I'm having trouble understanding the definition of the join operation. Near the bottom of page 3 of the book, there is an example showing the join of system A with three "unconnected" elements and system B with two connected elements and an unconnected element, *. The result of joining A with B is given as a two-element system with two the connected elements and one individual element, the *.
A join is defined as: "That is, we shall say the join of systems A and B, denote it A ∨ B, has a connection between points x and y if there are some points z1, . . . , zn such that, in at least one of A or B, it is true that x is connected to z1, zi is connected to zi+1, and zn is connected to y."
So why is * in the resulting joined system?
My working hypothesis is that in order to satisfy the definition of the join, we must assume that * is joined to itself, i.e., in the definition, the point x and the point y are the same point, namely *. In other words, the connection is reflexive in addition to being symmetric and transitive.
Since nobody seems to be having any problems with this, I suspect my hypothesis is unnecessary and I am overlooking some obvious, simple answer to my problem. Maybe I'm misinterpreting the definition of a join.