In these notes I came across a curious notation
Let $\overline d(x,y) = \min\{d(x,y), 1\}$ is a well known standard bounded metric on a metric space $(X,d)$
However, in the notes in linked, it was rewritten as: Given any metric space $(X, d)$ $\overline d(x,y) = d(x,y) \wedge1$, where $x\wedge y = \min\{x,y\}$
Is this way of writing well defined?
I am asking because on wikipedia https://en.wikipedia.org/wiki/Join_and_meet article , it says:
In a partially ordered set P, the join and meet of a subset S are respectively the supremum (least upper bound) of S, denoted ⋁S, and infimum (greatest lower bound) of S, denoted ⋀S. In general, the join and meet of a subset of a partially ordered set need not exist; when they do exist, they are elements of P.
Joint and meet are defined in terms of $\sup$ and $\inf$...which does not necessarily exist.
Can someone explain why it is formulated in terms of a single $\min$ in the definition above instead of $\inf$ in the definition of meet? How does being a metric space guarantee we have a $\min$?
Or maybe I am reading too much into this and $\wedge$ is not the meet operator. I know nothing about lattice theory if that helps.
The poset here is just $\mathbb{R}$. The meet symbol here is being used as a binary operator, so $x\wedge y$ means the same thing as $\bigwedge\{x,y\}$ (much the same as $x\cup y$ means a union of two sets $x$ and $y$ but $\bigcup X$ means the union of all the elements of $X$). So $d(x,y)\wedge 1$ means the infimum of the set containing the two real numbers $d(x,y)$ and $1$, which is just the lower of the two (in general, if a set has a least element, that is its infimum).