I am reading the book "Algebraic Graph Theory" by Chris Godsil and Gordon Royle and I got confused with Lemma 6.3.3:
The set of isomorphism classes of cores, partially ordered by "$\rightarrow$", is a lattice.
For which we need to prove that each pair of cores has a least upper bound and a greatest lower bound. To my understanding, I presume this means "upper class"$\rightarrow$"lower class". But the proof in the book seems reversing the meaning, i.e, "lower class"$\rightarrow$"upper class".
The following is the excerpt of proving the existence of least upper bound:
We start with the least upper bound. Let $X,Y$ be cores. For any core $Z$, if $X\rightarrow Z$ and $Y\rightarrow Z$, then $X\cup Y\rightarrow Z$. Hence $(X\cup Y)^\bullet$ is the least upper bound of $X$ and $Y$.
To understand the problem better, we take $X=C_7, Y=C_5$, then $X\rightarrow Y$ and so $(X\cup Y)^\bullet=C_5=Y$. On the other hand, $Y\nrightarrow X$. So how come $(X\cup Y)^\bullet$ is an upper bound of $X$ and $Y$ if my understanding was correct?
Note: Given graphs $X,Y$, the notation $X\rightarrow Y$ means $X$ is homomorphic to $Y$. And the core of $X$ is denoted by $X^\bullet$, is a subgraph of $X$ such that $X^\bullet$ is a core and $X\rightarrow X^\bullet$.