In this paper, we obtain （non-commutative） GrSbner-Shirshov bases for rect- angular bands, small extensions of semigroups and Bruck-Reilly extensions of monoids in terms of the general product.
A finite GrSbner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite GrSbner-Shirshov bases associated to the n...A finite GrSbner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite GrSbner-Shirshov bases associated to the natural degree-lexicographic ordering on the corresponding free algebra. The latter is in contrast with the case of a strongly related class of algebras, called Chinese algebras.展开更多
Bokut, Chen and Liu in 2010 gave a Composition-Diamond lemma for dialge- bras. In this paper, by introducing an arbitrary monomial-center ordering, we give a new Composition-Diamond lemma for dialgebras which makes th...Bokut, Chen and Liu in 2010 gave a Composition-Diamond lemma for dialge- bras. In this paper, by introducing an arbitrary monomial-center ordering, we give a new Composition-Diamond lemma for dialgebras which makes the two conditions in Bokut, Chen and Liu＇s result equivalent. The new lemma is more useful and convenient than the one Bokut, Chen and Liu got. We show that every ideal of the free dialgebra generated by a set X has a unique reduced GrSbner-Shirshov basis. As applications, we give a method to find normal forms of elements of an arbitrary disemigroup, in particular, two Zhuchoks＇ normal forms of the free commutative disemigroups and the free abelian disemigroups, and normal forms of the free left （right） commutative disemigroups.展开更多
In this paper, we construct the GrSbner-Shirshov bases for degenerate Ringel- Hall algebras of types A and G2 from the multiplication formulas of the corresponding generic extension monoid algebras.
In this paper, we find Hall-Shirshov type bases for free pre-Lie algebras. We show that Segal＇s basis of a free pre-Lie algebra is a type of these bases. We give a nonassociative GrSbner-Shirshov basis S for a free p...In this paper, we find Hall-Shirshov type bases for free pre-Lie algebras. We show that Segal＇s basis of a free pre-Lie algebra is a type of these bases. We give a nonassociative GrSbner-Shirshov basis S for a free pre-Lie algebra such that Irr（S） is a monomial basis （called normal words） of a free pre-Lie algebra, where Irr（S） is the set of all nonassociative words, not containing maximal nonassociative words of polynomials from S. We establish the Composition-Diamond lemma for free pre-Lie algebras over the basis of normal words and the degree breadth lexicographic ordering.展开更多
文摘In this paper, we obtain （non-commutative） GrSbner-Shirshov bases for rect- angular bands, small extensions of semigroups and Bruck-Reilly extensions of monoids in terms of the general product.
文摘A finite GrSbner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite GrSbner-Shirshov bases associated to the natural degree-lexicographic ordering on the corresponding free algebra. The latter is in contrast with the case of a strongly related class of algebras, called Chinese algebras.
文摘Bokut, Chen and Liu in 2010 gave a Composition-Diamond lemma for dialge- bras. In this paper, by introducing an arbitrary monomial-center ordering, we give a new Composition-Diamond lemma for dialgebras which makes the two conditions in Bokut, Chen and Liu＇s result equivalent. The new lemma is more useful and convenient than the one Bokut, Chen and Liu got. We show that every ideal of the free dialgebra generated by a set X has a unique reduced GrSbner-Shirshov basis. As applications, we give a method to find normal forms of elements of an arbitrary disemigroup, in particular, two Zhuchoks＇ normal forms of the free commutative disemigroups and the free abelian disemigroups, and normal forms of the free left （right） commutative disemigroups.
文摘In this paper, we construct the GrSbner-Shirshov bases for degenerate Ringel- Hall algebras of types A and G2 from the multiplication formulas of the corresponding generic extension monoid algebras.
文摘In this paper, we find Hall-Shirshov type bases for free pre-Lie algebras. We show that Segal＇s basis of a free pre-Lie algebra is a type of these bases. We give a nonassociative GrSbner-Shirshov basis S for a free pre-Lie algebra such that Irr（S） is a monomial basis （called normal words） of a free pre-Lie algebra, where Irr（S） is the set of all nonassociative words, not containing maximal nonassociative words of polynomials from S. We establish the Composition-Diamond lemma for free pre-Lie algebras over the basis of normal words and the degree breadth lexicographic ordering.
