In this post, we continue the discussion from the previous in the series.
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Let’s take another example, of constructed compatibility from semigroups to monoids.
Let S be a semigroup. We can define a monoid S’ based on S by adjoining an identity to S; explicitly, we can define something like
S’ = {(s, 0): s in S} 
where we define multiplication by (0, 1) such that this is the identity.
Conversely, if we start with a monoid S’, then the identity must be unique, and if S is the subset of non-identity elements of S, then the construction above on S yields something isomorphic to S’.
As a result, whenever we have a semigroup, we can immediately work with the corresponding monoid; no assumptions are made here, and thus the data encoded is the same.
With this in mind, do we need to redefine our goal with constructed compatibility theory? (Is the formalization we wrote down earlier really the one that corresponds to what we want?) This is related to: what data actually leads to a result? (See my post on The Unreasonable Effectiveness of Definitions in Mathematics.)
Given a model M of an axiom set A, if we can construct a model M’ of an axiom set A’ purely based off of M, and if such a construction is universal (any construction on top of M that yields a model of A’ is uniquely determined up to isomorphism), then we call this constructed compatibility from A to A’.
This is really just universal constructions! We’re asserting that a model of A’ can be universally constructed from a model of A. This allows us to immediately construct and use A’ whenever we have a model of A.
If we have “constructed compatibility” from A to B and from B to A, then in general this would really just be illuminating different points of view on the object A or equivalently B (for example, the algebraic and geometric views of lattices.) So how does this relate to my question about what data is needed to guarantee a result?
Really, the only data needed to guarantee a result are the axioms and assumptions. Thus, if we use intermediaries, then those have to be instances of: (1) abstraction, both in terms of repeated use as well as mental ease of use in focusing on a subset of properties, and (2) derived data, from the data appearing in the assumptions. Logically speaking, nothing else is possible.
So in order to illuminate exactly what data is leading to what results, we need to trace back what universal constructions are being performed and used as intermediaries, and we need to find the “minimal” set of data for which no universal constructions can be performed “within” it to “reduce the data further.”
Our constructed compatibility theory is thus equivalent to existence of universal constructions.
Going back to the starting example, we know that the data for a monoid is actually redundant: all the data is contained within the semigroup of non-identity elements. Thus, if a statement about a monoid doesn’t use the identity anywhere, then it applies to the semigroup of non-identity elements regardless of how much the identity is used to prove it.
It would still be interesting to try to “reduce the data as much as possible,” and more precisely to find “the minimal data,” for which there don’t exist any universal constructions within. It remains to investigate this.
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EDIT (2025): I don’t think I fully understood universal constructions here, so comparisons of my studies to that concept may be false. It remains to investigate this topic further.
