  The theory of "subalgebra basis" analogous to standard basis (the
generalization of Gr\"{o}bner bases to monomial ordering which are not
necessarily well ordering \cite{GP1}.) for ideals in polynomial rings over a
field is developed. We call these bases "SASBI Basis" for "Subalgebra Analogue
to Standard Basis for Ideals". The case of global orderings, here they are
called "SAGBI Basis" for "Subalgebra Analogue to Gr\"{o}bner Basis for Ideals",
is treated in \cite{RS1}. Sasbi bases may be infinite. In this paper we
consider subalgebras admitting a finite Sasbi basis and give algorithms to
compute them. The algorithms have been implemented as a library for the
computer algebra system SINGULAR \cite{GPS1}.
