  The presence of infinitesimals is traced back to some of the most general
algebraic structures, namely, semigroups, and in fact, magmas, [1], in which
none of the structures of linear order, field, or the Archimedean property need
to be present. Such a clarification of the basic structures from where
infinitesimals can in fact emerge may prove to have a special importance in
Physics, as seen in [4-16]. The relevance of the deeper and simpler roots of
infinitesimals, as they are given in Definitions 3.1 and 3.2, is shown by the
close connection in Theorem 4.1 and Corollary 4.1 between the presence of
infinitesimals and the non-Archimedean property, in the particular case of
linearly ordered monoids, a case which, however, has a wide applicative
interest.
