We apply the method of Grobner-Shirshov bases for replicated algebras devel-oped by Kolesnikov to offer a general approach for constructing free products of associative trialgebras(or trioids).In particular,the open p...We apply the method of Grobner-Shirshov bases for replicated algebras devel-oped by Kolesnikov to offer a general approach for constructing free products of associative trialgebras(or trioids).In particular,the open problem of Zhuchok on constructing free products of trioids is solved.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
By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schube...By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk＇s formula.展开更多
基金supported by the Natural Science Foundation of Huizhou University(2022JB035,HZU202001)the Guangdong Basic and Applied Basic Research Foundation(2023A1515011690)+3 种基金the Characteristic Innovation Project of Guangdong Provincial Department of Education(2023KTSCX145)supported by the NNSF of China(11571121,12071156)supported by the NNSF of China(12101248)by the China Postdoctoral Science Foundation(2021M691099).
文摘We apply the method of Grobner-Shirshov bases for replicated algebras devel-oped by Kolesnikov to offer a general approach for constructing free products of associative trialgebras(or trioids).In particular,the open problem of Zhuchok on constructing free products of trioids is solved.
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By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk＇s formula.
