  The notion of surreal number was introduced by J.H. Conway in the mid 1970's:
the surreal numbers constitute a linearly ordered (proper) class $No$
containing the class of all ordinal numbers ($On$) that, working within the
background set theory NBG, can be defined by a recursion on the class $On$.
Since then, have appeared many constructions of this class and was isolated a
full axiomatization of this notion that been subject of interest due to large
number of interesting properties they have, including model-theoretic ones.
Such constructions suggests strong connections between the class $No$ of
surreal numbers and the classes of all sets and all ordinal numbers.
  In an attempt to codify the universe of sets directly within the surreal
number class, we have founded some clues that suggest that this class is not
suitable for this purpose. The present work, that expounds parts of the PhD
thesis of the first author (\cite{Ran18}), establishes a basis to obtain an
"algebraic (set) theory for surreal numbers" along the lines of the Algebraic
Set Theory - a categorial set theory introduced in the 1990's based on the
concept of ZF-algebra: to establish abstract and general links between the
class of all surreal numbers and a universe of "surreal sets" similar to the
relations between the class of all ordinals ($On$) and the class of all sets
($V$), that also respects and expands the links between the linearly ordered
class of all ordinals and of all surreal numbers.
