Meet and join: Difference between revisions
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=== Different Subgroups === | === Different Subgroups === | ||
The set of subgroup temperaments also, similarly, forms a lattice, from which we can derive the same basic poset operation above. Again, we have that if A ≤ B iff A ⊓ B = A, or equivalently, A ≤ B iff A ⊔ B = B. But now, the minimal subgroup temperament is the "trivial subgroup temperament" which has no intervals except for 1/1 and tempers out nothing; we may simply call this subgroup <'''1'''>. For all subgroup temperaments, we have that A ⊓ <'''1'''> = <'''1'''>. Similarly, the maximal temperament is Q/Q, meaning the subgroup temperament formed by taking infinite-limit JI and tempering the entire thing out; we may call this '''OM'''. | The set of subgroup temperaments also, similarly, forms a lattice, from which we can derive the same basic poset operation above. Again, we have that if A ≤ B iff A ⊓ B = A, or equivalently, A ≤ B iff A ⊔ B = B. But now, the minimal subgroup temperament is the "trivial subgroup temperament" which has no intervals except for 1/1 and tempers out nothing; we may simply call this subgroup <'''1'''>. For all subgroup temperaments, we have that A ⊓ <'''1'''> = <'''1'''>. Similarly, the maximal temperament is Q/Q, meaning the subgroup temperament formed by taking infinite-limit JI and tempering the entire thing out; we may call this '''OM'''. | ||
Similarly, if we had gone with the val-join instead of kernel-join, we would have gotten the same result, except the minimal and maximal temperaments would be flipped. | |||
== A Note on Historical Terminology == | == A Note on Historical Terminology == | ||