Intermezzo

My own scheme of a complete direct attack would be as follows
One attacker, one target.

• Attacker
• Road to attacking square
• Attacking square
• Road to target square
• Target square
• Road from target to target square
• Target

With this scheme it is possible to describe every obstacle in an attack, and every preliminary move.

• Is the attacker already on the attacking square?
• Is the road to the attacking square blocked?
• Can the attacker reach the attacking square with tempo?
• Is the attacking square free from defense?
• Is the road from attacking square to target square blocked? 
• Is the target square free from defense?
• Is the target already on the target square?
• Is the road from the target to the target square free?
• Can I force the target towards the target square?

With a duplo attack the scheme gets more complicated.
One attacker, two targets.

• Attacker
• Road to attacking square
• Attacking square
• Road to target square 1
• Target square 1
• Road from target to target square 1
• Target 1
• Road to target square 2
• Target square 2
• Road from target to target square 2
• Target 2

And a discovered or simultaneous attack is even more complicated.
Two attackers, two targets

• Attacker 1
• Road to attacking square 1
• Attacking square 1
• Road to target square 1
• Target square 1
• Road from target to target square 1
• Target 1
• Attacker 2
• Road to target square 2
• Target square 2
• Road from target to target square 2
• Target 2

I left the trap out, but even then the schemes are already way too complicated to be handled  adequately by the human mind. This means that we must be prepared to sacrifice completeness for manageability. That is exactly what the system with the three motifs of mister Lasker does.

• PoP
• LoA
• Fun

By the way (btw), you can always find the commonly used abbreviations via the list in my sidebar. I introduce new terminology here and there. That frees my mind from a too rigid approach to Laskers system.

PoP
In the passed week, I had analyzed six positions, and every position has showed 1-3 PoPs. I think it is a safe bet to assume that there always will be at least one PoP in any tactical problem. Theoretically I can design a tactical combination with no Pops, but I don't think that has a practical  value, since the chance that you encounter such combination in practice will be very slim.
I associate a PoP with a piece. That piece can be a target or a defender (a piece with a function)

LoA
As with PoPs, I probably can design a combination without any straight LoA. Strictly speaking, Lasker says nowhere that a geometrical motif means a straight line of attack. But in practice, that is where you look for. But then again, such combination without any straight LoA would be extremely rare in practice. For now, I simply ignore those. Every position had 1-4 LoAs.

Fun
Pieces don't have necessarily functions. So a combination without any defending piece isn't necessarily rare. Yet beyond a certain complexity, you will not found combinations where no piece fulfills a function. Analyzing the functions is crucial to interpret what is going on. PoP and LoA are just identification methods of targets and roads, function describes the alteration of a piece. Unseen limitations of pieces.