathsf{BC}(X)/I$.

###### Proof.

For every $f\in I$, since $\mathsf{Irr}(T)=\mathsf{Irr}(\SS)$ and $\SS$ is a
Gröbner–Shirshov basis for $\mathsf{BC}(X)/I$, we have
$\overline{f}=\overline{h}=\overline{h^{\prime}}$ for some normal
$S$-polynomial $h$ and for some normal $T$-polynomial $h^{\prime}$. So we
obtain $f_{1}=f-\mathsf{lc}(f)h^{\prime}\in I$ and
$\overline{f_{1}}<\overline{f}$. By induction on $\overline{f}$, we deduce
that $f$ can be written as a linear combination of normal $T$-polynomials, in
particular, we have $f\in\mathsf{Id}(T)$. This shows that $I=\mathsf{Id}(T)$.
So the result follows from Theorem 3.10. ∎

Let $I$ be an ideal of $\mathsf{BC}(X)$. Then the set $\\{f\in
I\mid\mathsf{lc}(f)=1\\}$ forms a Gröbner-Shirshov basis for
$\mathsf{BC}(X)/I$, so Gröbner-Shirshov bases always exist. Now we show that
minimal Gröbner–Shirshov bases always exist.

######  Lemma 3.13.

Let $S$ be a Gröbner–Shirshov basis for $\mathsf{BC}(X)/\mathsf{Id}(S)$. Then
there exists a set $T\subseteq S$ such that $T$ is a minimal Gröbner–Shirshov
basis for the algebra $\mathsf{BC}(X)/\mathsf{Id}(S)$.

###### Proof.

Without loss of generality, assume that $S$ is a nonempty set. For each
$\mu\in\overline{\SS}:=\\{\overline{s}\in\mathsf{NF}(X)\mid s\in S\\}$, we fix
an arbitrary polynomial $f_{\mu}$ in $\SS$ such that $\overline{f_{\mu}}=\mu$.
Define

$\SS^{\prime}=\\{f_{\mu}\in\SS\mid\mu\in\overline{\SS}\\}.$

Then it is clear that $\SS\supseteq\SS^{\prime}$ and
$\mathsf{Irr}(\SS^{\prime})=\mathsf{Irr}(\SS)$. By Lemma 3.12, $\SS^{\prime}$
is a Gröbner–Shirshov basis for $\mathsf{BC}(X)/\mathsf{Id}(S)$. Define

$\SS_{1}=\\{s\in\SS^{\prime}\mid\ell(\bar{s})=1\\},\
\SS_{2}=\\{s\in\SS^{\prime}\mid\ell(\bar{s})>1\\}.$

Then we have $\SS^{\prime}=\SS_{1}\cup\SS_{2}$.

Let $\mu_{0}=\mathsf{min}\overline{\SS_{2}}$ be the minimal element in
$\overline{\SS_{2}}$. Define
$\SS_{\mu_{0}}=\\{s\in\SS_{2}\mid\bar{s}=\mu_{0}\\}$ and define
$\SS_{\mu}=\emptyset$ for every $\mu<\mu_{0}$ in $\mathsf{NF}(X)\setminus X$.
Suppose that for every $\nu<\mu$ in $\mathsf{NF}(X)\setminus X$, the set
$\SS_{\nu}$ has already been defined. Then we define

$\SS_{\mu}=\begin{cases}\cup_{\nu<\mu}\SS_{\nu}\ \ \ \ \ \ \ \ \ \ \ \ \
\text{ if
$\mu\not\in\mathsf{Irr}(\cup_{\nu<\mu}\SS_{\nu})\cap\overline{S}$,}\\\
\cup_{\nu<\mu}\SS_{\nu}\cup\\{f_{\mu}\\}\ \ \ \ \text{ if
$\mu\in\mathsf{Irr}(\cup_{\nu<\mu}\SS_{\nu})\cap\overline{S}$}.\end{cases}$

Define

$\SS^{\prime\prime}=\SS_{1}\cup(\bigcup_{\mu\in\mathsf{NF}(X)}\SS_{\mu}).$

We claim that $\SS^{\prime\prime}$ is a minimal Gröbner-Shirshov basis for
$\mathsf{BC}(X)/\mathsf{Id}(\SS)$. By Lemma 3.12, it suffices to show
$\mathsf{Irr}(\SS^{\prime\prime})=\mathsf{Irr}(\SS^{\prime})$. Since
$\SS^{\prime\prime}\subseteq\SS^{\prime}$, it suffices to show
$\mathsf{Irr}(\SS^{\prime\prime})\subseteq\mathsf{Irr}(\SS^{\prime})$, or
equivalently,
$\mathsf{NF}(X)\setminus\mathsf{Irr}(\SS^{\prime})\subseteq\mathsf{NF}(X)\setminus\mathsf{Irr}(\SS^{\prime\prime})$.
Assume that
$\mu=\overline{h^{\prime}}\in\mathsf{NF}(X)\setminus\mathsf{Irr}(\SS^{\prime})$
and assume that $h^{\prime}$ is a normal $\SS^{\prime}$-polynomial such that

$h^{\prime}=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[s^{\prime},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}},$

where $s^{\prime}\in\SS^{\prime}$, $z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},y_{1},...\leavevmode\hbox{}\hskip 0.56905pt,y_{q}\in X$ and
$p,q\in\mathbb{N}$. If $s^{\prime}\in\SS^{\prime\prime}$, then
$\mu\in\mathsf{NF}(X)\setminus\mathsf{Irr}(\SS^{\prime\prime})$. If
$s^{\prime}\in S^{\prime}\setminus S^{\prime\prime}$, then by the construction
of $S^{\prime\prime}$ we deduce $\ell(\overline{s^{\prime}})>1$ and
$\overline{s^{\prime}}=\overline{h^{\prime\prime}}$ for some normal
$\SS^{\prime\prime}$-polynomial $h^{\prime\prime}$. So we may assume

$h^{\prime\prime}=[a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{n},[s^{\prime\prime},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{m}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}},$

where $a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{n},b_{1},...\leavevmode\hbox{}\hskip 0.56905pt,b_{m}\in X$ and
$m,n\in\mathbb{N},m+n\in\mathbb{N}^{+}$, $s^{\prime\prime}\in
S^{\prime\prime}$ and $\ell(\overline{s^{\prime\prime}})\geq 2$. Set

$f=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},[h^{\prime\prime},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}.$

If $m\in\mathbb{N}^{+}$, then $f=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{n},[s^{\prime\prime},b_{1},...\leavevmode\hbox{}\hskip
0.56905pt,b_{m},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$ is a normal
$\SS^{\prime\prime}$-polynomial. Therefore we obtain
$\mu=\bar{f}\in\mathsf{NF}(X)\setminus\mathsf{Irr}(\SS^{\prime\prime})$. If
$m=0$, then since
$s^{\prime\prime}\in\SS^{\prime\prime}\subseteq\SS^{\prime}$, $\SS^{\prime}$
is a Gröbner–Shirshov basis in $\mathsf{BC}(X)$ and Lemma 3.8(ii) holds, we
have

$\displaystyle f$ $\displaystyle=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{n-1},[a_{n},s^{\prime\prime},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$
$\displaystyle=[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{n},[s^{\prime\prime},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}+\sum_{j}\alpha_{j}h_{j},$

where each $\alpha_{j}\in k$, each $h_{j}$ is a normal
$\SS^{\prime}$-polynomial and $\overline{h_{j}}<\bar{f}$. Therefore we have

$\mu=\bar{f}=\overline{[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{p},a_{1},...\leavevmode\hbox{}\hskip
0.56905pt,a_{n},[s^{\prime\prime},y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{q}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}}\in\mathsf{NF}(X)\setminus\mathsf{Irr}(\SS^{\prime\prime}).$

By Lemma 3.12, $\SS^{\prime\prime}$ is a Gröbner–Shirshov basis for
$\mathsf{BC}(X)/\mathsf{Id}(S)$. Moreover, by the construction of
$S^{\prime\prime}$, we obtain that $\SS^{\prime\prime}$ is a minimal
Gröbner–Shirshov basis for $\mathsf{BC}(X)/\mathsf{Id}(S)$. ∎

As a corollary of the Hilbert’s Theorem, see for example [1, Sect.1.6], every
finitely generated commutative algebra has a finite Gröbner basis.

######  Lemma 3.14.

Let $U$ be an infinite subset of $[X]^{+}$, where $X$ is a finite set. Then
there exists an infinite subset $\\{u_{i}\in U\mid i\in\mathbb{N}^{+}\\}$ of
$U$ such that $u_{i+1}$ is a multiple of $u_{i}$ for every
$i\in\mathbb{N}^{+}$.

###### Proof.

Let $\mathsf{Id}(U)$ be the ideal of $k[X]$ generated by $U$ and let $k[X|U]$
be the quotient algebra $k[X]/\mathsf{Id}(U)$. By Hilbert’s Theorem, there
exists a finite set $U_{1}\subset U$ that generates $\mathsf{Id}(U)$, namely,
we have $\mathsf{Id}(U_{1})=\mathsf{Id}(U)$. Moreover, it is easy to see that
$U_{1}$ is a Gröbner basis for $k[X]/\mathsf{Id}(U)$. Since $U$ is infinite,
there exists a monomial $u_{1}\in U_{1}$ such that infinite many multiples of
$u_{1}$ lies in $U$. Define $V_{2}=\\{u\in U\mid u_{1}|u,u\neq u_{1}\\}$ and
let $U_{2}$ be a finite subset of $V_{2}$ such that $V_{2}$ is a Gröbner basis
for $k[X|V_{2}]$. Then we can choose an element $u_{2}\in U_{2}$ such that
there are infinitely many multiples of $u_{2}$ in $V_{2}$. Continue this
process, we shall obtain the desired infinite set. ∎

######  Theorem 3.15.

Let $X$ be a finite set and let $I$ be an ideal of $\mathsf{BC}(X)$. Then
there exists a finite Gröbner–Shirshov basis for $\mathsf{BC}(X)/I$.

###### Proof.

By Lemma 3.13, we may assume that $\SS$ is a minimal Gröbner–Shirshov basis
for $\mathsf{BC}(X)/I$. We shall prove that $S$ is finite. Suppose for the
contrary that $S$ is an infinite set.

Define $\SS_{1}=\\{s\in\SS\mid\ell(\bar{s})=1\\},\
{\SS_{2}=\\{s\in\SS\mid\ell(\bar{s})>1\\}}$. Then we have
$\SS=\SS_{1}\cup\SS_{2}$. Since $\SS$ is a minimal Gröbner–Shirshov basis for
$\mathsf{BC}(X)/I$ and $X$ is a finite set, we obtain that $\SS_{1}$ is a
finite set while $\SS_{2}$ is infinite. Define

$L=\\{u\in[X]^{+}\mid\mbox{ there exists }v\in[X]^{+}\mbox{ such that
}[u;v]\in\overline{\SS_{2}}\\}$

and

$R=\\{v\in[X]^{+}\mid\mbox{ there exists }u\in[X]^{+}\mbox{ such that
}[u;v]\in\overline{\SS_{2}}\\}.$

Without loss of generality, we assume that $L$ is an infinite set. By
Corollary 3.14, there exists an infinite set

$L_{1}=\\{u_{i}\in L\mid i\in\mathbb{N}^{+}\\}$

such that $u_{i+1}$ is a multiple of $u_{i}$ for every $i\in\mathbb{N}^{+}$.
Define

$\SS^{\prime}=\\{[u;v]\mid u\in L_{1},v\in[X]^{+},[u;v]\in\overline{\SS}\\}.$

Obviously, the set $\SS^{\prime}$ is infinite. Define

$R_{1}=\\{v\in[X]^{+}\mid\mbox{ there exists }u\in[X]^{+}\mbox{ such that
}[u;v]\in\SS^{\prime}\\}.$

If $R_{1}$ is finite, then for some monomial $v\in R_{1}$, there exist some
monmomials $u_{i},u_{j}\in L_{1}$ such that
$[u_{i};v],[u_{j};v]\in\SS^{\prime}\subseteq\overline{\SS}$ and $u_{i}|u_{j}$.
Thus $[u_{j};v]$ is a multiple of $[u_{i};v]$, which contradicts to the
assumption that $\SS$ is a minimal Gröbner–Shirshov basis for
$\mathsf{BC}(X)/I$.

If $R_{1}$ is infinite, then by Corollary 3.14 there exists an infinite set
$R_{2}=\\{v_{i}\in R_{1}\mid i\in\mathbb{N}^{+}\\}$ such that $v_{i+1}$ is a
multiple of $v_{i}$ in $[X]^{+}$ for every $i\in\mathbb{N}^{+}$. We claim that
for every monomial $u_{i}$ in $L_{1}$, there exists at most one monomial
$v_{j_{i}}\in R_{2}$ such that $[u_{i};v_{j_{i}}]\in S^{\prime}$. For
otherwise, say $[u_{i};v_{p}],[u_{i};v_{q}]\in S^{\prime}\subseteq\bar{\SS}$
and $p<q$, then we have $[u_{i};v_{p}]|[u_{i};v_{q}]$, which contradicts to
the assumption that $S$ is a minimal Gröbner–Shirshov basis. Similarly, for
every monomial $v_{i}$ in $R_{2}$, there exists exactly one monomial
$u_{j_{i}}\in L_{1}$ such that $[u_{j_{i}};v_{i}]\in S^{\prime}$. Assume
$[u_{p};v_{1}]\in\SS^{\prime}\subseteq\overline{\SS}$. Since
$\\{u_{1},...\leavevmode\hbox{}\hskip 0.56905pt,u_{p}\\}$ is a finite set, by
the above reasoning, for some large enough integer $m>1$, there exists some
element $[u_{n};v_{m}]\in S^{\prime}\subseteq\bar{\SS}$ such that $n>p$.
Therefore $[u_{n};v_{m}]$ is a multiple of $[u_{p};v_{1}]$, which contradicts
to the assumption that $\SS$ is a minimal Gröbner–Shirshov basis for
$\mathsf{BC}(X)/I$. ∎

As a corollary, we obtain an alternative proof of the following results.

######  Corollary 3.16.

[12] Finitely generated bicommutative algebras are weakly noetherian, i.e.,
satisfy the ascending chain condition for two-sided ideals.

######  Corollary 3.17.

The word problem of an arbitrary finitely presented bicommutative algebra is
solvable.

We conclude this section with a few comments on the Gröbner-Shirshov bases
theory we established for bicommutative algebras. In Definition 3.7(i), when
$\bar{f}$ is a letter, certainly we can define
$[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{m},[f,y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$ and
$[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{m-1},[z_{m},f,y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$ to be normal
$S$-polynomials, where $z_{1},...\leavevmode\hbox{}\hskip 0.56905pt,y_{n}\in
X$. But then many of the proofs in the article will become much more lengthy
with boring discussions due to this alternative way of defining normal
$S$-polynomials. And in our way, there is also some disadvantage. For example,
let $S=\\{f\\}$ and $\bar{f}=x\in X$. Then one naturally expects $S$ to be a
Gröbner–Shirshov basis. However, in our way, this is not the case. We have to
add all the relations of the form $[f;y]$ and $[y;f]$ for every $y\in X$ to
$S$ in order to form a Gröbner–Shirshov basis. We accept this defect because
adding these polynomials just means that every time we can rewrite $\bar{f}$
into $\bar{f}-f$ in its appearance in $[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{m},[f,y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$ and
$[z_{1},...\leavevmode\hbox{}\hskip
0.56905pt,z_{m-1},[z_{m},f,y_{1},...\leavevmode\hbox{}\hskip
0.56905pt,y_{n}]{{}_{{}_{\mathsf{L}}}}]{{}_{{}_{\mathsf{R}}}}$, which is in
fact the same as the ordinary way of dealing with Gröbner–Shirshov basis.

##  4\. Gelfand-Kirillov dimension of bicommutative algebras

Our aim in this section is to prove the main result of this article, namely,
the Gelfand-Kirillov dimension of an arbitrary finitely generated
bicommutative algebra is a nonnegative integer. The method we applied here is
the Gröbner-Shirshov bases theory for bicommutative algebras, which is
established in Section 3.1. Moreover, we show that, in order to calculate the
Gelfand-Kirillov dimension of a finitely generated bicommutative algebra, it
suffices to investigate some property of a finite set instead of constructing
a possibly infinite linear basis for the bicommutative algebra under
consideration.

We begin with some notation. Let $V,V_{1}$ and $V_{2}$ be vector subspaces of
a bicommutative algebra $\mathcal{B}$. We define

$(V_{1}V_{2})=\mathsf{span}_{k}\\{(xy)\mid x\in V_{1},y\in V_{2}\\}.$

Then we define $V^{1}=V$ and $V^{n}=\sum_{1\leq i\leq n-1}(V^{i}V^{n-i})$ for
every $n\geq 2$. Finally, we define

$V^{\leq n}=V^{1}+V^{2}+\mathrel{\leavevmode\hbox{}\hskip
0.56905pt{\cdot}{\cdot}{\cdot}\leavevmode\hbox{}\hskip 0.56905pt}+V^{n}.$

Let $\mathcal{B}$ be a bicommutative algebra. Then the Gelfand-Kirillov
dimension of $\mathcal{B}$ is defined to be

$\mathsf{GKdim}(\mathcal{B})=\sup\limits_{V}\overline{\lim\limits_{n\to\infty}}\log_{n}\mathsf{dim}(V^{\leq
n}),$

where the supremum is taken over all finite dimensional subspaces of
$\mathcal{B}$. It follows immediately that, for a bicommutative algebra
$\mathcal{B}=\mathsf{BC}(X|\SS)$, where $X$ is a finite set and $\SS$ is a
Gröbner–Shirshov basis for $\mathsf{BC}(X|\SS)$, we have

$\mathsf{GKdim}(\mathcal{B})=\overline{{\lim\limits_{n\to\infty}}}\log_{n}\\#\\{\mu\in\mathsf{Irr}(\SS)\mid\ell(\mu)\leq
n\\}.$

Let $X=\\{x_{1},\dots,x_{r}\\}$ be a nonempty finite set and let $\SS$ be a
finite Gröbner–Shirshov basis for $\mathsf{BC}(X|\SS)$. Define

$\displaystyle N_{\SS}=\begin{cases}1,&\mbox{if }\SS=\emptyset\\\
\mathsf{min}\\{n\mid\bar{s}|[x_{1}^{n}\cdots x_{r}^{n};x_{1}^{n}\cdots
x_{r}^{n}],s\in\SS\\},&\mbox{if }\SS\neq\emptyset.\end{cases}$

For every $x\in X$, we define

$\mathsf{Sdeg}(x)=0,$

and for every monomial $\mu=[x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots x_{r}^{m_{r}}]\in\mathsf{NF}(X)\setminus X$,
we define

$\mathsf{Sdeg}(\mu)=\\#\\{i\mid
n_{i}>N_{\SS},i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}+\\#\\{i\mid
m_{i}>N_{\SS},i\in\\{1,...\leavevmode\hbox{}\hskip 0.56905pt,r\\}\\}.$

Finally, for an arbitrary nonempty set $B\subseteq\mathsf{Irr}(\SS)$, we
define

$\mathsf{Sdeg}(B)=\max\\{\mathsf{Sdeg}(\mu)\mid\mu\in B\\}$

and define

$\mathsf{Sdeg}(\emptyset)=0.$

We shall soon see that $\mathsf{Sdeg}(B)$ is always finite if $X$ is a finite
set. Now we investigate the Gelfand-Kirillov dimension of a finitely generated
bicommutative algebra.

######  Lemma 4.1.

Let $X$ be a nonempty finite set and let $\SS$ be a finite Gröbner–Shirshov
basis for $\mathsf{BC}(X|S)$. Then we have
$\mathsf{GKdim}(\mathsf{BC}(X|S))=\mathsf{Sdeg}(\mathsf{Irr}(\SS))$.

###### Proof.

Assume $X=\\{x_{1},\dots,x_{r}\\}$. Define
$M=\mathsf{Sdeg}(\mathsf{Irr}(\SS))$. If $M=0$, then we have
$\mathsf{Irr}(\SS)\subseteq X\cup\\{[x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots x_{r}^{m_{r}}]\in\mathsf{NF}(X)\setminus
X\mid n_{i},m_{j}\leq N_{S},1\leq i,j\leq r\\}$. It follows that
$\mathsf{GKdim}(\mathsf{BC}(X|S))=0=\mathsf{Sdeg}(\mathsf{Irr}(\SS)\setminus
X)$.

Now we suppose $M\geq 1$. Since $M=\mathsf{Sdeg}(\mathsf{Irr}(\SS))$, for
every $\mu\in\mathsf{Irr}(\SS)\setminus X\subseteq\mathsf{NF}(X)\setminus X$,
we have $\mathsf{Sdeg}(\mu)\leq M$. Therefore, we have

$\mathsf{Irr}(\SS)\subseteq\\{\mu\in\mathsf{NF}(X)\mid\mathsf{Sdeg}(\mu)\leq
M\\}.$

It follows immediately that we have

(4.1) $\mathsf{GKdim}(\mathsf{BC}(X|\SS))\leq M.$

Moreover, since $\mathsf{Sdeg}(\mathsf{Irr}(S)\setminus X)=M$, there exists
some monomial $\mu\in\mathsf{Irr}(\SS)\setminus X$ such that
$\mathsf{Sdeg}(\mu)=M$. Without loss of generality, we assume
$\mu=[x_{1}^{n_{1}}\cdots x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots x_{r}^{m_{r}}]$,
where $n_{i_{1}},\dots,n_{i_{p}},m_{j_{1}},\dots,m_{j_{q}}>N_{\SS}$, $p+q=M$,
$p,q\in\mathbb{N}$, $n_{l},m_{l^{\prime}}\in\\{0,1\\}$ for all
$l\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\setminus\\{i_{1},...\leavevmode\hbox{}\hskip
0.56905pt,i_{p}\\}$ and $l^{\prime}\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\setminus\\{j_{1},...\leavevmode\hbox{}\hskip
0.56905pt,j_{q}\\}$. There are three cases to consider.

If $q=0$, then we have $p=M$. Without loss of generality, we may assume

$\mu=[x_{1}^{n_{1}}\cdots x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots x_{r}^{m_{r}}],$

where ${n_{1},\dots,n_{{}_{M}}>N_{\SS}}$, and there exists some integer
$t\in\\{1,...,r\\}$ such that $m_{t}\geq 1$. For every monomial
$\nu\in\mathsf{Irr}(\SS)$ and for every monomial
$\nu^{\prime}\in\mathsf{NF}(X)$ with $\nu^{\prime}|\nu$, it is clear that
$\nu^{\prime}$ also lies in $\mathsf{Irr}(\SS)$. So we deduce

$[x_{1}^{N_{\SS}+1}\cdots x_{{}_{M}}^{N_{\SS}+1};x_{t}]\in\mathsf{Irr}(\SS).$

Now we show that

$\mathsf{Irr}(\SS)\supseteq\\{[x_{1}^{l_{1}}\cdots
x_{{}_{M}}^{l_{{}_{M}}};x_{t}]\in\mathsf{NF}(X)\setminus X\mid
l_{1},\dots,l_{{}_{M}}\geq 0\\}.$

It suffices to show that $[x_{1}^{n}\cdots x_{{}_{M}}^{n};x_{t}]$ belongs to
$\mathsf{Irr}(\SS)$ for every $n\geq N_{\SS}+1$. Suppose for the contrary that
$[x_{1}^{n}\cdots x_{{}_{M}}^{n};x_{t}]\notin\mathsf{Irr}(\SS)$ for some
$n>N_{\SS}+1$. Then we define

$n_{0}=\mathsf{min}\\{n\mid[x_{1}^{n}\cdots
x_{{}_{M}}^{n};x_{t}]\notin\mathsf{Irr}(\SS)\\}.$

Clearly, we have

$n_{0}>N_{\SS}+1$

and $[x_{1}^{n_{0}-1}\cdots x_{{}_{M}}^{n_{0}-1};x_{t}]\in\mathsf{Irr}(\SS)$.
Therefore, there exists some element $s\in\SS$ such that we have
$\bar{s}|[x_{1}^{n_{0}}\cdots x_{{}_{M}}^{n_{0}};x_{t}]$ and
$\bar{s}\nmid[x_{1}^{n_{0}-1}\cdots x_{{}_{M}}^{n_{0}-1};x_{t}]$. By the
definition of $N_{\SS}$, we deduce $N_{\SS}\geq n_{0}$, which contradicts the
fact that $n_{0}>N_{\SS}+1$. So we have

$\\{[x_{1}^{l_{1}}\cdots
x_{{}_{M}}^{l_{{}_{M}}};x_{t}]\in\mathsf{NF}(X)\setminus X\mid
l_{1},\dots,l_{{}_{M}}\geq 0\\}\subseteq\mathsf{Irr}(\SS).$

Therefore, if $q=0$, then there exist $M$ distinct letters
$x_{i_{1}},...\leavevmode\hbox{}\hskip 0.56905pt,x_{i_{M}}\in X$, and
$x_{t}\in X$ such that

(4.2) $\\{[x_{i_{1}}^{l_{1}}\cdots
x_{i_{M}}^{l_{{}_{M}}};x_{t}]\in\mathsf{NF}(X)\setminus X\mid
l_{1},\dots,l_{{}_{M}}\geq 0\\}\subseteq\mathsf{Irr}(S).$

If $p=0$, then similarly there exist $M$ distinct letters
$x_{i_{1}},...\leavevmode\hbox{}\hskip 0.56905pt,x_{i_{M}}\in X$, and
$x_{t}\in X$ such that

(4.3) $\\{[x_{t};x_{i_{1}}^{l_{1}}\cdots
x_{i_{M}}^{l_{{}_{M}}}]\in\mathsf{NF}(X)\setminus X\mid
l_{1},\dots,l_{{}_{M}}\geq 0\\}\subseteq\mathsf{Irr}(\SS).$

If $p,q\geq 1$, then by a similar proof as for the case when $q=0$, we can
show that

(4.4) $\\{[x_{i_{1}}^{n_{i_{1}}}\cdots
x_{i_{p}}^{n_{i_{p}}};x_{j_{1}}^{m_{j_{1}}}\cdots
x_{j_{q}}^{m_{j_{q}}}]\in\mathsf{NF}(X)\setminus X\mid
n_{i_{1}},...\leavevmode\hbox{}\hskip 0.56905pt,m_{j_{q}}\geq
0\\}\subseteq\mathsf{Irr}(\SS).$

Combining (4.2),(4.3) and (4.4), we have
$\mathsf{GKdim}(\mathsf{BC}(X|\SS))\geq M$. Finally, by (4.1), we obtain
$\mathsf{GKdim}(\mathsf{BC}(X|\SS))=M$. ∎

######  Theorem 4.2.

Let $X=\\{x_{1},\dots,x_{r}\\}$ be a nonempty finite set and let
$\mathcal{B}=\mathsf{BC}(X|\SS)$, where $\SS$ is a finite Gröbner–Shirshov
basis for $\mathsf{BC}(X|S)$. Then we have
$\mathsf{GKdim}(\mathcal{B})=\mathsf{Sdeg}(\\{\mu\mid\mu=[x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots x_{r}^{m_{r}}]\in\mathsf{Irr}(\SS)\setminus
X,n_{i},m_{j}\in\\{0,1,N_{\SS}+1\\},i,j\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}).$ In particular, the Gelfand-Kirillov dimension of a
finitely generated bicommutative algebra is a nonnegative integer.

###### Proof.

Define

$M^{\prime}=\mathsf{Sdeg}(\\{\mu\mid\mu=[x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots x_{r}^{m_{r}}]\in\mathsf{Irr}(\SS)\setminus
X,n_{i},m_{j}\in\\{0,1,N_{\SS}+1\\},1\leq i,j\leq r\\})$

and

$M=\mathsf{Sdeg}(\mathsf{Irr}(\SS)).$

Clearly we have $M^{\prime}\leq M$. By Lemma 4.1, it suffices to show $M\leq
M^{\prime}$.

If $M^{\prime}=0$, then we have

$\mathsf{Irr}(\SS)\subseteq X\cup\\{[x_{1}^{p_{1}}\cdots
x_{r}^{p_{r}};x_{1}^{q_{1}}\cdots x_{r}^{q_{r}}]\in\mathsf{NF}(X)\setminus
X\mid p_{i},q_{j}\leq N_{\SS},1\leq i,j\leq r\\}.$

Then it follows immediately that we have
$M=\mathsf{GKdim}(\mathsf{BC}(X|\SS))=0=M^{\prime}$.

If $M^{\prime}>0$, then we have $M\geq M^{\prime}>0$, and thus there exists
some monomial $\nu=[x_{1}^{n_{1}}\cdots x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots
x_{r}^{m_{r}}]\in\mathsf{Irr}(\SS)\setminus X$ satisfying
$\mathsf{Sdeg}(\nu)=M>0$. By the construction of $\mathsf{Irr}(S)$, every
monomial $\nu^{\prime}$ satisfying $\nu^{\prime}|\nu$ lies in
$\mathsf{Irr}(S)$. For every $i\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}$, we define

$\displaystyle n^{\prime}_{i}=\begin{cases}N_{S}+1,&\mbox{if }n_{i}>N_{S},\\\
1,&\mbox{if }n_{i}\in\\{1,...\leavevmode\hbox{}\hskip 0.56905pt,N_{S}\\},\\\
0,&\mbox{if }n_{i}=0;\end{cases}\ \ \ \ \ \ \ \ \
m^{\prime}_{i}=\begin{cases}N_{S}+1,&\mbox{if }m_{i}>N_{S},\\\ 1,&\mbox{if
}m_{i}\in\\{1,...\leavevmode\hbox{}\hskip 0.56905pt,N_{S}\\},\\\ 0,&\mbox{if
}m_{i}=0.\end{cases}$

Then it follows that we have

$\mathsf{Sdeg}(\nu^{\prime})=M$

and

$\nu^{\prime}=[x_{1}^{n_{1}^{\prime}}\cdots
x_{r}^{n_{r}^{\prime}};x_{1}^{m_{1}^{\prime}}\cdots x_{r}^{m_{r}^{\prime}}]\in
B,$

where

$B:=\\{\mu\mid\mu=[x_{1}^{n_{1}}\cdots x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots
x_{r}^{m_{r}}]\in\mathsf{Irr}(\SS)\setminus
X,n_{i},m_{j}\in\\{0,1,N_{\SS}+1\\},i,j\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}.$

Therefore we have $M^{\prime}\geq\mathsf{Sdeg}(\nu)=M$. The proof is
completed. ∎

Theorem 4.2 offers a fast algorithm for calculating the Gelfand-Kirillov
dimension of a finitely presented bicommutative algebra by investigating the
finite set

$\\{\mu\mid\mu=[x_{1}^{n_{1}}\cdots x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots
x_{r}^{m_{r}}]\in\mathsf{Irr}(\SS)\setminus
X,n_{i},m_{j}\in\\{0,1,N_{\SS}+1\\},i,j\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}$

rather than $\mathsf{Irr}(\SS)$.

######  Example 4.3.

Let $X=\\{x_{1},\dots,x_{r}\\}$ be a nonempty finite set. Then the Gelfand-
Kirillov dimension of the free bicommutative algebra $\mathsf{BC}(X)$
generated by $X$ is $2r$.

###### Proof.

As a special case of Theorem 4.2, say $X=\\{x_{1},\dots,x_{r}\\}$ and
$\SS=\emptyset$, then we have $N_{S}=1$ and $[x_{1}^{2}\cdots
x_{r}^{2};x_{1}^{2}\cdots x_{r}^{2}]\in\mathsf{Irr}(\SS)$, and thus
$\mathsf{GKdim}(\mathsf{BC}(X))=2r$. ∎

######  Corollary 4.4.

Let $X=\\{x_{1},\dots,x_{r}\\}$ be a nonempty finite set. Let $\SS$ be a
nonempty finite subset of $\mathsf{NF}(X)\setminus X$ and
$\mu\in\mathsf{NF}(X)\setminus X$. Suppose that
$\mathsf{GKdim}(\mathsf{BC}(X|\SS))=n\geq 3$. Let

$B=\\{\mu^{\prime}\mid\mu^{\prime}=[x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots x_{r}^{m_{r}}]\in\mathsf{NF}(X)\setminus
X,n_{i},m_{j}\in\\{0,1,N_{\SS}+1\\},i,j\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}.$

If there exists some monomial $\nu\in B\cap\mathsf{Irr}(\SS\cup\\{\mu\\})$
with $\mathsf{Sdeg}(\nu)=n$, then we have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS\cup\\{\mu\\}))=\mathsf{GKdim}(\mathsf{BC}(X|\SS));$

Otherwise, we have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS\cup\\{\mu\\}))=\mathsf{GKdim}(\mathsf{BC}(X|\SS))-1.$

###### Proof.

Obviously, every nonempty subset of $\mathsf{NF}(X)\setminus X$ is a
Gröbner–Shirshov basis in $\mathsf{BC}(X)$ and thus $S$ is a Gröbner–Shirshov
basis for $\mathsf{BC}(X|S)$. Then we have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS))=\mathsf{Sdeg}(B\cap\mathsf{Irr}(\SS))=n$

for some positive integer $n\geq 3$.

If there exists some monomial $\nu\in B\cap\mathsf{Irr}(\SS\cup\\{\mu\\})$
with $\mathsf{Sdeg}(\nu)=n$, then we have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS\cup\\{\mu\\}))=n=\mathsf{GKdim}(\mathsf{BC}(X|\SS)).$

If for every monomial $\nu^{\prime}\in B\cap\mathsf{Irr}(S)$ with
$\mathsf{Sdeg}(\nu^{\prime})=n$, $\nu^{\prime}$ does not lie in
$\mathsf{Irr}(\SS\cup\\{\mu\\})$, then we have $\mu|\nu^{\prime}$. Suppose
$\nu_{0}\in B\cap\mathsf{Irr}(S)$ and $\mathsf{Sdeg}(\nu_{0})=n$. Then we may
assume

$\nu_{0}=[x_{1}^{n_{1}}\cdots x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots
x_{r}^{m_{r}}]\in\mathsf{Irr}(\SS)\setminus X,$

where $n_{i_{1}}=\cdots=n_{i_{p}}=m_{j_{1}}=\cdots=m_{j_{q}}=N_{\SS}+1$,
$p+q=n$, $p,q\in\mathbb{N}$, $n_{l},m_{l^{\prime}}\in\\{0,1\\}$ for all
$l\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\setminus\\{i_{1},...\leavevmode\hbox{}\hskip
0.56905pt,i_{p}\\}$ and $l^{\prime}\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\setminus\\{j_{1},...\leavevmode\hbox{}\hskip
0.56905pt,j_{q}\\}$. Since $n\geq 3$, we have $p>1$ or $q>1$. Without loss of
generality, we assume $p>1$ and suppose $i_{1}=1$, $i_{2}=2$ and
$[x_{1};x_{t}]|\mu$ for some $x_{t}\in X$. Then $[x_{2}^{n_{2}}\cdots
x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots
x_{r}^{m_{r}}]\in\mathsf{Irr}(\SS\cup\\{\mu\\})\setminus X$. Therefore, we
have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS\cup\\{\mu\\}))=n-1=\mathsf{GKdim}(\mathsf{BC}(X|\SS))-1.$

The proof is completed. ∎

If we have $\mathsf{GKdim}(\mathsf{BC}(X|S))=n>0$, then in general we cannot
have a good estimation on $\mathsf{GKdim}(\mathsf{BC}(X|S\cup\\{x\\}))$, where
$x$ lies in $X$. For instance, let $X=\\{x_{1},...\leavevmode\hbox{}\hskip
0.56905pt,x_{r}\\}\ (r>1)$ and let $S=\\{[x;y]\mid x,y\in X,x\neq x_{1}\\}$.
Then we have $N_{S}=1$ and
$[x_{1}^{2};x_{1}^{2}...x_{r}^{2}]\in\mathsf{Irr}(S)$. Clearly, for every
$\mu\in\mathsf{Irr}(S)$, we have $\mathsf{Sdeg}(\mu)\leq r+1$, so
$\mathsf{GKdim}(\mathsf{BC}(X|S))=r+1$. Moreover, we have
$\mathsf{GKdim}(\mathsf{BC}(X|S\cup\\{x_{1}\\}))=0$ and
$\mathsf{GKdim}(\mathsf{BC}(X|S\cup\\{x_{2}\\}))=r$.

The following example shows that if $\mathsf{GKdim}(\mathsf{BC}(X|\SS))=2$,
then we cannot obtain a similar result as in Corollary 4.4.

######  Example 4.5.

Let $X=\\{x_{1},x_{2}\\}$ and
$\SS=\\{[x_{1};x_{1}],[x_{1};x_{2}],[x_{2};x_{1}]\\}$. Then we obtain that
$N_{S}=1$. Let
$B:=\\{\mu\mid\mu=[x_{1}^{n_{1}}x_{2}^{n_{2}};x_{1}^{m_{1}}x_{2}^{m_{2}}]\in\mathsf{NF}(X)\setminus
X,n_{1},n_{2},m_{1},m_{2}\in\\{0,1,2\\}\\}$. Clearly $\SS$ is a
Gröbner–Shirshov basis for $\mathsf{BC}(X|\SS)$ and we have
$B\cap\mathsf{Irr}(\SS)=\\{[x_{2}^{2};x_{2}^{2}]\\}$. Thus
$\mathsf{GKdim}(\mathsf{BC}(X|\SS))=2$. Let $\nu=[x_{2};x_{2}]$. Then we have
$B\cap\mathsf{Irr}(\SS\cup\\{\nu\\})=\emptyset$. Therefore

$\mathsf{GKdim}(\mathsf{BC}(X|\SS\cup\\{\nu\\}))=0=\mathsf{GKdim}(\mathsf{BC}(X|\SS))-2.$

The following proposition offers a rough estimation of the Gelfand-Kirillov
dimension of a finitely presented bicommutative algebra.

######  Proposition 4.6.

Let $X=\\{x_{1},\dots,x_{r}\\}$ be a nonempty finite set and
$S=\\{f_{1},...\leavevmode\hbox{}\hskip 0.56905pt,f_{t}\\}$ a finite
Gröbner–Shirshov basis for $\mathsf{BC}(X|\SS)$, where $r>1$. Then the
following hold:

(i) If $1\leq t\leq r$ and $\overline{\SS}\cap X=\emptyset$, then we have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS))\in\\{2r-t,...\leavevmode\hbox{}\hskip
0.56905pt,2r-1\\};$

(ii) If $t>r$, then we have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS))\in\\{0,...\leavevmode\hbox{}\hskip
0.56905pt,2r-1\\}.$

###### Proof.

(i) We use induction on $t$. If $t=1$, then we may assume
$f_{1}=[x_{1}^{n_{1}}\cdots x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots x_{r}^{m_{r}}]$,
where $n_{1}>0$ and $m_{i}>0$ for some integer $i\leq r$. Thus we obtain that

$[x_{2}^{N_{S}+1}\cdots x_{r}^{N_{S}+1};x_{1}^{N_{S}+1}\cdots
x_{r}^{N_{S}+1}]\in\mathsf{Irr}(S)\mbox{ and }[x_{1}^{N_{S}+1}\cdots
x_{r}^{N_{S}+1};x_{1}^{N_{S}+1}\cdots x_{r}^{N_{S}+1}]\notin\mathsf{Irr}(S).$

Therefore by Theorem 4.2, we have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS))=2r-1.$

Now we assume $t>1$. Obviously,
$\\{\overline{f_{1}},...\leavevmode\hbox{}\hskip
0.56905pt,\overline{f_{t-1}}\\}$ is a Gröbner–Shirshov basis in
$\mathsf{BC}(X)$ and $\mathsf{Irr}(\SS)=\mathsf{Irr}(\overline{\SS})$. By
induction hypothesis, we have

$p:=\mathsf{GKdim}(\mathsf{BC}(X|\\{\overline{f_{1}},...\leavevmode\hbox{}\hskip
0.56905pt,\overline{f_{t-1}}\\}))\in\\{2r-t+1,...\leavevmode\hbox{}\hskip
0.56905pt,2r-1\\}.$

Since $2\leq t\leq r$, we have $2r-t+1\geq 3$. So by Corollary 4.4, we have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS))=\mathsf{GKdim}(\mathsf{BC}(X|\overline{\SS}))\in\\{p-1,p\\}\subseteq\\{2r-t,...\leavevmode\hbox{}\hskip
0.56905pt,2r-1\\}.$

(ii) Since $[x_{1}^{N_{S}+1}\cdots x_{r}^{N_{S}+1};x_{1}^{N_{S}+1}\cdots
x_{r}^{N_{S}+1}]\notin\mathsf{Irr}(S)$, by Theorem 4.2, the result follows
immediately. ∎

######  Example 4.7.

Let $X=\\{x_{1},\dots,x_{r}\\}$, $r>1$ and let $t$ be a positive integer. For
every positive integer $p\leq\min\\{r,t\\}$, we define $m=t-p$ and define

$\SS=\\{[x_{i};x_{i}]\mid 1\leq i\leq p\\}\cup\\{[x_{r}^{i};x_{1}^{3m-i}]\mid
m\leq i\leq 2m-1\\}.$

Let $B:=\\{[x_{1}^{n_{1}}\cdots x_{r}^{n_{r}};x_{1}^{m_{1}}\cdots
x_{r}^{m_{r}}]\in\mathsf{NF}(X)\setminus X\mid
n_{i},m_{j}\in\\{0,1,N_{S}+1\\},i,j\in\\{1,...\leavevmode\hbox{}\hskip
0.56905pt,r\\}\\}$. Clearly $\SS$ is a Gröbner–Shirshov basis for
$\mathsf{BC}(X|\SS)$ with $\\#\SS=t$ and we have $N_{S}=\max\\{2m,1\\}$. Then
we obtain

$[x_{1}^{N_{S}+1}x_{p+1}^{N_{S}+1}\cdots x_{r}^{N_{S}+1};x_{2}^{N_{S}+1}\cdots
x_{r}^{N_{S}+1}]\in B\cap\mathsf{Irr}(\SS),$

and for every monomial $\mu\in B$ with $\mathsf{Sdeg}(\mu)=2r-p+1$, $\mu$ does
not lie in $\mathsf{Irr}(\SS)$. Thus we have

$\mathsf{GKdim}(\mathsf{BC}(X|\SS))=2r-p.$

### Acknowledgment

The authors are grateful to L. A. Bokut and Yu Li for bringing the topic of
bicommutative algebras to our attention.

### Disclosure statement

No potential conflict of interest was reported by the authors.

### Funding

Y.C. is supported by the NNSF of China (11571121, 12071156); Z.Z. is supported
by the fellowship of China Postdoctoral Science Foundation 2021M691099 and the
Young Teacher Research and Cultivation Foundation of South China Normal
University 20KJ02.

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