MATH.2:11 - SoTA-Echoing

Question: when can operations be inherited by classes without depending on an arbitrary representative?

Adopt congruence and quotient algebra from Burris and Sankappanavar, A Course in Universal Algebra, corrected 2012 edition, Chapter II §5, pp.35-36. The compatibility condition and quotient definition supply :4.3-:4.4 for total operations. This is a constructive answer whose general reason applies beyond any one example.

The serious alternative is an equivalence relation chosen only by a shared property. It costs less to state, but the absolute-value example cannot support the required addition. The compatibility argument adds work where the quotient operation is claimed and resolves that ambiguity.

Adapt the construction to the partial-operation use by preserving domain membership as well as output classes. The alternative existential convention permits an operation when some class member permits it. That is useful for a possible-transition question but loses availability at the represented state. The permission case selects the stronger convention and places its added condition in :4.3. The cited total-algebra passage does not supply this partial-operation convention.

Adopt the composition consequences of imposed path equations from Fong and Spivak’s Seven Sketches in Compositionality, §3.2.2, pp.84-85, in :5.3. Reopen the choice if the receiving question accepts a weaker bound or set of possibilities at lower cost, or a new operation distinguishes members of an existing class. Neither source establishes that a particular physical or organizational interpretation is adequate.