What is a Decomposed Word™?

Question

_{This is in the spirit of the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles.}

If a word has a certain property, I call it a Decomposed Word™.

You can use the examples below to find the property:

Decomposed™	Not Decomposed™
JOINING	ADHERING
AGGRAVATE	EXACERBATE
ANNEXING	APPENDING
WATCHLIST	REGISTER
VANQUISHING	DEFEATING
PLAINTEXT	CLEARTEXT
DESEXUALIZE	EMASCULATE
MOCCASIN	SLIPPER
LITIGATION	LAWSUIT
MIX	STIR
CODIFY	TABULATE
METHAMPHETAMINE	OPIOID
CONQUERING	SUBJUGATING

Here is a CSV version:

Decomposed™,Not Decomposed™
JOINING,ADHERING
AGGRAVATE,EXACERBATE
ANNEXING,APPENDING
WATCHLIST,REGISTER
VANQUISHING,DEFEATING
PLAINTEXT,CLEARTEXT
DESEXUALIZE,EMASCULATE
MOCCASIN,SLIPPER
LITIGATION,LAWSUIT
MIX,STIR
CODIFY,TABULATE
METHAMPHETAMINE,OPIOID
CONQUERING,SUBJUGATING

@randal'thor I would argue it's different, since one has to do with the sum of the contained RNs, while the other contains only a single RN. We can let the community decide, though. — GentlePurpleRain, Sep 26 '16 at 17:13
@randal'thor I was not aware of the other question when I posted mine (or I probably wouldn't have posted it). — GentlePurpleRain, Sep 26 '16 at 17:14
Yes, there's a very slight difference, but in most cases the two would come to the same thing. And the idea needed - the interesting part - is exactly the same ("think of Roman numerals"). They're both worthy of upvotes though, of course :-) — Rand al'Thor, Sep 26 '16 at 17:17

Gareth McCaughan · Accepted Answer · 2016-09-19T21:26:00.753

I think Decomposed Words are

those for which, if you extract the letters that are used for Roman numerals, make a prime number

and Not Decomposed Words are accordingly

those which yield either a composite number or 1.

The given examples don't let us tell

how to classify words that contain no such letters.

Per GentlePurpleRain's suggestion in comments: the name is because

when you decompose a number as far as possible into factors, the components you end up with are the prime numbers (what about 1? I hear you cry; well, when you decompose 1 into factors you get the empty set of factors, all of whose elements are in fact prime).

And here are the actual values:

2 II JOINING | 501 DI ADHERING 5 V AGGRAVATE | 90 XC EXACERBATE 11 XI ANNEXING | 501 DI APPENDING 151 CLI WATCHLIST | 1 I REGISTER 7 VII VANQUISHING | 501 DI DEFEATING 59 LIX PLAINTEXT | 160 CLX CLEARTEXT 541 DXLI DESEXUALIZE | 1150 MCL EMASCULATE 1201 MCCI MOCCASIN | 51 LI SLIPPER 53 LIII LITIGATION | 51 LI LAWSUIT 1009 MIX MIX | 1 I STIR 401 CDI CODIFY | 50 L TABULATE 3001 MMMI METHAMPHETAMINE | 498 IID OPIOID 101 CI CONQUERING | 1 I SUBJUGATING

You may notice that the second-last one of those on the right

is nonstandard because you can't really put two smaller "decrementing" Is before a single D. DII is also composite (of course; it's even and doesn't equal 2) but if we just add up Roman numerals without regard to position then MIX no longer works because MXI = 1011 which, being a multiple of 3, is not prime.

But the questioner indicates in comments that

his intention was that anything not containing a valid Roman numeral should be a Not Decomposed Word, so OPIOID is fine for that reason.

Perhaps you could flesh out your answer by providing a) some examples using the words provided above and b) an explanation of why they are called "decomposed"? — GentlePurpleRain, Sep 19 '16 at 20:26
@GentlePurpleRain, done. I'm a little puzzled by OPIOID, but perhaps I'm just on drugs or something. — Gareth McCaughan, Sep 19 '16 at 21:15
Wow! It wasn't my intention for the non-decomposed words to even generate anything valid at all. I'm surprised that (just by chance) all but one of them did... — GentlePurpleRain, Sep 19 '16 at 21:17

What is a Decomposed Word™?

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