KAES: Critical point

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Dwight Read Sept 9 to 2008 to Doug,

When you click on Self in the kin term map (or Father or Mother) and then go to the algebra constructor, you are (implicitly) assuming that the algebra is constructed by mapping terms from the kin term to the algebra.  This is not correct and doesn't work because the algebra constructor is not building an algebra by transferring or translating kin terms to the algebra. This is a critical point.

Rather, the algebra is a stand-alone construction, with the exception that the algebra constructor will examine the simplified kin term map (and only the simplified kin term map as only the simplified kin term map has "unambiguous" structural form) for a structural property such as: Does the simplified kin term map have an identity element?  Note that the algebra constructor does not determine that the simplified kin term map has an identity element from the presence of the Self term in the kin term map, but from its discovery that the Self element is, in fact, an identity element in the simplified kin term map.  You could have used "Quiggle" instead of "Self" as the name in the kin term map and the algebra constructor would still deduce from examining the structure of the kin term map and how kin term products work that there is an identity term in the simplified kin term map and so it should add an identity element, I,  to the algebra generating set.  In so doing, the algebra constructor "forgets" that I corresponds to Self (or Quiggle or whatever name was used in the kin term map).

This is a critical point.  Nowhere does the program "carry over" anything other than structural properties from the kin term map to the algebra.  When it tests for an isomorphism, for example, it must literally construct a mapping from each algebra element generator to a correspoinding kin term based on the structural property of the algebra and the structural property of the kin term map.  It has to discover that I logically corresponds to Self as the link I --> Self is not a structural property.  Rather the link is from identity element to identity element and from this the name of the identity element in the algebra (namely, I) is linked to the name of the identity kin term (namely Self).

The next step, Enter Generators, uses the same logic.  The algebra constructor looks at the structure of the simplified kin term map and determines how many generating elements it has.  In the case of the AKT simplified kin term map, it determines that there is a single generating term.  Then it adds a generating element, P,  to the set of generators.  It does NOT carry over from the kin term map the information that the generating term is Parent.  Rather, the algebra constructor has no information about the kin term map except it has an identity element and a single generator as determined from the structure of the simplified kin term map.  Later, it will deduce that P must correspond to Parent from the structure of the algebra that is being generated and the structure of the kin term map to which the algebra is being compared.  (Of course, if the algebra does not have the same structure as the kin term map it may well make a "wrong" deduction from our assumption that Parent should be the generating term.  This is where the power of the algebraic modeling comes in -- the model is not a re-write in some sense of the kin term map, but an independent construction based on the fundamental structural properties that makes one kinship terminology different than another kinship terminology.)

So this is why clicking on Self and then going to Algebra Operation doesn't work.  Do what it says -- simplify the kin term map and then go to the Algebra Operations.

As the construction continues (e.g., when the Reciprocal Structure for the Base Algebra is constructed) the algebra constructor does not use any information from the kin term map.  It simply makes an isomorphic copy of the generating set {I, P}, namely {I, C} (with the same identify element --it is non-trivial whether the identity element in the isomorphic generating is the same or a different element than the identity element in the Base Algebra as occurs with the classificatory terminologies).  If you make a graph at this point (before adding the reciprocal equation), you are making the graph for the free algebra with generators {I, P, C} and no structural equations (beyond those that make I an identity element) and that free algebra has the form of a fractal as you will see. When you click on "Add Reciprocal Element Equations" it invokes the structural definition of reciprocal elements, namely that PC = I.  This is part of the theory of kin ship terminology structures -- that this is universally the structural definition for making P and C reciporcal elements as we understand reciprocal kinship terms. (That the equation is PC = I and not CP = I follows from the Base algebra being a structure for the ascending kin terms.)

There are some properties that are terminology specific, such as PPS = 0 (parent of parent of spouse is not a kin term) such an equation has to be determined by examining the kin term map. (The equation can be deleted and it is interesting to see how much more complex the affinal part of the algebra becomes without this equation).

Dwight