A New Composition-Diamond Lemma for Dialgebras

Abstract

Bokut, Chen and Liu in 2010 gave a Composition-Diamond lemma for dialgebras. 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 Gröbner-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.