ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

This is a booklet in a series with the title ”Abstract algebraic structures”, which I have
written for the algebra courses (E1,BE2) part of the mathematics education at RUC
These booklets should be seen in connection with another series with the title ”Concrete
algebraic structures”. These two series should be viewed as the jaws of a pair of pinchers.
It would obviously be futile (and frustrating) to teach abstract algebra without involving
concrete examples and it would be poor (and devoid of perspective) only to build concrete
examples without involving the underlying abstract structures.
And then again, there is a certain beauty in emphasizing the abstract character by isolating it
and let its top-down nature appear clearly as in done in the controversial idol ”The Elements
of Mathematics” by Bourbaki. Starting with the coarsest structures and then step by step
refining by adding structural elements. All results which occur in many circumstances are
proved once and for ever in its most elementary form.
In the same way there is some satisfaction combined with letting each concrete structure stand
as simple as possible without to much ado, das Ding an sich. And the pleasure by recognizing
the essentially same type of argument recurring over and over in different disguises.
Besides having the aesthetic satisfaction from the pure abstraction and the pure pleasure
from the concrete details, both perspectives have a great cognitive influence and contributes
to the development of competencies which are essential to any mathematician.
I have chosen to emphasize these to oppositely directed but coordinated perspectives which
is implied in the two series.
The individual concrete structures are displayed in separate expositions without mutual
references. Subjects which are needed at more places are repeated at each place. But the
choice of details is made in such a way as to best possible deliver stuff to be used in examples
in the abstract part.
The text of the abstract part is not printed but can be found at
http://milne.ruc.dk/~am/algebra
Anders Madsen, may 2012
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