Title: A new Composition-Diamond lemma for dialgebras,
Abstract: 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
Gröbner-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 Gröbner-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.