MATH.22:5 - Archetypal Grounding

MATH.22:5.1 - Drop commutativity while preserving invertible composition

Suppose a collection has an associative operation, an identity e and an inverse for every element. These are the group assumptions. Add commutativity, xy=yx, and familiar rearrangements become available. For example, MATH.19’s argument proves (xy)^n=x^n y^n.

The intended new use composes invertible operations whose order matters. Remove commutativity and retain the group assumptions.

Cancellation survives, although its proof may change. A commutative proof multiplies ax=ay on the right by the inverse of a, then rearranges axa⁻¹ and aya⁻¹ to obtain x=y. This uses commutativity. A replacement proof multiplies on the left: a⁻¹(ax)=a⁻¹(ay). Associativity gives (a⁻¹a)x=(a⁻¹a)y, hence x=y by the inverse and identity laws. The theorem therefore survives after removing commutativity.

Unrestricted rearrangement fails. Consider all permutations of {1,2,3}, composed with the rightmost permutation acting first. Composition is associative, the identity leaves each element fixed, and each bijection has an inverse. Let f swap 1 and 2, and let g swap 2 and 3. Then fg sends 1 to 2 to 3 to 1 as a cycle, while f² and g² are identities. Thus (fg)² sends 1 to 3, but f²g² sends 1 to 1.

This is a model of the group assumptions with noncommuting elements. The two-element group {e,h}, with h²=e, is a model where every pair commutes. Under ordinary classical group logic, the two models show that commutativity is independent of the retained group axioms.

The revised theory now admits ordered compositions. It retains cancellation and inverses. Rearrangement remains usable for particular pairs whose commutation is established; it has ceased to be a general permission.

MATH.22:5.2 - Remove total comparability and inspect a proposed new operation

A partial order is reflexive, antisymmetric and transitive. A total order additionally compares every pair: x<=y or y<=x. Suppose the work needs to retain pairs for which neither direction holds.

Remove total comparability. On pairs of natural numbers, define (a,b)<=(c,d) when a<=c and b<=d. The three partial-order laws follow coordinate by coordinate. The pairs (1,0) and (0,1) are incomparable. The ordinary order on natural numbers supplies a model with total comparability; the pair order supplies a model without it.

Now ask for a least common upper bound of two objects. In the pair order, it is the coordinatewise maximum. It is above both inputs, and any common upper bound is above each coordinatewise maximum. This proves both the construction and its leastness.

The partial-order axioms alone do not supply that operation. Take four distinct objects a,b,u,v. Besides reflexive comparisons, require a<=u, a<=v, b<=u and b<=v, and no others. This is a partial order. Both u and v are upper bounds of a and b, but neither is below the other; a and b are not upper bounds. There is no least upper bound of the pair.

Consequently, writing “x join y” for every pair in an arbitrary partial order adds an existence requirement unless a construction has already supplied it. The work can select an order where joins exist, add the join assumption and accept its narrower class of models, or undertake a construction that enlarges the objects. Choosing one of u or v alone supplies an upper bound, not the missing least one.

This separates two changes that a generic instruction to “relax the order” would hide: allowing incomparability and requiring a combining operation. A model of alternatives can use the resulting order; any decision to prefer one alternative remains a separate choice.

MATH.22:5.3 - Introduce total reciprocal notation without retaining a false law

Suppose arithmetic uses field laws, including 0!=1 and distributivity, and the reciprocal law x*inv(x)=1 for x!=0. The proposed change makes inv a total operation and asks that same law to hold at zero.

The retained laws already imply 0*y=0. Indeed, (0+0)*y=0*y+0*y by distributivity, and cancelling one 0*y from 0*y=0*y+0*y leaves 0=0*y. The proposed new reciprocal law at zero would therefore imply 0=1. The retained assumptions and the new law conflict.

A different change works: over the rational numbers, define inv(0)=0 and inv(x)=1/x for x!=0. This gives a total operation, retains the nonzero reciprocal law, and leaves the field operations unchanged. It has not made cancellation of a zero factor valid.

For a use involving the equation x*y=x*z, cancellation still requires x!=0. At x=0, all rational y,z satisfy the equation. A calculation using total reciprocal notation must retain that condition or inspect the zero case.

The result is a consistent construction within rational arithmetic and a revised law with its condition, rather than the initially requested unconditional inverse law. Total notation and stronger algebraic permission have been separated.