A new Composition-Diamond lemma for dialgebras

Let $Di\langle X\rangle$ be the free dialgebra over a field generated by a set $X$. Let $S$ be a monic subset of $Di\langle X\rangle$. A Composition-Diamond lemma for dialgebras is firstly established by Bokut, Chen and Liu in 2010 \cite{Di} which claims that if (i) $S$ is a Gr\"{o}bner-Shirshov basis in $Di\langle X\rangle$, then (ii) the set of $S$-irreducible words is a linear basis of the quotient dialgebra $Di\langle X \mid S \rangle$, but not conversely. Such a lemma based on a fixed ordering on normal diwords of $Di\langle X\rangle$ and special definition of composition trivial modulo $S$. In this paper, by introducing an arbitrary monomial-center ordering and the usual definition of composition trivial modulo $S$, we give a new Composition-Diamond lemma for dialgebras which makes the conditions (i) and (ii) equivalent. We show that every ideal of $Di\langle X\rangle$ has a unique reduced Gr\"{o}bner-Shirshov basis. The new lemma is more useful and convenient than the one in \cite{Di}. As applications, we give a method to find normal forms of elements of an arbitrary disemigroup, in particular, A.V. Zhuchok's (2010) and Y.V. Zhuchok's (2015) normal forms of the free commutative disemigroups and the free abelian disemigroups, and normal forms of the free left (right) commutative disemigroups.