March 16th, 2012
“Let R be a ring and M be a left R-module. M is called an amply weak radical supplemented module in case M = U + V implies that U has a weak Rad-supplement L ? V. In this paper various properties of this module is developed and it is shown that R is semilocal if and only if every left R-module is amply weak Rad-supplemented.” — Amply Weak Radical Supplemented Modules by Burcu Nisanci
“In ring theory, a branch of mathematics, a radical of a ring is an ideal of ‘bad’ elements of the ring. The strong radical is defined as either the intersection of the maximal two-sided ideals, the intersection of all maximal modular ideals, or the upper radical of the class of all simple rings with identity… The weak radical coincides with the both radical class and the semiradical class.” – Wikipedia and math-scienced journal article (partially paraphrased and combined)
Since I’ve been so critical of you, I thought I’d be nice
And provide you with some free advice.
You say Newt dropping out would be good for you,
But it turns out that just might not be true.
For you, it might be best for Newt to stay in the race,
But in a weakened state in a distant third place.
Why? If Newt drops out, his supporters will be split,
If he stays in, you can attract more of Newt’s supporters than Mitt.
If you want the GOP nomination to win
(And if Newt continues to stay in),
The key’s not where Newtites jump ship to, but when.
For reasons why Newtites who favor you might be the first ones to go,
Please read my theoretical analysis below…
In advanced mathematics, a “strong radical” is the intersection of the maximal two-sided ideals, the intersection of all maximal modular ideals, or the upper radical of the class of all simple rings with identity,” or in political terms, the intersection of different parts or “identities” of the maximal radical base. A “weak radical” is the opposite, and simultaneously attempts to be both radical and semiradical, i.e., flip-flopping back and forth, even if that “weak radical” was previously considered the conservative R option instead of the left R-module, or M (for Mitt and/or moderate?).
Based on the poll results released today, Romney and Santorum almost exactly split (40% versus 39% respectively) split Gingrich supporters’ second-choice vote (Ron Paul gets 12%). While those numbers are important in figuring out what happens to Romney’s and Santorum’s respective electoral prospects if Gingrich drops out, it can be misleading in figuring what happens if Gingrich stays in, but in a weakened state that causes him to start bleeding supporters. In that case, when and which Gingrich supporters jump ship becomes as important as whom they switch to. Given an approximately 50-50 split, the impact of a weakening Gingrich then depends completely on who jumps ship first.
If former Gingrich supporters who secondarily prefer Santorum jump ship first and in greater numbers, the best of all possible worlds for Santorum then becomes neither Gingrich’s exit, nor a strong Gingrich. The best outcome for Santorum may be what now looks like the most likely one: a weakened Gingrich whose Santorum-leaning supporters desert, but who’s still strong enough to hold onto his Romney-leaning supporters.
Note however that the opposite might also be true: If Newt’s Santorum-leaning supporters are more committed and less likely to abandon Newt, then the earlier conventional wisdom is again correct: a weakened Newt’s remaining in the race disproportionately benefits Romney, while his departure would benefit Santorum by also freeing up the Santorum-leaners. To take that a step further, the kind of Gingrich candidacy that most helps Santorum might be a stronger, almost evenly-matched Gingrich. A closer match-up might also have the benefit to Santorum of providing more motivation to Santorum supporters, resulting in greater Santorum turnout.
Some anecdotal evidence may actually support the weak-Newt-radical-benefits-Romney view. One can surmise that if Gingrich’s supporters are divided into two groups by practicality versus ideology, then the practical ones would be more likely to support Romney over Santorum, and to do so sooner, thus being among the first to desert Gingrich for Romney when Newt’s victory looked unlikely. (Either way, as I advanced in Harsh Math and Algebra II, Santorum and not Romney is still the one who stands to benefit from Gingrich’s staying in the race, and the determining factor then becomes whether Newt’s candidacy is a strong or weak one.)
But in my view, the benefit of a weak-but-still-alive Gingrich at least temporarily tips towards Santorum, because of what I would call the front-runner effect. As long as Romney remains the putative front-runner, Gingrich supporters who favor Romney over Santorum are more likely to hang onto Gingrich for the simple reason that they believe that Romney, as front-runner, does not need their support. They therefore tell themselves that they can stick with their current first choice, at least for now. On the other hand, Gingrich-supporters who favor Santorum as their second choice are more likely to tell themselves that Santorum needs their support right away, and are therefore more likely to jump ship as the utility of their support for Gingrich vanishes.
This tendency is reinforced by the higher degree of excitement and voter attraction which Santorum generates, relative to Romney’s blandness. Hence, I believe that on balance, Santorum-leaners are more likely to switch first as Gingrich weakens. But it’s a tough call, since many factors are in play. In the absence of polls that specifically try to ferret out the answer to that question, the only way to figure this out may be to… wait and see what happens.
Until then, my current working hypothesis is that Santorum is better of with Newt in the race… as long as it’s a weak Newt, and as long as Romney remains the putative front-runner. So here’s my version of Nisanci’s proposition:
“Let R be a ring of Republicans and M be a left R-module. M is called an amply weak radical supplemented module in case M! = G – S, where M! is the M’s share of voters attracted after ring-leader G drops out, less votes attracted by strong radical S, where initial S > initial G and M! has a weak Rad-supplement. R is semilocal if and only if every left R-module is amply weak Rad-supplemented.”