Subalgebra analogue of Standard bases for ideals in $K[[t_1,t_2,\dots ,t_m]][x_1,x_2,\dots ,x_n]$

In this paper, we develop a theory for Standard bases of $K$-subalgebras in $K[[t_1,t_2,\dots ,t_m]][x_1,x_2,...,x_n]$ over a field $K$ with respect to a monomial ordering which is local on $t$ variables and we call them Subalgebra Standard bases. We give an algorithm to compute subalgebra homogeneous normal form and an algorithm to compute weak subalgebra normal form which we use to develop an algorithm to construct Subalgebra Standard bases. Throughout this paper, we assume that subalgebras are finitely generated.