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====2.4.2.  Algebraic Aspects====

====2.4.2.  Algebraic Aspects====

Finally, there is a body of mathematical work that investigates algebraic and differential geometry over finite fields.  This usually takes place at such high levels of abstraction that the field of two elements is just another special case.  In this work the principal focus is on the field operations of sum (<math>+</math>) and product (&nbsp;<math>\cdot</math>&nbsp;), which correspond to the logical operations of exclusive disjunction (xor, neq) and conjunction (and), respectively.  The stress laid on these special operations creates a covert bias in the algebraic field.  Unfortunately for the purposes of logic, the totality of boolean operations is given short shrift on the scaffold affecting this algebraic slant.  For example, there are sixteen operations just at the level of binary connectives, not to mention the exploding population of k-ary operations, all of which deserve in some sense to be treated as equal citizens of the logical realm.

Finally, there is a body of mathematical work that investigates algebraic and differential geometry over finite fields.  This usually takes place at such high levels of abstraction that the field of two elements is just another special case.  In this work the principal focus is on the field operations of sum (<math>+</math>) and product (&nbsp;<math>\cdot</math>&nbsp;), which correspond to the logical operations of exclusive disjunction (xor, neq) and conjunction (and), respectively.  The stress laid on these special operations creates a covert bias in the algebraic field.  Unfortunately for the purposes of logic, the totality of boolean operations is given short shrift on the scaffold affecting this algebraic slant.  For example, there are sixteen operations just at the level of binary connectives, not to mention the exploding population of ''k''-ary operations, all of which deserve in some sense to be treated as equal citizens of the logical realm.

Moreover, from an algebraic perspective the dyadic or boolean case exhibits several features peculiar to itself.  Binary addition (<math>+</math>) and subtraction (<math>-</math>) amount to the same operation, making each element its own additive inverse.  This circumstance in turn exacts a constant vigilance to avert the verbal confusion between algebraic negatives and logical negations.  The property of being invertible under products (&nbsp;<math>\cdot</math>&nbsp;) is neither a majority nor a typical possession, since only the element 1 has a multiplicative inverse, namely itself.  On account of these facts the strange case of the two element field is often set aside, or set down as a "degenerate" situation in algebraic studies.  Obviously, in turning to take it up from a differential standpoint, any domain that confounds "plus" and "minus" and "not equal to" is going to play havoc with our automatic intuitions about difference operators, linear approximations, inequalities and thresholds, and many other critical topics.

Moreover, from an algebraic perspective the dyadic or boolean case exhibits several features peculiar to itself.  Binary addition (<math>+</math>) and subtraction (<math>-</math>) amount to the same operation, making each element its own additive inverse.  This circumstance in turn exacts a constant vigilance to avert the verbal confusion between algebraic negatives and logical negations.  The property of being invertible under products (&nbsp;<math>\cdot</math>&nbsp;) is neither a majority nor a typical possession, since only the element 1 has a multiplicative inverse, namely itself.  On account of these facts the strange case of the two element field is often set aside, or set down as a "degenerate" situation in algebraic studies.  Obviously, in turning to take it up from a differential standpoint, any domain that confounds "plus" and "minus" and "not equal to" is going to play havoc with our automatic intuitions about difference operators, linear approximations, inequalities and thresholds, and many other critical topics.