I'm reading about lattices and order relations. I came up with a property that says.
$a \land b \lt a$ and $a \land b \lt b$ iff $a$ and $b$ are incomparable.
This confuses me up a litle because I think $a \land b \lt a$ and $a \land b \lt b$ is always true since $\land$ denotes the maximum lower bound. Can anyone show me a proof and explain to me why the above is true?
Easiest examples are positive integers, comparison is by divisor and strict divisor.
2 and 3 are not comparable. The gcd is 1, which (strictly) divides both.
2 and 6 are comparable. the gcd is 2. This strictly divides 6, but does not strictly divide 2 because it is equal to 2.