  In \cite{PrzytyskiTraczyk} J.H.Przytyski and P.Traczyk introduced an
algebraic structure, called {\it a Conway algebra,} and constructed an
invariant of oriented links, which is a generalization of the Homflypt
polynomial invariant. On the other hand, in \cite{KauffmanLambropoulou} L. H.
Kauffman and S. Lambropoulou introduced new 4-variable invariants of oriented
links, which are obtained by two computational steps: in the first step we
apply a skein relation on every mixed crossing to produce unions of unlinked
knots. In the second step, we apply the skein relation on crossings of the
unions of unlinked knots, which introduces a new variable. In this paper, we
will introduce a generalization of the Conway algebra $\widehat{A}$ with two
binary operations and we construct an invariant valued in $\widehat{A}$ by
applying those two binary operations to mixed crossings and self crossings
respectively. Moreover, the generalized Conway algebra gives us an invariant of
oriented links, which satisfies non-linear skein relations.
