My Favorit slide:
Algebraic Properties Are Important!
• Associative: grouping doesn’t matter!
• Commutative: order doesn’t matter!
• Idempotent: duplicates don’t matter!
• Identity: this value doesn’t matter!
• Zero: other values don’t matter!
Invariants give the implementation wiggle room, that is, the
freedom to exploit alternate representations and implmentations.
In particular, associativity gives implementations the necessary
wiggle room to use parallelism—or not—as resources dictate.
My favorite comment was about the C language having the address of operator (pointers). Whereas a Java program has no such thing and thus JVM can garbage collect knowing that the program is not depending on the memory location as it may have been in C. By giving up the address of operator, the JVM has the wiggle room to implement the garbage collection.
Abstractions are made based on the properties (aka "wiggle room") of the underlying concept.