Would I be correct in assuming that each Minor Magical Focus can be thought of as a subset of a Major Magical Focus?
For instance, a Minor MF in canines is a sunset of a Major MF in Mammals. Or maybe even a Major MF in Land Animals. Maybe even a subset of Major MF in Carnivorous Quadripeds?
Likewise a Major MF in Weather encompasses multiple Minor MF in wind, rain and lightning.
Is there any Minor MF that doesn't fit that pattern? Possibly Certamen.
Or would Certamen be covered by some hypothetical Major MF in Counterspelling? Major MF in (combat) Metamagic? Major MF in Magical Duels?
Also, what MF covers both the various "Mirror of Opposition [Form]" MuViGen spells
I think you're better off asking a mathematician to explain this as an exercise in set theory than asking players of the game.
A minor focus covers a narrow area. A major focus covers a bigger area. What counts as major or minor depends a lot on your SG/troupe. Yes, every small area can be said to be part of a bigger area. No, they don't always overlap neatly - the Major MF of "domestic animals" will cover a wide range of mammals and birds, but each minor speciality of "horses" or "ducks" would cover both wild and domestic versions.
A Major MF in "counterspelling" is given in the Merinita section of Houses of Hermes: Mystery Cults, what it actually covers is up to your troupe.
I think you are best just deciding for yourself, based on what works for you, since the distinction between major and minor seems to shift from book to book, and since people will likely consider a very useful focus as Major, even if what it actually covers is tiny, such as a focus in "magi" or "me and mine."
I pulled together a list of Major and Minor foci from all the RAW books a year ago and didn't see any planned or logical subsets; they appear to be highly variable, so YSMV.
Nice, but wasn't there a Mythic Bloodline that granted a Minor Magical Focus in Arms and Armour? I seem to recall adding that to Mythic Blood in Project Redcap.
And what is a Minor Magical Focus in Longevity Potions? I thought purely Lab activities were excluded from being the subject a Magical Focus.
I must admit, the ArM5 book's sample Major and Minor Magical Focuses did suggest that most Minor Magical Focus could be completely encompassed by the appropriate Major Magical Focus. Not so much a rule, as a logical extension of the difference between Major and Minor.
Longevity may not be canon, but ArM5 p46 does have a minor focus in "Aging", which I would argue should cover forcibly aging people with PeCo, spells to resolve aging crises and longevity rituals.
"Arms and armour" seems a bit broad for a minor, when the confraternity of Roland only gets "swords" and Faerie Sympathy "Weapons" is a thing.
EDIT: ironboundtome missed out the Merinita chapter from HoH:MC - the different Major Magical Foci listed (each magical society in Merinita seems to have one) are Faerie
Charms
Counterspells
Wilderness
Illusion
The Bjornaer section of HoH:MC has several minor ones on page 37.
The Verditius section has the minor foci (Swords) and (Wooden Wands) in the confraternities.
GotF has a minor one in Emotions on page 40.
DI has a minor focus in the moon on page 16 and living trees on page 83.
I wouldn't use either of those two as guidelines for what a Minor Magical Focus should cover. First, logic doesn't work that way. There is no reason a Minor Magical Focus cannot be ultra-small in scope. Particularly, compare "wooden wands" (also from a Confraternity) to "wood" (canon example of a Minor Magical Focus from the core book itself). Wooden wands is far, far more restrictive, which keeps it less than the limit, but wood is also less than the limit. It's like choosing an whole number less than 100, and someone giving 1 as an example and based on that saying 90 cannot possibly be true because it is so much bigger than 1. It does work the other way, though: the example of 90 is greater than 1, so if 90 is valid so must be 1. Since "wood" is valid, "wooden wands" must be. Since "arms and armor" is valid, so must "swords" be. Second, Faerie Sympathies vary widely in scope from the breadth of Major Magical Focus (e.g. "fish," "men," "women") to narrower than a Minor Magical Focus (e.g. "hounds" v. "canines"). "Weapons" v. "arms and armour" is just like "hounds" v. "canines," and "canines" comes right from the examples in the core book.
Plus "familiars." For the life of me I cannot figure out why it's a Major Magical Focus. It's certainly allowed to be Major, but it's comparable to yet narrower than many similar Minor Magical Foci.
That's really odd since "emotions" is given as an example of a Major Magical Focus in the core book. Yes, "emotions" is very narrow for a Major Magical Focus. We could break Mentem down into memories, emotions, thoughts, ghosts, and handful of other little things like improving mental Characteristics, communication a few ways, making a mind solid, other Mentem spirits, etc. A little less than a fifth of a Form is roughly the upper limit of Minor Magical Focus. In this case, it would apply to animal emotions, too, which would make it broader than that roughly fifth of a Form, though. But it does seem to be on the narrow end of a Major Magical Focus.
Since there is a Major Magical Focus in "Destruction", conceivably there could be a Major Magical Focus in "Magical Battles", that comes into effect when exchanging magical energies with an opponent. Such a Major MF in "Magical Battles" could conceivably encompass everything a Minor MF in Certamen does.
But then I take a step back and remember that before the invention of the Parma Magica, the first wizard to "shoot" tended to win by default. And after the Parma Magica was invented, the Order of Hermes tended to stomp on wizards who engage excessively in magical battles.
Hence it is somewhat unlikely that anybody ever learned a Magical MF in "Magical Battles".
Unless it comes from a Hedge Tradition where magical combat was not immediately fatal, maybe those Hounds of God Benadati (sp?)
Minor Magical Focus in Aging is also presented in GotF page 38 as a step on the path of aging; associated with difficult longevity ritual, drinking from the fountain of youth, and Immortality of the Forrest so you'd really suspect that the intention was to make it useful for longevity rituals.
Just recalling that in the description of both Major Magical Focus and Minor Magical Focus virtues in the Arm5 book, it states
Which I always took to mean that you can't have a focus purely in Longevity Rituals, but you can have a focus in Aging, which should include the subset Longevity rituals.