Gröbner-Shirshov Bases for Semirings

Gr¨obner-Shirshovbases for semirings

Yuqun Chen

(Joint work with L. A. Bokut and Qiuhui Mo)

South China Normal University, Guangzhou, China

CMS Summer Meeting

Winnipeg, 9 June 2014 M-CD-lemma Let M be a class of (in general, non-associative) Ω-algebras, M(X ) a free Ω-algebra in M generated by X with a linear basis consisting of “normal (non-associative Ω-) words” [u], S ⊂ M(X ) a monic subset, < a monomial ordering on normal words

Introduction

In 1962, A. I. Shirshov established the theory of one-relator Lie algebras Lie(X |s = 0) and provided the algorithmic decidability of the word problem for any one-relator Lie algebra. In order to proceed his ideas, he first created the so called Gröbner-Shirshovbases theory for Lie algebras Lie(X |S) which are presented by generators and defining relations. Introduction

M-CD-lemma Let M be a class of (in general, non-associative) Ω-algebras, M(X ) a free Ω-algebra in M generated by X with a linear basis consisting of “normal (non-associative Ω-) words” [u], S ⊂ M(X ) a monic subset, < a monomial ordering on normal words In many cases, each of conditions (a) and (b) is equivalent to the condition

But in some of our “new CD-lemmas”, this is not the case. For example, dialgebra, conformal algebra, etc.

Introduction and Id(S) the ideal of M(X ) generated by S. Let S be a Gr¨obner-Shirshovbasis. Then (a) If f ∈ Id(S), then [f¯] = [a¯sb], where [f¯] is the leading word of f and [asb] is a normal S-word. (b) Irr(S) ={[u]|[u] 6= [a¯sb], s ∈ S, [asb] is a normal S − word} is a linear basis of the quotient algebra M(X |S) = M(X )/Id(S). But in some of our “new CD-lemmas”, this is not the case. For example, dialgebra, conformal algebra, etc.

In many cases, each of conditions (a) and (b) is equivalent to the condition

(c) S is a Gr¨obner-Shirshov basis in M(X ). Introduction and Id(S) the ideal of M(X ) generated by S. Let S be a Gr¨obner-Shirshovbasis. Then (a) If f ∈ Id(S), then [f¯] = [a¯sb], where [f¯] is the leading word of f and [asb] is a normal S-word. (b) Irr(S) ={[u]|[u] 6= [a¯sb], s ∈ S, [asb] is a normal S − word} is a linear basis of the quotient algebra M(X |S) = M(X )/Id(S).

In many cases, each of conditions (a) and (b) is equivalent to the condition

But in some of our “new CD-lemmas”, this is not the case. For example, dialgebra, conformal algebra, etc. Gr¨obner-Shirshovbasis is NOT a linear basis but a GOOD set of deﬁning relations: S ⊂ M(X )

Roughly speaking, S is a Gr¨obner-Shirshov basis for the ideal I of M(X ) iﬀ

S is a GOOD ideal generators: I = Id(S);

S has GOOD leading terms:

∀f ∈ Id(S), f¯ = a¯sb for some words a, b.

Introduction

What is a Gr¨obner-Shirshovbasis?

. Roughly speaking, S is a Gr¨obner-Shirshov basis for the ideal I of M(X ) iﬀ

S is a GOOD ideal generators: I = Id(S);

S has GOOD leading terms:

∀f ∈ Id(S), f¯ = a¯sb for some words a, b.

Introduction

What is a Gr¨obner-Shirshovbasis?

Gr¨obner-Shirshovbasis is NOT a linear basis but a GOOD set of deﬁning relations: S ⊂ M(X ). S is a GOOD ideal generators: I = Id(S);

S has GOOD leading terms:

∀f ∈ Id(S), f¯ = a¯sb for some words a, b.

Introduction

What is a Gr¨obner-Shirshovbasis?

Gr¨obner-Shirshovbasis is NOT a linear basis but a GOOD set of deﬁning relations: S ⊂ M(X ).

Roughly speaking, S is a Gr¨obner-Shirshov basis for the ideal I of M(X ) iﬀ S has GOOD leading terms:

∀f ∈ Id(S), f¯ = a¯sb for some words a, b.

Introduction

What is a Gr¨obner-Shirshovbasis?

Gr¨obner-Shirshovbasis is NOT a linear basis but a GOOD set of deﬁning relations: S ⊂ M(X ).

Roughly speaking, S is a Gr¨obner-Shirshov basis for the ideal I of M(X ) iﬀ

S is a GOOD ideal generators: I = Id(S); Introduction

What is a Gr¨obner-Shirshovbasis?

Gr¨obner-Shirshovbasis is NOT a linear basis but a GOOD set of deﬁning relations: S ⊂ M(X ).

Roughly speaking, S is a Gr¨obner-Shirshov basis for the ideal I of M(X ) iﬀ

S is a GOOD ideal generators: I = Id(S);

S has GOOD leading terms:

∀f ∈ Id(S), f¯ = a¯sb for some words a, b. • normal forms; • word problems; conjugacy problem; • rewriting system; • automata theory; • embedding of algebras into simple algebras and two-generated algebras; PBW type theorems; • extensions of groups and algebras; homology; • Dehn function; complexity; growth function; Hilbert series; etc.

Introduction

Main applications • word problems; conjugacy problem; • rewriting system; • automata theory; • embedding of algebras into simple algebras and two-generated algebras; PBW type theorems; • extensions of groups and algebras; homology; • Dehn function; complexity; growth function; Hilbert series; etc.

Introduction

Main applications

• normal forms; • rewriting system; • automata theory; • embedding of algebras into simple algebras and two-generated algebras; PBW type theorems; • extensions of groups and algebras; homology; • Dehn function; complexity; growth function; Hilbert series; etc.

Introduction

Main applications

• normal forms; • word problems; conjugacy problem; • automata theory; • embedding of algebras into simple algebras and two-generated algebras; PBW type theorems; • extensions of groups and algebras; homology; • Dehn function; complexity; growth function; Hilbert series; etc.

Introduction

Main applications

• normal forms; • word problems; conjugacy problem; • rewriting system; • embedding of algebras into simple algebras and two-generated algebras; PBW type theorems; • extensions of groups and algebras; homology; • Dehn function; complexity; growth function; Hilbert series; etc.

Introduction

Main applications

• normal forms; • word problems; conjugacy problem; • rewriting system; • automata theory; • extensions of groups and algebras; homology; • Dehn function; complexity; growth function; Hilbert series; etc.

Introduction

Main applications

Introduction

Main applications

• normal forms; • word problems; conjugacy problem; • rewriting system; • automata theory; • embedding of algebras into simple algebras and two-generated algebras; PBW type theorems; • extensions of groups and algebras; homology; • Dehn function; complexity; growth function; Hilbert series; etc. • Deﬁne all possible compositions: intersection, inclusion, etc.

• Construct a free object M(X ) generated by X in M. Suppose N is a linear basis (normal words) of M(X ). • Find an admissible (monomial) well ordering on N.

• Let S ⊂ M(X ). Deﬁne normal S-words: An S-word us := ux7→s

is normal if ux7→s = ux7→s .

Introduction

How to establish a CD-lemma in a variety?

Let M be a class (variety) of (in general, non-associative) Ω-algebras and X a set. • Deﬁne all possible compositions: intersection, inclusion, etc.

• Find an admissible (monomial) well ordering on N.

• Let S ⊂ M(X ). Deﬁne normal S-words: An S-word us := ux7→s

is normal if ux7→s = ux7→s .

Introduction

How to establish a CD-lemma in a variety?

Let M be a class (variety) of (in general, non-associative) Ω-algebras and X a set.

• Construct a free object M(X ) generated by X in M. Suppose N is a linear basis (normal words) of M(X ). • Deﬁne all possible compositions: intersection, inclusion, etc.

• Let S ⊂ M(X ). Deﬁne normal S-words: An S-word us := ux7→s

is normal if ux7→s = ux7→s .

Introduction

How to establish a CD-lemma in a variety?

Let M be a class (variety) of (in general, non-associative) Ω-algebras and X a set.

• Construct a free object M(X ) generated by X in M. Suppose N is a linear basis (normal words) of M(X ). • Find an admissible (monomial) well ordering on N. • Deﬁne all possible compositions: intersection, inclusion, etc.

Introduction

How to establish a CD-lemma in a variety?

Let M be a class (variety) of (in general, non-associative) Ω-algebras and X a set.

• Construct a free object M(X ) generated by X in M. Suppose N is a linear basis (normal words) of M(X ). • Find an admissible (monomial) well ordering on N.

• Let S ⊂ M(X ). Deﬁne normal S-words: An S-word us := ux7→s

is normal if ux7→s = ux7→s . Introduction

How to establish a CD-lemma in a variety?

Let M be a class (variety) of (in general, non-associative) Ω-algebras and X a set.

• Construct a free object M(X ) generated by X in M. Suppose N is a linear basis (normal words) of M(X ). • Find an admissible (monomial) well ordering on N.

• Let S ⊂ M(X ). Deﬁne normal S-words: An S-word us := ux7→s

is normal if ux7→s = ux7→s . • Deﬁne all possible compositions: intersection, inclusion, etc. S is called a Gr¨obner-Shirshovbasis in M(X ) if any composition in S is trivial modulo S and corresponding w.

• Prove key lemma: Suppose S is a Gr¨obner-Shirshov basis in M(X ). Then 1. Every polynomial of the ideal Id(S) is a linear combination of some normal S-words.

2. If w := us = vs0 , then

us − vs0 ≡ 0 mod(S, w).

Introduction

Let h ∈ M(X ) and w ∈ N. Then h is trivial mod(S, w), denoted by

h ≡ 0 mod(S, w), P if h = αi uisi for some αi ∈ k and normal S-words uisi with uisi < w. • Prove key lemma: Suppose S is a Gr¨obner-Shirshov basis in M(X ). Then 1. Every polynomial of the ideal Id(S) is a linear combination of some normal S-words.

2. If w := us = vs0 , then

us − vs0 ≡ 0 mod(S, w).

Introduction

Let h ∈ M(X ) and w ∈ N. Then h is trivial mod(S, w), denoted by

h ≡ 0 mod(S, w), P if h = αi uisi for some αi ∈ k and normal S-words uisi with uisi < w.

S is called a Gr¨obner-Shirshovbasis in M(X ) if any composition in S is trivial modulo S and corresponding w. Introduction

Let h ∈ M(X ) and w ∈ N. Then h is trivial mod(S, w), denoted by

h ≡ 0 mod(S, w), P if h = αi uisi for some αi ∈ k and normal S-words uisi with uisi < w.

S is called a Gr¨obner-Shirshovbasis in M(X ) if any composition in S is trivial modulo S and corresponding w.

• Prove key lemma: Suppose S is a Gr¨obner-Shirshov basis in M(X ). Then 1. Every polynomial of the ideal Id(S) is a linear combination of some normal S-words.

2. If w := us = vs0 , then

us − vs0 ≡ 0 mod(S, w). For Lie algebras (Shirshov 1962).

For (commutative, anti-commutative) non-associative algebras (Shirshov 1962).

For associative algebras (Shirshov 1962, Bokut 1976, Bergman 1978, Mora 1986).

For inﬁnite series algebras (both formal and convergent) (Hironaka 1964).

For commutative algebra (Buchberger 1965).

For free Ω-algebras (Knuth, Bendix 1968).

For restricted Lie algebras (A. A. Mikhalev, 1989).

Introduction

We list some known CD-lemmas: For (commutative, anti-commutative) non-associative algebras (Shirshov 1962).

For associative algebras (Shirshov 1962, Bokut 1976, Bergman 1978, Mora 1986).

For inﬁnite series algebras (both formal and convergent) (Hironaka 1964).

For commutative algebra (Buchberger 1965).

For free Ω-algebras (Knuth, Bendix 1968).

For restricted Lie algebras (A. A. Mikhalev, 1989).

Introduction

We list some known CD-lemmas:

For Lie algebras (Shirshov 1962). For associative algebras (Shirshov 1962, Bokut 1976, Bergman 1978, Mora 1986).

For inﬁnite series algebras (both formal and convergent) (Hironaka 1964).

For commutative algebra (Buchberger 1965).

For free Ω-algebras (Knuth, Bendix 1968).

For restricted Lie algebras (A. A. Mikhalev, 1989).

Introduction

We list some known CD-lemmas:

For Lie algebras (Shirshov 1962).

For (commutative, anti-commutative) non-associative algebras (Shirshov 1962). For inﬁnite series algebras (both formal and convergent) (Hironaka 1964).

For commutative algebra (Buchberger 1965).

For free Ω-algebras (Knuth, Bendix 1968).

For restricted Lie algebras (A. A. Mikhalev, 1989).

Introduction

We list some known CD-lemmas:

For Lie algebras (Shirshov 1962).

For (commutative, anti-commutative) non-associative algebras (Shirshov 1962).

For associative algebras (Shirshov 1962, Bokut 1976, Bergman 1978, Mora 1986). For commutative algebra (Buchberger 1965).

For free Ω-algebras (Knuth, Bendix 1968).

For restricted Lie algebras (A. A. Mikhalev, 1989).

Introduction

We list some known CD-lemmas:

For Lie algebras (Shirshov 1962).

For (commutative, anti-commutative) non-associative algebras (Shirshov 1962).

For associative algebras (Shirshov 1962, Bokut 1976, Bergman 1978, Mora 1986).

For inﬁnite series algebras (both formal and convergent) (Hironaka 1964). For free Ω-algebras (Knuth, Bendix 1968).

For restricted Lie algebras (A. A. Mikhalev, 1989).

Introduction

We list some known CD-lemmas:

For Lie algebras (Shirshov 1962).

For (commutative, anti-commutative) non-associative algebras (Shirshov 1962).

For associative algebras (Shirshov 1962, Bokut 1976, Bergman 1978, Mora 1986).

For inﬁnite series algebras (both formal and convergent) (Hironaka 1964).

For commutative algebra (Buchberger 1965). For restricted Lie algebras (A. A. Mikhalev, 1989).

Introduction

We list some known CD-lemmas:

For Lie algebras (Shirshov 1962).

For (commutative, anti-commutative) non-associative algebras (Shirshov 1962).

For associative algebras (Shirshov 1962, Bokut 1976, Bergman 1978, Mora 1986).

For inﬁnite series algebras (both formal and convergent) (Hironaka 1964).

For commutative algebra (Buchberger 1965).

For free Ω-algebras (Knuth, Bendix 1968). Introduction

We list some known CD-lemmas:

For Lie algebras (Shirshov 1962).

For (commutative, anti-commutative) non-associative algebras (Shirshov 1962).

For associative algebras (Shirshov 1962, Bokut 1976, Bergman 1978, Mora 1986).

For inﬁnite series algebras (both formal and convergent) (Hironaka 1964).

For commutative algebra (Buchberger 1965).

For free Ω-algebras (Knuth, Bendix 1968).

For restricted Lie algebras (A. A. Mikhalev, 1989). For tensor product of a polynomial algebra and an exterior (Grassman) algebra (T. Stokes 1990, A. A. Mikhalev and A. A. Zolotykh 2007).

For path algebras (D. R. Farkas, C. D. Feustel and E. L. Green 1993).

For associative algebras over commutative algebras (A. A. Mikhalev and A. A. Zolotykh 1998).

For modules (S.-J. Kang and K.-H. Lee 2000, E. S. Chibrikov 2007).

For non-commutative power series algebras (L. Hellstr¨om2002).

For associative conformal algebras (L. A. Bokut, Y. Fong and W. F. Ke 2004).

For algebras based on well-ordered semigroups (Y. Kobayashi).

Introduction

For Lie superalgebras (A. A. Mikhalev 1989). For path algebras (D. R. Farkas, C. D. Feustel and E. L. Green 1993).

For associative algebras over commutative algebras (A. A. Mikhalev and A. A. Zolotykh 1998).

For modules (S.-J. Kang and K.-H. Lee 2000, E. S. Chibrikov 2007).

For non-commutative power series algebras (L. Hellstr¨om2002).

For associative conformal algebras (L. A. Bokut, Y. Fong and W. F. Ke 2004).

For algebras based on well-ordered semigroups (Y. Kobayashi).

Introduction

For Lie superalgebras (A. A. Mikhalev 1989).

For tensor product of a polynomial algebra and an exterior (Grassman) algebra (T. Stokes 1990, A. A. Mikhalev and A. A. Zolotykh 2007). For associative algebras over commutative algebras (A. A. Mikhalev and A. A. Zolotykh 1998).

For modules (S.-J. Kang and K.-H. Lee 2000, E. S. Chibrikov 2007).

For non-commutative power series algebras (L. Hellstr¨om2002).

For associative conformal algebras (L. A. Bokut, Y. Fong and W. F. Ke 2004).

For algebras based on well-ordered semigroups (Y. Kobayashi).

Introduction

For Lie superalgebras (A. A. Mikhalev 1989).

For tensor product of a polynomial algebra and an exterior (Grassman) algebra (T. Stokes 1990, A. A. Mikhalev and A. A. Zolotykh 2007).

For path algebras (D. R. Farkas, C. D. Feustel and E. L. Green 1993). For modules (S.-J. Kang and K.-H. Lee 2000, E. S. Chibrikov 2007).

For non-commutative power series algebras (L. Hellstr¨om2002).

For associative conformal algebras (L. A. Bokut, Y. Fong and W. F. Ke 2004).

For algebras based on well-ordered semigroups (Y. Kobayashi).

Introduction

For Lie superalgebras (A. A. Mikhalev 1989).

For tensor product of a polynomial algebra and an exterior (Grassman) algebra (T. Stokes 1990, A. A. Mikhalev and A. A. Zolotykh 2007).

For path algebras (D. R. Farkas, C. D. Feustel and E. L. Green 1993).

For associative algebras over commutative algebras (A. A. Mikhalev and A. A. Zolotykh 1998). For non-commutative power series algebras (L. Hellstr¨om2002).

For associative conformal algebras (L. A. Bokut, Y. Fong and W. F. Ke 2004).

For algebras based on well-ordered semigroups (Y. Kobayashi).

Introduction

For Lie superalgebras (A. A. Mikhalev 1989).

For tensor product of a polynomial algebra and an exterior (Grassman) algebra (T. Stokes 1990, A. A. Mikhalev and A. A. Zolotykh 2007).

For path algebras (D. R. Farkas, C. D. Feustel and E. L. Green 1993).

For associative algebras over commutative algebras (A. A. Mikhalev and A. A. Zolotykh 1998).

For modules (S.-J. Kang and K.-H. Lee 2000, E. S. Chibrikov 2007). For associative conformal algebras (L. A. Bokut, Y. Fong and W. F. Ke 2004).

For algebras based on well-ordered semigroups (Y. Kobayashi).

Introduction

For Lie superalgebras (A. A. Mikhalev 1989).

For tensor product of a polynomial algebra and an exterior (Grassman) algebra (T. Stokes 1990, A. A. Mikhalev and A. A. Zolotykh 2007).

For path algebras (D. R. Farkas, C. D. Feustel and E. L. Green 1993).

For associative algebras over commutative algebras (A. A. Mikhalev and A. A. Zolotykh 1998).

For modules (S.-J. Kang and K.-H. Lee 2000, E. S. Chibrikov 2007).

For non-commutative power series algebras (L. Hellstr¨om2002). For algebras based on well-ordered semigroups (Y. Kobayashi).

Introduction

For Lie superalgebras (A. A. Mikhalev 1989).

For tensor product of a polynomial algebra and an exterior (Grassman) algebra (T. Stokes 1990, A. A. Mikhalev and A. A. Zolotykh 2007).

For path algebras (D. R. Farkas, C. D. Feustel and E. L. Green 1993).

For associative algebras over commutative algebras (A. A. Mikhalev and A. A. Zolotykh 1998).

For modules (S.-J. Kang and K.-H. Lee 2000, E. S. Chibrikov 2007).

For non-commutative power series algebras (L. Hellstr¨om2002).

For associative conformal algebras (L. A. Bokut, Y. Fong and W. F. Ke 2004). Introduction

For Lie superalgebras (A. A. Mikhalev 1989).

For tensor product of a polynomial algebra and an exterior (Grassman) algebra (T. Stokes 1990, A. A. Mikhalev and A. A. Zolotykh 2007).

For path algebras (D. R. Farkas, C. D. Feustel and E. L. Green 1993).

For associative algebras over commutative algebras (A. A. Mikhalev and A. A. Zolotykh 1998).

For modules (S.-J. Kang and K.-H. Lee 2000, E. S. Chibrikov 2007).

For non-commutative power series algebras (L. Hellstr¨om2002).

For associative conformal algebras (L. A. Bokut, Y. Fong and W. F. Ke 2004).

For algebras based on well-ordered semigroups (Y. Kobayashi). Lie algebras over commutative algebras, metabelian Lie algebras, pre-Lie (Vinberg) algebras, semirings, Rota-Baxter algebras, (Loday) dialgebras, associative n-conformal algebras, associative algebras with multiple operators (associative Ω-algebras), small category algebras, tensor products of free associative algebras, associative diﬀerential algebras, associative λ-diﬀerential Ω-algebras, L-algebras, non-associative Ω-algebras over commutative algebras, strict monoidal categories, conformal modules, Novikov algebras, etc.

There are some applications particularly to new proofs of some known theorems.

New CD-lemmas

Last years, we proved CD-lemmas for: There are some applications particularly to new proofs of some known theorems.

New CD-lemmas

Last years, we proved CD-lemmas for:

Lie algebras over commutative algebras, metabelian Lie algebras, pre-Lie (Vinberg) algebras, semirings, Rota-Baxter algebras, (Loday) dialgebras, associative n-conformal algebras, associative algebras with multiple operators (associative Ω-algebras), small category algebras, tensor products of free associative algebras, associative diﬀerential algebras, associative λ-diﬀerential Ω-algebras, L-algebras, non-associative Ω-algebras over commutative algebras, strict monoidal categories, conformal modules, Novikov algebras, etc. New CD-lemmas

Last years, we proved CD-lemmas for:

There are some applications particularly to new proofs of some known theorems. X ∗: the free monoid generated by the set X .

[X ]: the free commutative monoid generated by the set X .

N: the set of natural numbers which is regarded as a semiring.

k: a ﬁeld.

RighX i = (X ∗, ·, ◦): the free semiring generated by X .

Rig[X ] = ([X ], ·, ◦): the free commutative semiring generated by X .

kRighX i: the free semiring algebra over k generated by X .

Id(S): the ideal generated by the set S.

Introduction

Notations [X ]: the free commutative monoid generated by the set X .

N: the set of natural numbers which is regarded as a semiring.

k: a ﬁeld.

RighX i = (X ∗, ·, ◦): the free semiring generated by X .

Rig[X ] = ([X ], ·, ◦): the free commutative semiring generated by X .

kRighX i: the free semiring algebra over k generated by X .

Id(S): the ideal generated by the set S.

Introduction

Notations

X ∗: the free monoid generated by the set X . N: the set of natural numbers which is regarded as a semiring.

k: a ﬁeld.

RighX i = (X ∗, ·, ◦): the free semiring generated by X .

Rig[X ] = ([X ], ·, ◦): the free commutative semiring generated by X .

kRighX i: the free semiring algebra over k generated by X .

Id(S): the ideal generated by the set S.

Introduction

Notations

X ∗: the free monoid generated by the set X .

[X ]: the free commutative monoid generated by the set X . k: a ﬁeld.

RighX i = (X ∗, ·, ◦): the free semiring generated by X .

Rig[X ] = ([X ], ·, ◦): the free commutative semiring generated by X .

kRighX i: the free semiring algebra over k generated by X .

Id(S): the ideal generated by the set S.

Introduction

Notations

X ∗: the free monoid generated by the set X .

[X ]: the free commutative monoid generated by the set X .

N: the set of natural numbers which is regarded as a semiring. RighX i = (X ∗, ·, ◦): the free semiring generated by X .

Rig[X ] = ([X ], ·, ◦): the free commutative semiring generated by X .

kRighX i: the free semiring algebra over k generated by X .

Id(S): the ideal generated by the set S.

Introduction

Notations

X ∗: the free monoid generated by the set X .

[X ]: the free commutative monoid generated by the set X .

N: the set of natural numbers which is regarded as a semiring. k: a ﬁeld. Rig[X ] = ([X ], ·, ◦): the free commutative semiring generated by X .

kRighX i: the free semiring algebra over k generated by X .

Id(S): the ideal generated by the set S.

Introduction

Notations

X ∗: the free monoid generated by the set X .

[X ]: the free commutative monoid generated by the set X .

N: the set of natural numbers which is regarded as a semiring. k: a ﬁeld.

RighX i = (X ∗, ·, ◦): the free semiring generated by X . kRighX i: the free semiring algebra over k generated by X .

Id(S): the ideal generated by the set S.

Introduction

Notations

X ∗: the free monoid generated by the set X .

[X ]: the free commutative monoid generated by the set X .

N: the set of natural numbers which is regarded as a semiring. k: a ﬁeld.

RighX i = (X ∗, ·, ◦): the free semiring generated by X .

Rig[X ] = ([X ], ·, ◦): the free commutative semiring generated by X . Id(S): the ideal generated by the set S.

Introduction

Notations

X ∗: the free monoid generated by the set X .

[X ]: the free commutative monoid generated by the set X .

N: the set of natural numbers which is regarded as a semiring. k: a ﬁeld.

RighX i = (X ∗, ·, ◦): the free semiring generated by X .

Rig[X ] = ([X ], ·, ◦): the free commutative semiring generated by X . kRighX i: the free semiring algebra over k generated by X . Introduction

Notations

X ∗: the free monoid generated by the set X .

[X ]: the free commutative monoid generated by the set X .

N: the set of natural numbers which is regarded as a semiring. k: a ﬁeld.

RighX i = (X ∗, ·, ◦): the free semiring generated by X .

Rig[X ] = ([X ], ·, ◦): the free commutative semiring generated by X . kRighX i: the free semiring algebra over k generated by X .

Id(S): the ideal generated by the set S. 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 readers call “rig” instead of “semiring”.

Free semiring

S∞ Let A be an Ω-system, Ω = n=1 Ωn where Ωn is the set of n-ary operations, for example, ary (δ) = n if δ ∈ Ωn. We will call A an Ω-groupoid and each ω ∈ Ω would be called a product. Let k be a ﬁeld and kA the groupoid algebra over k, i.e. the k-linear space with a k-basis A and linear multiple products ω ∈ Ω that are extended by linearity from A to kA. Such kA is called an Ω-algebra. Some readers call “rig” instead of “semiring”.

Free semiring

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. Free semiring

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 readers call “rig” instead of “semiring”. ∗ For any u1, u2,..., un ∈ X we have u1 ◦ u2 ◦ · · · ◦ un ∈ RighX i and

u1 ◦ u2 ◦ · · · ◦ un = ui1 ◦ ui2 ◦ · · · ◦ uin where

{u1, u2,..., un} = {ui1 , ui2 ,..., uin } and ui1 ≤ ui2 ≤ ... ≤ uin .

RighX i = (X ∗, ·, ◦) is a Ω-groupoid, where Ω = {·, ◦}. From now on, we assume that Ω = {·, ◦}.

Free semiring

The class of semirings is a variety. So a free semiring RighX i generated by a set X is deﬁned as usual as for any variety of universal systems. Let (X ∗, ·, 1) be the free monoid generated by X . If one ﬁxes some linear ordering < on X ∗, then any element of RighX i has a unique form w = u1 ◦ u2 ◦ · · · ◦ un, where ∗ ui ∈ X , u1 ≤ u2 ≤ ... ≤ un, n ≥ 0 and w = θ if n = 0. RighX i = (X ∗, ·, ◦) is a Ω-groupoid, where Ω = {·, ◦}. From now on, we assume that Ω = {·, ◦}.

Free semiring

∗ For any u1, u2,..., un ∈ X we have u1 ◦ u2 ◦ · · · ◦ un ∈ RighX i and u1 ◦ u2 ◦ · · · ◦ un = ui1 ◦ ui2 ◦ · · · ◦ uin where

{u1, u2,..., un} = {ui1 , ui2 ,..., uin } and ui1 ≤ ui2 ≤ ... ≤ uin . Free semiring

∗ For any u1, u2,..., un ∈ X we have u1 ◦ u2 ◦ · · · ◦ un ∈ RighX i and u1 ◦ u2 ◦ · · · ◦ un = ui1 ◦ ui2 ◦ · · · ◦ uin where

{u1, u2,..., un} = {ui1 , ui2 ,..., uin } and ui1 ≤ ui2 ≤ ... ≤ uin .

RighX i = (X ∗, ·, ◦) is a Ω-groupoid, where Ω = {·, ◦}. From now on, we assume that Ω = {·, ◦}. ∗ For any u1 ◦ u2 ◦ · · · ◦ un ∈ RighX i, ui ∈ X , we denote |u|◦ = n.

∗ For any u ◦ u ◦ u ◦ · · · ◦ u where |u ◦ u ◦ u ◦ · · · ◦ u|◦ = n, u ∈ X , we will denote it by u◦n.

For any u = w1 ◦ w2 ◦ · · · ◦ wm ◦ um+1 ◦ · · · ◦ un, ∗ v = w1 ◦ w2 ◦ · · · ◦ wm ◦ vm+1 ◦ · · · ◦ vt , where wl , ui , vj ∈ X , such that

ui 6= vj for any i = m + 1,..., n, j = m + 1,..., t,

we denote

lcm◦(u, v) = w1 ◦ w2 ◦ · · · ◦ wm ◦ um+1 ◦ · · · ◦ un ◦ vm+1 ◦ · · · ◦ vt

the least common multiple of u and v in RighX i with respect to ◦.

Free semiring

Let k be a ﬁeld. We call the groupoid algebra kRighX i a semiring algebra. Any element in RighX i is called a monomial and any element in kRighX i is called a polynomial. ∗ For any u ◦ u ◦ u ◦ · · · ◦ u where |u ◦ u ◦ u ◦ · · · ◦ u|◦ = n, u ∈ X , we will denote it by u◦n.

For any u = w1 ◦ w2 ◦ · · · ◦ wm ◦ um+1 ◦ · · · ◦ un, ∗ v = w1 ◦ w2 ◦ · · · ◦ wm ◦ vm+1 ◦ · · · ◦ vt , where wl , ui , vj ∈ X , such that

ui 6= vj for any i = m + 1,..., n, j = m + 1,..., t,

we denote

lcm◦(u, v) = w1 ◦ w2 ◦ · · · ◦ wm ◦ um+1 ◦ · · · ◦ un ◦ vm+1 ◦ · · · ◦ vt

the least common multiple of u and v in RighX i with respect to ◦.

Free semiring

Let k be a ﬁeld. We call the groupoid algebra kRighX i a semiring algebra. Any element in RighX i is called a monomial and any element in kRighX i is called a polynomial.

∗ For any u1 ◦ u2 ◦ · · · ◦ un ∈ RighX i, ui ∈ X , we denote |u|◦ = n. For any u = w1 ◦ w2 ◦ · · · ◦ wm ◦ um+1 ◦ · · · ◦ un, ∗ v = w1 ◦ w2 ◦ · · · ◦ wm ◦ vm+1 ◦ · · · ◦ vt , where wl , ui , vj ∈ X , such that

ui 6= vj for any i = m + 1,..., n, j = m + 1,..., t,

we denote

lcm◦(u, v) = w1 ◦ w2 ◦ · · · ◦ wm ◦ um+1 ◦ · · · ◦ un ◦ vm+1 ◦ · · · ◦ vt

the least common multiple of u and v in RighX i with respect to ◦.

Free semiring

Let k be a ﬁeld. We call the groupoid algebra kRighX i a semiring algebra. Any element in RighX i is called a monomial and any element in kRighX i is called a polynomial.

∗ For any u1 ◦ u2 ◦ · · · ◦ un ∈ RighX i, ui ∈ X , we denote |u|◦ = n.

∗ For any u ◦ u ◦ u ◦ · · · ◦ u where |u ◦ u ◦ u ◦ · · · ◦ u|◦ = n, u ∈ X , we will denote it by u◦n. Free semiring

Let k be a ﬁeld. We call the groupoid algebra kRighX i a semiring algebra. Any element in RighX i is called a monomial and any element in kRighX i is called a polynomial.

∗ For any u1 ◦ u2 ◦ · · · ◦ un ∈ RighX i, ui ∈ X , we denote |u|◦ = n.

∗ For any u ◦ u ◦ u ◦ · · · ◦ u where |u ◦ u ◦ u ◦ · · · ◦ u|◦ = n, u ∈ X , we will denote it by u◦n.

For any u = w1 ◦ w2 ◦ · · · ◦ wm ◦ um+1 ◦ · · · ◦ un, ∗ v = w1 ◦ w2 ◦ · · · ◦ wm ◦ vm+1 ◦ · · · ◦ vt , where wl , ui , vj ∈ X , such that

ui 6= vj for any i = m + 1,..., n, j = m + 1,..., t, we denote

lcm◦(u, v) = w1 ◦ w2 ◦ · · · ◦ wm ◦ um+1 ◦ · · · ◦ un ◦ vm+1 ◦ · · · ◦ vt the least common multiple of u and v in RighX i with respect to ◦. Let > be any monomial ordering on RighX i, i.e. > is a well ordering such that for any v, w ∈ RighX i and u a ?-monomial,

w > v ⇒ u|w > u|v .

For every polynomial f ∈ kRighX i, f has the leading monomial f¯. If the coeﬃcient of f¯ is 1, then we call f to be monic.

CD-lemma for semirings

Let ?/∈ X . By a ?-monomial we mean a monomial in RighX ∪ ?i with only one occurrence of ?. Let u be a ?-monomial and s ∈ kRighX i. Then we call u|s = u|?7→s an s-monomial. For every polynomial f ∈ kRighX i, f has the leading monomial f¯. If the coeﬃcient of f¯ is 1, then we call f to be monic.

CD-lemma for semirings

Let ?/∈ X . By a ?-monomial we mean a monomial in RighX ∪ ?i with only one occurrence of ?. Let u be a ?-monomial and s ∈ kRighX i. Then we call u|s = u|?7→s an s-monomial.

Let > be any monomial ordering on RighX i, i.e. > is a well ordering such that for any v, w ∈ RighX i and u a ?-monomial,

w > v ⇒ u|w > u|v . CD-lemma for semirings

Let ?/∈ X . By a ?-monomial we mean a monomial in RighX ∪ ?i with only one occurrence of ?. Let u be a ?-monomial and s ∈ kRighX i. Then we call u|s = u|?7→s an s-monomial.

Let > be any monomial ordering on RighX i, i.e. > is a well ordering such that for any v, w ∈ RighX i and u a ?-monomial,

w > v ⇒ u|w > u|v .

For every polynomial f ∈ kRighX i, f has the leading monomial f¯. If the coeﬃcient of f¯ is 1, then we call f to be monic. ∗ (I) If there exist a, b ∈ X , such that |lcm◦(f¯ a, bg¯)|◦ < |f¯ a|◦ + |bg¯|◦ then we call (f , g)w = fa ◦ u − bg ◦ v the intersection composition of f and g with respect to w where w = lcm◦(f¯ a, bg¯) = f¯ a ◦ u = bg ◦ v. ∗ (II) If there exist a, b ∈ X , such that |lcm◦(f¯, agb¯ )|◦ < |f¯|◦ + |agb¯ |◦ 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¯, agb¯ ) = f¯ ◦ u = agb ◦ v.

In the above deﬁnition, clearly,

(f , g)w ∈ Id(f , g) and (f , g)w < w,

where Id(f , g) is the Ω-ideal of kRighX i generated by f , g.

CD-lemma for semirings

Deﬁnition Let > be a monomial ordering on RighX i. Let f , g be two monic polynomials in kRighX i. ∗ (II) If there exist a, b ∈ X , such that |lcm◦(f¯, agb¯ )|◦ < |f¯|◦ + |agb¯ |◦ 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¯, agb¯ ) = f¯ ◦ u = agb ◦ v.

In the above deﬁnition, clearly,

(f , g)w ∈ Id(f , g) and (f , g)w < w,

where Id(f , g) is the Ω-ideal of kRighX i generated by f , g.

CD-lemma for semirings

(f , g)w ∈ Id(f , g) and (f , g)w < w,

where Id(f , g) is the Ω-ideal of kRighX i generated by f , g.

CD-lemma for semirings

Deﬁnition Let > be a monomial ordering on RighX i. Let f , g be two monic polynomials in kRighX i. ∗ (I) If there exist a, b ∈ X , such that |lcm◦(f¯ a, bg¯)|◦ < |f¯ a|◦ + |bg¯|◦ then we call (f , g)w = fa ◦ u − bg ◦ v the intersection composition of f and g with respect to w where w = lcm◦(f¯ a, bg¯) = f¯ a ◦ u = bg ◦ v. ∗ (II) If there exist a, b ∈ X , such that |lcm◦(f¯, agb¯ )|◦ < |f¯|◦ + |agb¯ |◦ 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¯, agb¯ ) = f¯ ◦ u = agb ◦ v. CD-lemma for semirings

In the above deﬁnition, clearly,

(f , g)w ∈ Id(f , g) and (f , g)w < w, where Id(f , g) is the Ω-ideal of kRighX i generated by f , g. The set S is called a Gr¨obner-Shirshovbasis in kRighX i if any composition in S is trivial modulo S and corresponding to w.

CD-lemma for semirings

Deﬁnition Suppose that w is a monomial, S a set of monic polynomials in kRighX i and h a polynomial. Then h is trivial modulo (S, w), P denoted by h ≡ 0 mod(S, w), if h = αi ai si bi ◦ ui , where each i ∗ αi ∈ k, ai , bi ∈ X , ui ∈ RighX i, si ∈ S and ai si bi ◦ ui < w. CD-lemma for semirings

The set S is called a Gr¨obner-Shirshovbasis in kRighX i if any composition in S is trivial modulo S and corresponding to w. Denote Irr(S) = {w ∈ RighX i | w 6= asb ◦ u for any a, b ∈ X ∗, u ∈ RighX i, s ∈ S}.

A Gr¨obner-Shirshovbasis S in kRighX i is reduced if for any s ∈ S, supp(s) ⊆ Irr(S − {s}), where supp(s) = {u1, u2,..., un} if Pn s = i=1 αi ui , 0 6= αi ∈ k, ui ∈ RighX i.

If a subset S of kRighX i is not a Gröbner-Shirshovbasis for Id(S) then one can add to S a nontrivial composition (f , g)w of f , g ∈ S and continue this process repeatedly (actually using the transfinite induction) in order to obtain a set Scomp of generators of Id(S) such that Scomp is a Gröbner-Shirshov basis in kRighX i. Such a process is called Shirshov algorithm.

CD-lemma for semirings

A set S is called a minimal Gröbner-Shirshovbasis in kRighX i if S is a Gröbner-Shirshovbasis in kRighX i and for any f , g ∈ S with f 6= g, 6 ∃ a, b ∈ X ∗, u ∈ RighX i, s.t., f = agb ◦ u. A Gröbner-Shirshovbasis S in kRighX i is reduced if for any s ∈ S, supp(s) ⊆ Irr(S − {s}), where supp(s) = {u1, u2,..., un} if Pn s = i=1 αi ui , 0 6= αi ∈ k, ui ∈ RighX i.

CD-lemma for semirings

A set S is called a minimal Gr¨obner-Shirshovbasis in kRighX i if S is a Gr¨obner-Shirshovbasis in kRighX i and for any f , g ∈ S with f 6= g, 6 ∃ a, b ∈ X ∗, u ∈ RighX i, s.t., f = agb ◦ u.

Denote Irr(S) = {w ∈ RighX i | w 6= asb ◦ u for any a, b ∈ X ∗, u ∈ RighX i, s ∈ S}. If a subset S of kRighX i is not a Gröbner-Shirshovbasis for Id(S) then one can add to S a nontrivial composition (f , g)w of f , g ∈ S and continue this process repeatedly (actually using the transfinite induction) in order to obtain a set Scomp of generators of Id(S) such that Scomp is a Gröbner-Shirshov basis in kRighX i. Such a process is called Shirshov algorithm.

CD-lemma for semirings

Denote Irr(S) = {w ∈ RighX i | w 6= asb ◦ u for any a, b ∈ X ∗, u ∈ RighX i, s ∈ S}.

A Gr¨obner-Shirshovbasis S in kRighX i is reduced if for any s ∈ S, supp(s) ⊆ Irr(S − {s}), where supp(s) = {u1, u2,..., un} if Pn s = i=1 αi ui , 0 6= αi ∈ k, ui ∈ RighX i. CD-lemma for semirings

Denote Irr(S) = {w ∈ RighX i | w 6= asb ◦ u for any a, b ∈ X ∗, u ∈ RighX i, s ∈ S}.

A Gr¨obner-Shirshovbasis S in kRighX i is reduced if for any s ∈ S, supp(s) ⊆ Irr(S − {s}), where supp(s) = {u1, u2,..., un} if Pn s = i=1 αi ui , 0 6= αi ∈ k, ui ∈ RighX i.

CD-lemma for semirings

Theorem (Bokut-Chen-Mo, Composition-Diamond lemma for semirings) Let S be a set of monic polynomials in kRighX i, > a monomial ordering on RighX i and Id(S) the Ω-ideal of kRighX i generated by S. Then the following statements are equivalent. (1) S is a Gr¨obner-Shirshovbasis in kRighX i. (2) f ∈ Id(S) ⇒ f¯ = asb ◦ u for some a, b ∈ X ∗, u ∈ RighX i and s ∈ S. (3) Irr(S) = {w ∈ RighX i | w 6= asb ◦ u for any a, b ∈ X ∗, u ∈ RighX i, s ∈ S} is a k-basis of kRighX |Si = kRighX i/Id(S). CD-lemma for semirings

Theorem (Bokut-Chen-Mo) Let I be an ideal of kRighX i and > a monomial ordering on RighX i. Then there exists a unique reduced Gr¨obner-Shirshovbasis S for I . For any u, v ∈ [X ], we denote lcm·(u, v) the least common multiple of u and v in [X ]. Then there exist uniquely a, b ∈ [X ] such that lcm·(u, v) = au = bv.

CD-lemma for semirings

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 ﬁxes some linear ordering < on the set [X ], then any element of Rig[X ] has an unique form θ, or w = u1 ◦ u2 ◦ · · · ◦ un, where ui ∈ [X ], u1 ≤ u2 ≤ ... ≤ un, n ≥ 1. CD-lemma for semirings

For any u, v ∈ [X ], we denote lcm·(u, v) the least common multiple of u and v in [X ]. Then there exist uniquely a, b ∈ [X ] such that lcm·(u, v) = au = bv. CD-lemma for semirings

Deﬁnition Let < be a monomial ordering on Rig[X ]. Let f , g be two monic polynomials in kRig[X ] and f = u1 ◦ u2 ◦ · · · ◦ un, g = v1 ◦ v2 ◦ · · · ◦ vm where each ui , vj ∈ [X ]. For any pair (a, b) ∈ {(aij , bij ) | 1 ≤ i ≤ n, 1 ≤ j ≤ m} where aij , bij ∈ [X ] such that lcm·(ui , vj ) = aij ui = bij vj , we call (f , g)w = af ◦ u − bg ◦ v the composition of f and g with respect to w where w = lcm◦(af , bg) = af ◦ u = bg¯ ◦ v.

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-Shirshovbasis Scomp for a finite set S in kRig[X ]. However, the reduced Gröbner-Shirshov basis of Id(S) is generally infinite even if both S and X are finite. Theorem (Bokut-Chen-Mo) Let I be an ideal of kRig[X ] and > a monomial ordering on Rig[X ]. Then there exists a unique reduced Gröbner-Shirshovbasis S for I .

CD-lemma for semirings

Theorem (Bokut-Chen-Mo, Composition-Diamond lemma for commutative semirings) Let S be a set of monic polynomials in kRig[X ] and > a monomial ordering on Rig[X ]. Then the following statements are equivalent. (1) S is a Gr¨obner-Shirshovbasis in kRig[X ]. (2) f ∈ Id(S) ⇒ f¯ = as ◦ u for some a ∈ [X ], u ∈ Rig[X ] and s ∈ S. (3) Irr(S) = {w ∈ Rig[X ] | w 6= as ◦ u for any a ∈ [X ], u ∈ Rig[X ], s ∈ S} is a k-basis of kRig[X |S] = kRig[X ]/Id(S). CD-lemma for semirings

Theorem (Bokut-Chen-Mo) Let I be an ideal of kRig[X ] and > a monomial ordering on Rig[X ]. Then there exists a unique reduced Gr¨obner-Shirshovbasis S for I . Fiore-Leinster ﬁnd a strongly normalizing reduction system and a 2 normal form of the semiring N[x]/(x = 1 + x + x ).

2 2 N[x]/(x = 1 + x + x ) = Rig[x|x = 1 ◦ x ◦ x ].

Applications

In the paper

M. Fiore and T. Leinster, An objective representation of the Gaussian integers, Journal of Symbolic Computation, 37(2004), 707-716. 2 2 N[x]/(x = 1 + x + x ) = Rig[x|x = 1 ◦ x ◦ x ].

Applications

In the paper

M. Fiore and T. Leinster, An objective representation of the Gaussian integers, Journal of Symbolic Computation, 37(2004), 707-716.

Fiore-Leinster ﬁnd a strongly normalizing reduction system and a 2 normal form of the semiring N[x]/(x = 1 + x + x ). Applications

In the paper

M. Fiore and T. Leinster, An objective representation of the Gaussian integers, Journal of Symbolic Computation, 37(2004), 707-716.

Fiore-Leinster ﬁnd a strongly normalizing reduction system and a 2 normal form of the semiring N[x]/(x = 1 + x + x ).

2 2 N[x]/(x = 1 + x + x ) = Rig[x|x = 1 ◦ x ◦ x ]. We order [x] by degree ordering ≺: xn ≺ xm ⇔ n < m.

For any u ∈ Rig[x], u can be uniquely expressed as u = u1 ◦ u2 ◦ · · · ◦ un, where u1, u2,..., un ∈ [x], and u1 u2 ... un. Denote wt(u) = (un, un−1,..., u1).

We order Rig[x] as follows: for any u, v ∈ Rig[x], if one of the sequences is not a preﬁx of other, then

u < v ⇐⇒ wt(u) < wt(v) lexicographically;

if the sequence of u is a preﬁx of the sequence of v, then u < v.

Then, it is clear that < on Rig[x] is a monomial ordering.

Applications

We deﬁne a monomial ordering on Rig[x] ﬁrst. For any u ∈ Rig[x], u can be uniquely expressed as u = u1 ◦ u2 ◦ · · · ◦ un, where u1, u2,..., un ∈ [x], and u1 u2 ... un. Denote wt(u) = (un, un−1,..., u1).

We order Rig[x] as follows: for any u, v ∈ Rig[x], if one of the sequences is not a preﬁx of other, then

u < v ⇐⇒ wt(u) < wt(v) lexicographically;

if the sequence of u is a preﬁx of the sequence of v, then u < v.

Then, it is clear that < on Rig[x] is a monomial ordering.

Applications

We deﬁne a monomial ordering on Rig[x] ﬁrst.

We order [x] by degree ordering ≺: xn ≺ xm ⇔ n < m. We order Rig[x] as follows: for any u, v ∈ Rig[x], if one of the sequences is not a preﬁx of other, then

u < v ⇐⇒ wt(u) < wt(v) lexicographically;

if the sequence of u is a preﬁx of the sequence of v, then u < v.

Then, it is clear that < on Rig[x] is a monomial ordering.

Applications

We deﬁne a monomial ordering on Rig[x] ﬁrst.

We order [x] by degree ordering ≺: xn ≺ xm ⇔ n < m.

For any u ∈ Rig[x], u can be uniquely expressed as u = u1 ◦ u2 ◦ · · · ◦ un, where u1, u2,..., un ∈ [x], and u1 u2 ... un. Denote wt(u) = (un, un−1,..., u1). Then, it is clear that < on Rig[x] is a monomial ordering.

Applications

We deﬁne a monomial ordering on Rig[x] ﬁrst.

We order [x] by degree ordering ≺: xn ≺ xm ⇔ n < m.

For any u ∈ Rig[x], u can be uniquely expressed as u = u1 ◦ u2 ◦ · · · ◦ un, where u1, u2,..., un ∈ [x], and u1 u2 ... un. Denote wt(u) = (un, un−1,..., u1).

We order Rig[x] as follows: for any u, v ∈ Rig[x], if one of the sequences is not a preﬁx of other, then

u < v ⇐⇒ wt(u) < wt(v) lexicographically; if the sequence of u is a preﬁx of the sequence of v, then u < v. Applications

We deﬁne a monomial ordering on Rig[x] ﬁrst.

We order [x] by degree ordering ≺: xn ≺ xm ⇔ n < m.

For any u ∈ Rig[x], u can be uniquely expressed as u = u1 ◦ u2 ◦ · · · ◦ un, where u1, u2,..., un ∈ [x], and u1 u2 ... un. Denote wt(u) = (un, un−1,..., u1).

We order Rig[x] as follows: for any u, v ∈ Rig[x], if one of the sequences is not a preﬁx of other, then

u < v ⇐⇒ wt(u) < wt(v) lexicographically; if the sequence of u is a preﬁx of the sequence of v, then u < v.

Then, it is clear that < on Rig[x] is a monomial ordering. Corollary (Fiore-Leinster) A normal form of the semiring Rig[x|x = 1 ◦ x ◦ x2] is the set

{1◦(m+1)◦x2, 1◦m◦x◦n, 1◦m◦(x3)◦n, x◦m◦(x2)◦n, (x2)◦m◦(x3)◦n|n, m ≥ 0}.

Applications

Theorem (Bokut-Chen-Mo) Let the ordering on Rig[x] be as above. Then kRig[x|x = 1 ◦ x ◦ x2] = kRig[x|S] and S is a Gr¨obner-Shirshovbasis in kRig[x], where S consists of the following relations 1. x4 = 1 ◦ 1 ◦ x2, 2. x ◦ x3 = 1 ◦ x2, 3.1 ◦ x2 ◦ xn = xn (1 ≤ n ≤ 3). Applications

Corollary (Fiore-Leinster) A normal form of the semiring Rig[x|x = 1 ◦ x ◦ x2] is the set

{1◦(m+1)◦x2, 1◦m◦x◦n, 1◦m◦(x3)◦n, x◦m◦(x2)◦n, (x2)◦m◦(x3)◦n|n, m ≥ 0}. Theorem (Bokut-Chen-Mo) Let the ordering on Rig[x] be as the above. Then kRig[x|x = 1 ◦ x2] = kRig[x|S] and S is a Gr¨obner-Shirshovbasis in kRig[x], where S consists of the following relations 1.1 ◦ x2 = x, 2. x ◦ x4 = 1 ◦ x3, 3. x5 = 1 ◦ x4, 4.1 ◦ x3 ◦ xn = xn (3 ≤ n ≤ 4)}.

Applications

In the paper

A. Blass, Seven trees in one, Journal of Pure and Applied Algebra, 103(1995), 1-21.

2 Blass ﬁnds a normal form of the semiring N[x]/(x = 1 + x ). Clearly, 2 2 N[x]/(x = 1 + x ) = Rig[x|x = 1 ◦ x ]. Applications

In the paper

A. Blass, Seven trees in one, Journal of Pure and Applied Algebra, 103(1995), 1-21.

2 Blass ﬁnds a normal form of the semiring N[x]/(x = 1 + x ). Clearly, 2 2 N[x]/(x = 1 + x ) = Rig[x|x = 1 ◦ x ].

Theorem (Bokut-Chen-Mo) Let the ordering on Rig[x] be as the above. Then kRig[x|x = 1 ◦ x2] = kRig[x|S] and S is a Gr¨obner-Shirshovbasis in kRig[x], where S consists of the following relations 1.1 ◦ x2 = x, 2. x ◦ x4 = 1 ◦ x3, 3. x5 = 1 ◦ x4, 4.1 ◦ x3 ◦ xn = xn (3 ≤ n ≤ 4)}. By replacing x, x3 for 1 ◦ x2, x2 ◦ x4 respectively, we have

Corollary (Blass) A normal form of the semiring Rig[x|x = 1 ◦ x2] is the set

{1◦n ◦x2 ◦x4, 1◦n ◦(x2)◦m, (x2)◦m ◦(x4)◦t , 1◦n ◦(x4)◦t | n, m, t ≥ 0}.

Applications

Corollary (Bokut-Chen-Mo) A normal form of the semiring Rig[x|x = 1 ◦ x2] is the set

{(1◦n ◦ x◦m)xt , 1◦n ◦ x3, 1◦n ◦ (x4)◦m | n, m ≥ 0, 0 ≤ t ≤ 3}. Corollary (Blass) A normal form of the semiring Rig[x|x = 1 ◦ x2] is the set

{1◦n ◦x2 ◦x4, 1◦n ◦(x2)◦m, (x2)◦m ◦(x4)◦t , 1◦n ◦(x4)◦t | n, m, t ≥ 0}.

Applications

Corollary (Bokut-Chen-Mo) A normal form of the semiring Rig[x|x = 1 ◦ x2] is the set

{(1◦n ◦ x◦m)xt , 1◦n ◦ x3, 1◦n ◦ (x4)◦m | n, m ≥ 0, 0 ≤ t ≤ 3}.

By replacing x, x3 for 1 ◦ x2, x2 ◦ x4 respectively, we have Applications

Corollary (Bokut-Chen-Mo) A normal form of the semiring Rig[x|x = 1 ◦ x2] is the set

{(1◦n ◦ x◦m)xt , 1◦n ◦ x3, 1◦n ◦ (x4)◦m | n, m ≥ 0, 0 ≤ t ≤ 3}.

By replacing x, x3 for 1 ◦ x2, x2 ◦ x4 respectively, we have

Corollary (Blass) A normal form of the semiring Rig[x|x = 1 ◦ x2] is the set

{1◦n ◦x2 ◦x4, 1◦n ◦(x2)◦m, (x2)◦m ◦(x4)◦t , 1◦n ◦(x4)◦t | n, m, t ≥ 0}. 2 n Let In = Id(x ◦ 1, x ◦ 1,..., x ◦ 1). Then

I1 ( I2 ( ... ( In ( ....

is an inﬁnit ascending chain of ideals in kRig[x] as follows.

Let us deﬁne congruence relation ρn on N[x] generated by the set {(xi ◦ 1, x ◦ 1), 2 ≤ i ≤ n}. Then we have an inﬁnite ascending chain of congruences ρ1 ( ρ2 ( ... ( ρn ( ....

Applications

Example Considering the semiring N[x]/(x + 1 = x) = Rig[x| x ◦ 1 = x], it is easy to have that kRig[x| x ◦ 1 = x] = kRig[x|S] where S = {xn ◦ 1 = xn| n ≥ 1} is the reduced Gr¨obner-Shirshovbasis in kRig[x] with the ordering as above. Then

I1 ( I2 ( ... ( In ( ....

is an inﬁnit ascending chain of ideals in kRig[x] as follows.

Let us deﬁne congruence relation ρn on N[x] generated by the set {(xi ◦ 1, x ◦ 1), 2 ≤ i ≤ n}. Then we have an inﬁnite ascending chain of congruences ρ1 ( ρ2 ( ... ( ρn ( ....

Applications

2 n Let In = Id(x ◦ 1, x ◦ 1,..., x ◦ 1). Let us deﬁne congruence relation ρn on N[x] generated by the set {(xi ◦ 1, x ◦ 1), 2 ≤ i ≤ n}. Then we have an inﬁnite ascending chain of congruences ρ1 ( ρ2 ( ... ( ρn ( ....

Applications

2 n Let In = Id(x ◦ 1, x ◦ 1,..., x ◦ 1). Then

I1 ( I2 ( ... ( In ( .... is an inﬁnit ascending chain of ideals in kRig[x] as follows. Then we have an inﬁnite ascending chain of congruences ρ1 ( ρ2 ( ... ( ρn ( ....

Applications

2 n Let In = Id(x ◦ 1, x ◦ 1,..., x ◦ 1). Then

I1 ( I2 ( ... ( In ( .... is an inﬁnit ascending chain of ideals in kRig[x] as follows.

Let us deﬁne congruence relation ρn on N[x] generated by the set {(xi ◦ 1, x ◦ 1), 2 ≤ i ≤ n}. Applications

2 n Let In = Id(x ◦ 1, x ◦ 1,..., x ◦ 1). Then

I1 ( I2 ( ... ( In ( .... is an inﬁnit ascending chain of ideals in kRig[x] as follows.

Let us deﬁne congruence relation ρn on N[x] generated by the set {(xi ◦ 1, x ◦ 1), 2 ≤ i ≤ n}. Then we have an inﬁnite ascending chain of congruences ρ1 ( ρ2 ( ... ( ρn ( .... Applications

Corollary Each congruence on the semiring N is generated by one element. In particular, N is noetherian. N[x] is not noetherian. Some new CD-lemmas

L.A. Bokut, Yuqun Chen and Qiuhui Mo, Groebner-Shirshov bases for semirings, Journal of Algebra, 385(2013) 47-63. arXiv:1208.0538

L.A. Bokut, Yuqun Chen and Yu Li, Lyndon-Shirshov basis and anti-commutative algebras, Journal of Algebra, Journal of Algebra, 378(2013), 173-183. arXiv:1110.1264

Yongshan Chen and Yuqun Chen, Groebner-Shirshov bases for matabelian Lie algebras, Journal of Algebra, 358£2012), 143-161. arXiv:1106.2362

L.A. Bokut, Yuqun Chen and Yongshan Chen, Groebner-Shirshov bases for Lie algebras over a commutative algebra, Journal of Algebra, 337(2011), 82-102. arXiv:1006.3217v3 Some new CD-lemmas

L.A. Bokut, Yuqun Chen and Yongshan Chen, Composition-Diamond lemma for tensor product of free algebras, Journal of Algebra, 323(2010), 2520-2537. arXiv:0804.2115

L.A. Bokut, Yuqun Chen and Jianjun Qiu, Groebner-Shirshov bases for associative algebras with multiple operations and free Rota-Baxter algebras, Journal of Pure and Applied Algebra, 214(2010), 89-100. arXiv:0805.0640v2

L.A. Bokut, Yuqun Chen and Jiapeng Huang, Groebner-Shirshov bases for L-algebras, International Journal of Algebra and Computation, International Journal of Algebra and Computation, 23£3¤(2013) 547-571. arXiv:1005.0118 Some new CD-lemmas

Chen Yuqun, Yongshan Chen and Yu Li, Composition-Diamond lemma for diﬀerential algebras, The Arabian Journal for Science and Engineering, 34(2A)(2009),135-145. arXiv:0805.2327v2

Jianjun Qiu and Yuqun Chen, Composition-Diamond lemma for λ-diﬀerential associative algebras with multiple operators, Journal of Algebra and its Applications, 9(2010), 223-239.

L.A. Bokut, Yuqun Chen and Yu Li, Groebner-Shirshov bases for categories, Nankai Series in Pure, Applied Mathematics and Theoretical Physics,Operads and Universal Algebra,Vol.9(2012), 1-23. arXiv:1101.1563

Yuqun Chen, Groebner-Shirshov bases for strick monoidal categories, submitted. Some new applications

Yuqun Chen and Chanyan Zhong, Groebner-Shirshov bases for braid groups in Adyan-Thurston generators, Algebra Colloq., to appear. arXiv:0909.3639

Yuqun Chen, Groebner-Shirshov basis for Schreier extensions of groups, Comm. Algebra, 36(5)(2008), 1609-1625. arXiv:0804.0641

Yuqun Chen, Groebner-Shirshov basis for extensions of algebras, Algebra Colloq., 16(2)(2009), 283-292. arXiv:0804.0643

L.A. Bokut, Yuqun Chen and Yu Li, Anti-commutative Groebner-Shirshov bases of a free Lie algebra, Science in China, 52(2)(2009), 244-253. arXiv:0804.0914 Some new applications

Yuqun Chen and Zhong Chanyan, Groebner-Shirshov basis for some one-relator groups, Algebra Colloq., 19(2011), 99-116. arXiv:0804.0643

Yuqun Chen and Zhong Chanyan, Groebner-Shirshov basis for HNN extensions of groups and for the alternative group, Comm. Algebra, 36(1)(2008), 94-103. arXiv:0804.0642

Yuqun Chen and Jianjun Qiu, Groebner-Shirshov basis for the Chinese monoid, Journal of Algebra and its Applications,7(5)(2008), 623-628. arXiv:0804.0972

L.A. Bokut, Yuqun Chen and Xiangui Zhao, Groebner-Shirshov beses for free inverse semigroups, International Journal of Algebra and Computation, 19(2)(2009), 129-143. arXiv:0804.0959 Some new applications

Yuqun Chen and Qiuhui Mo, Embedding dendriform algebra into its universal enveloping Rota-Baxter algebra, Proc. Amer. Math. Soc., 139(2011), 4207-4216. arXiv:1005.2717

L.A. Bokut, Yuqun Chen and Qiuhui Mo, Groebner-Shirshov bases and embeddings of algebras, International Journal of Algebra and Computation, 20(2010), 875-900. arXiv:0908.1992

Yuqun Chen, Wenshu Chen and Runai Luo, Word problem for Novikov’s and Boone’s group via Groebner-Shirshov bases, Southeast Asian Bull Math., 32(5)(2008), 863-877.

Yuqun Chen, Hongshan Shao and K. P. Shum, On Rosso-Yamane theorem on PBW basis of Uq(AN ), CUBO A Mathematical Journal, 10(3)(2008), 171-194. arXiv:0804.0954 Some new applications

Yuqun Chen and Bin Wang, Groebner-Shirshov Bases and Hilbert Series of Free Dendriform Algebras. Southeast Asian Bull. Math., 34(2010), 639-650. arXiv:1009.0185

Yuqun Chen and Qiuhui Mo, Groebner-Shirshov bases for free partially commutative Lie algebras, Comm. Algebra, 41(2013), 3753-3761. arXiv:1105.4796

L.A. Bokut, Yuqun Chen, Weiping Chen and Jing Li, New approaches to plactic monoid via Gr¨obner-Shirshovbases. arXiv:1106.4753

Yuqun Chen, Yu Li and Qingyan Tang, Gr¨obner-Shirshovbases for some Lie algebras. Siberian Math. J. to appear. arXiv:1303.5366 Some new surveys

Surveys:

L. A. Bokut, Yuqun Chen and K. P. Shum, Some new results on Groebner-Shirshov bases, Proceedings of International Conference on Algebra 2010, Advances in Algebraic Structures, 2012, pp.53-102. arXiv:1102.0449.

L. A. Bokut and Yuqun Chen, Gr¨obner-Shirshovbases and their calculation, submitted. (70 pages) Thank you !