Fundamental issues of abstraction UTK M351 Algebra I Jochen Denzler This is in particular for those of you who find the mere list of field and ring axioms abstract and confusing already from the beginning A Parable The FBI puts out a poster WANTED FOR hwhateveri John Doe and a picture They seize somebody who matches the picture and is otherwise suspected for the charge He says I can t be the guy you were looking for John Doe but my name is Charly Miller Pretty dumb excuse isn t it John Doe was not meant as the name of one particular individual but represents an unknown unspecified name A certain man identified by a drivers license as named John Doe his SSN is 123 45 6789 complains since he has nothing to do with the charges by the FBI A dimwit His name John Doe is the name of an individual and thus distinct from the John Doe on the FBI poster Distinct means they are not necessarily the same even though the person John Doe SSN 123 45 6789 might just happen to be the wanted man by a quirky accident Abstract Algebra In the axiom about the identity or neutral element for addition which reads There is some element 0 X such that for every a X it holds a 0 0 a a N the 0 is like the FBI s John Doe Expanded version of the axiom There is some element in X we ll call it 0 such that etc Of course we call it 0 because it plays a similar role like the number 0 plays among the real numbers Like the FBI thinks John Doe is kind of a typical name for suspects living here They won t call their unidentified suspects Maximilian Hinterhuber For those with some computer science background the double use of symbols like 0 or 1 is the same issue as using metacharacters How to avoid thinking outside the box There is some benefit either way but right at the beginning you think out of the box so naturally that I must train your ability not to do it Consider the set X of odd integers with the usual addition and multiplication of integers I ask you Does it satisfy the axiom N You answer No because 0 the neutral alias identity element for addition is not an odd number i e not in X In some sense you are right but you are already thinking outside the box If you thought inside the box you would say No because there is no odd integer which when added to any odd integer a returns that same integer a Rather than saying outside the box there is some 0 but it s disqualified you say inside the box no there isn t any I don t know nor care what s outside In some sense I have already worsened the difficulty by posing the very problem I seduced you to think outside the box when I said set X of odd integers with the usual addition Usual addition In posing the very problem I have thought outside the box Inside the box there is no addition at all 5 3 No not 8 because there is no such a thing called 8 in X Strictly speaking I should not have given this problem But I had to perpetrate such inaccuracy first so I could teach you this distinction I asked to check the commutative law C You say True order is not of essence when adding any integers so in particular it is not of essence when adding odd integers You are right of course but you are again thinking outside the box If you thought inside the box you d say C not applicable After all how could I claim or deny a b b a if M351 p 1 Fundamentals the very in this statement already fails to make sense An occasion where you would automatically think inside the box I gave you the example of the set E O with the definitions for and E E E E O O E O O O E E O O E E E E O O O This set does not come as a subset of some other larger set unlike the previous example In this case there is no outside the box If I give you a problem of this kind I could not even make up one that fails to satify the closure axioms If you want to check if Inv holds in this set you cannot start to find out what E would be and then see if it s one of E and O Instead you have to examine the elements E and O and see which of them if any qualifies as an additive inverse of E These bloody closure axioms are a foggy obfuscation at this stage They wouldn t even be in the textbooks were it not for the sole purpose to adapt to preexistent difficulties with abstraction When I say There is a function X X X that maps a b to something called a b then I have already included the Clo axiom it s buried in the X behind the arrow here marked boldface for emphasis X X X No need to repeat it Note that our textbook honorably omits closure axioms I perpetrated them viciously to connect this class to what we had in 300 and to clarify a fundamental point Subsets Subrings and Subhwhateveri s If you already have a ring like Z or R or a field like R then you can consider a subset as we did before when considering the set of odd integers subset of Z or the set of rationals Q subset of R Then you can ask if this subset is a ring or a field in its own right with the and defined already in the larger set This is quite familiar to you and it is in this spirit that you studied the problems which axioms are verified in Z in the set of even integers in the set of odd integers etc When this is how you start your exploration the laws of associativity commutativity and distributivity become easy They hold for all integers therefore in particular for the odd ones When the set you are studying comes as a subset of some larger set then it is appropriate to think out of the box as you naturally do in this case So I did not mean above to say it s wrong to think of a certain set as a subset of a larger set that is known to be a field or a ring I only mean to say you are on the wrong track if you can think of it only that way When you are studying a subset of a ring and ask if this subset is a ring in it s own right and if so we ll call it a subring then closedness is the big issue And other issues are whether the special objects whose existence is required by certain …