Talk:Meet and join: Difference between revisions

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I like this meet and join concept a lot! But I have some concerns about the symbols used. They’re used nowhere else on the wiki, so I think this shouldn’t be that big of a deal. I know that Mike Battaglia is working on other materials using meet and join but other than that this would have little impact on anyone I think.

My initial concern is that these symbols ⋏ and ⋎ are easy to confuse with the symbols used for the wedge and vee products, ∧ and ∨ (the latter is currently used as the symbol for an interior product but that's another issue). Gene states these curvy versions were chosen over the straight versions to avoid confusion, which is well-intentioned, but I'm concerned that it's not enough. Especially since there is tremendous historical confusion between the symbols ∧ and ∨ as they are used in exterior algebra (from where Gene brought to us the wedge and vee product we commonly use now) and ∧ and ∨ as they are used in order theory (from where Gene draws the meet and join concept). It surprises many to find that wedge ∧ is akin to join ∨, in that they both increase grade (rank or nullity), and that veeing ∨ is akin to meet ∧ in that they both decreases grade. We suspect that the symbols for wedge and vee were chosen by the direction they point, while the symbols for join and meet were chosen for their resemblance to the union and intersection symbols (join ∨ is like union ∪; meet ∧ is like intersection ∩), and that’s how similar operations ended up with opposite symbols. Even Gene seemed to get this confused (see my other suggestions here). So I suggest that it would be best to leave the ∧ and ∨ type symbols in the RTT domain to the one job they already popularly do for us — wedge (and maybe vee or interior product) — and find some other symbols for meet and join.

My initial concern is that these symbols ⋏ and ⋎ are easy to confuse with the symbols used for the wedge and vee products, ∧ and ∨ (the latter is currently used as the symbol for an interior product but that's another issue; see: [[Talk:Interior_product#Questions.2C_observations.2C_suggestions]]). Gene states these curvy versions were chosen over the straight versions to avoid confusion, which is well-intentioned, but I'm concerned that it's not enough. Especially since there is tremendous historical confusion between the symbols ∧ and ∨ as they are used in exterior algebra (from where Gene brought to us the wedge and vee product we commonly use now) and ∧ and ∨ as they are used in order theory (from where Gene draws the meet and join concept). It surprises many to find that wedge ∧ is akin to join ∨, in that they both increase grade (rank or nullity), and that veeing ∨ is akin to meet ∧ in that they both decreases grade. We suspect that the symbols for wedge and vee were chosen by the direction they point, while the symbols for join and meet were chosen for their resemblance to the union and intersection symbols (join ∨ is like union ∪; meet ∧ is like intersection ∩), and that’s how similar operations ended up with opposite symbols. Even Gene seemed to get this confused (see my other suggestions here). So I suggest that it would be best to leave the ∧ and ∨ type symbols in the RTT domain to the one job they already popularly do for us — wedge (and maybe vee or interior product) — and find some other symbols for meet and join.

And there is an excellent choice for join already available! The join operation may be understood as an extension of a well-established operation in RTT which was given the symbol & and is sometimes called "cross-breeding" because it is used to combine equal temperament maps, or "breeds". With & you concat rank-1 mappings into higher rank mappings. And with "join" you concat mapping matrices of any rank into a higher rank mapping matrix. So it’s the same idea, now generalized to any rank. Therefore, & should be used for join: 19&31 could be read "19 join 31" and meantone&marvel would be read "meantone join marvel".

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 I like this meet and join concept a lot! But I have some concerns about the symbols used. They’re used nowhere else on the wiki, so I think this shouldn’t be that big of a deal. I know that Mike Battaglia is working on other materials using meet and join but other than that this would have little impact on anyone I think.
-My initial concern is that these symbols ⋏ and ⋎ are easy to confuse with the symbols used for the wedge and vee products, ∧ and ∨ (the latter is currently used as the symbol for an interior product but that's another issue). Gene states these curvy versions were chosen over the straight versions to avoid confusion, which is well-intentioned, but I'm concerned that it's not enough. Especially since there is tremendous historical confusion between the symbols ∧ and ∨ as they are used in exterior algebra (from where Gene brought to us the wedge and vee product we commonly use now) and ∧ and ∨ as they are used in order theory (from where Gene draws the meet and join concept). It surprises many to find that wedge ∧ is akin to join ∨, in that they both increase grade (rank or nullity), and that veeing ∨ is akin to meet ∧ in that they both decreases grade. We suspect that the symbols for wedge and vee were chosen by the direction they point, while the symbols for join and meet were chosen for their resemblance to the union and intersection symbols (join ∨ is like union ∪; meet ∧ is like intersection ∩), and that’s how similar operations ended up with opposite symbols. Even Gene seemed to get this confused (see my other suggestions here). So I suggest that it would be best to leave the ∧ and ∨ type symbols in the RTT domain to the one job they already popularly do for us — wedge (and maybe vee or interior product) — and find some other symbols for meet and join.
+My initial concern is that these symbols ⋏ and ⋎ are easy to confuse with the symbols used for the wedge and vee products, ∧ and ∨ (the latter is currently used as the symbol for an interior product but that's another issue; see: [[Talk:Interior_product#Questions.2C_observations.2C_suggestions]]). Gene states these curvy versions were chosen over the straight versions to avoid confusion, which is well-intentioned, but I'm concerned that it's not enough. Especially since there is tremendous historical confusion between the symbols ∧ and ∨ as they are used in exterior algebra (from where Gene brought to us the wedge and vee product we commonly use now) and ∧ and ∨ as they are used in order theory (from where Gene draws the meet and join concept). It surprises many to find that wedge ∧ is akin to join ∨, in that they both increase grade (rank or nullity), and that veeing ∨ is akin to meet ∧ in that they both decreases grade. We suspect that the symbols for wedge and vee were chosen by the direction they point, while the symbols for join and meet were chosen for their resemblance to the union and intersection symbols (join ∨ is like union ∪; meet ∧ is like intersection ∩), and that’s how similar operations ended up with opposite symbols. Even Gene seemed to get this confused (see my other suggestions here). So I suggest that it would be best to leave the ∧ and ∨ type symbols in the RTT domain to the one job they already popularly do for us — wedge (and maybe vee or interior product) — and find some other symbols for meet and join.
 And there is an excellent choice for join already available! The join operation may be understood as an extension of a well-established operation in RTT which was given the symbol & and is sometimes called "cross-breeding" because it is used to combine equal temperament maps, or "breeds". With & you concat rank-1 mappings into higher rank mappings. And with "join" you concat mapping matrices of any rank into a higher rank mapping matrix. So it’s the same idea, now generalized to any rank. Therefore, & should be used for join: 19&31 could be read "19 join 31" and meantone&marvel would be read "meantone join marvel".