The order versions of the Hahn--Banach Theorem and envelopes. I. Homogeneous functions

We present here the General formulation of the problem of existence and construction of upper and lower envelope for an arbitrary function with values from the completion of the ordered set $\mathrm{S}$ for a certain class of functions with values in $\mathrm{S}$. The task is parsed only for the simplest case of model class of homogeneous functions. Consider only order-algebraic version without the involvement of the topology.