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\centerline {\bf Dialgebra}
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A {\it diassociative algebra}, or {\it dialgebra} for short, over the field $K$,
 is a $K$-module $D$ equipped with two $K$-linear maps
$$\dashv\ : D\otimes D\longrightarrow D,\qquad {\rm and}\qquad 
\vdash\ : D\otimes D\longrightarrow D,$$
called respectively the {\it left product} and {\it the
right product}, 
satisfying the following di-associativity axioms
$$
 \Biggl\{ \matrix { x\dashv(y\dashv z) =(x\dashv y)\dashv
z  =x\dashv (y\vdash z)
\cr  (x\vdash y)\dashv z = x\vdash (y\dashv z)\hfill\cr
(x\dashv y)\vdash z = x\vdash (y\vdash z) =
(x\vdash y)\vdash z .\cr}
$$
A {\it bar-unit} in $D$ is an element $e\in D$ such that $
x\dashv e=x=e\vdash x\quad \hbox{for\ all}\quad x\in D$.
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\noindent {\bf Examples.}

\noindent 0. An associative algebra is a particular case of dialgebra for which $\dashv\, =\, \vdash
$.

\noindent 1. Let $(A, d)$ be a differential associative algebra. So, by hypothesis, $d(ab) = da\,
b+a\, db$. Define left and right products on $A$
 by the formulas
$$
x\dashv y:= x\ dy \quad {\rm and}\quad x\vdash y := dx\ y.
$$
It is immediate to check that $A$, equipped with these two products, is a dialgebra.

\noindent 2. Let $A$ be an associative algebra and let $M$ be an $A$-bimodule. Let $f:M\to A$ be
an $A$-bimodule map. Then one can put a dialgebra structure on $M$ as follows :
$$m\dashv m':= mf(m'),\qquad {\rm and } \qquad m\vdash m':= f(m)m'.
$$
\noindent {\bf Leibniz algebra.} It is well-known that any associative algebra gives rise to a Lie
algebra by the formula
 $[a,b] = ab-ba$. Similarly, any dialgebra gives rise to a {\it Leibniz algebra} by the
formula $[a,b] = a\dashv b-b\vdash a.$
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\noindent {\bf Homology of dialgebras.} The chain-complex of a dialgebra $D$ involves the set of
 {\it planar binary trees} with $n$ vertices denoted $Y_n$ :
$$
CY_n(D) := k[Y_n]\otimes D^{\otimes n}.
$$
The boundary map is of Hochschild type: $d= \sum_{i=1}^{i=n} (-1)^id_i$ where
$$
d_i(y;x_1, \ldots, x_n) = (d_i(y); x_1,\ldots,
x_i\,{}^{\vdash}_{\dashv}\, x_{i+1}, \ldots, x_n). 
$$
The tree $d_i(y)$ is obtained by removing the $i$th leaf from $y$, and the operation is either 
 $\dashv$ or $\vdash$ depending on the orientation of the  $i$th leaf.
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\noindent {\bf Ariane's thread:}

\noindent A. Frabetti, {\it Dialgebra homology of associative algebras}, C.
R. Acad. Sci. Paris 325, (1997), 135-140. 

\noindent J.-L. Loday, {\it Alg\`ebres ayant deux op\' erations associatives (dig\`ebres)}, C.
R. Acad. Sci. Paris 321 (1995), 141-146.

\hfill ${ }_{\rm JLL/mek/Dialgebra/12.97}$

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