In George Grätzer book, Lattice Theory : Foundation, there is some fundamental result I am not sure to interpret correctly. The owner of the book will find it page 32. It is the second statement of the Lemma 5. Here it is :
Consider a lattice $L$. Consider $H$ a subset of $L$. Then consider $I$ the ideal generated by $H$, being the smallest ideal of $L$ (from inclusion point of view) containing $H$.
The lemma states that :
$$I = Id(H) \iff I = \{x | x \leq h_0 \lor \ldots \lor h_{n-1},\;for\;some\;n\geq1\;and\;h_0,\ldots,h_{n-1}\in H\}$$
From what I understand, it means :
$$\forall x \in I, \exists n \in \Bbb{N}^*, \exists \{h_0,\ldots,h_{n-1}\}\in H^n,x \leq h_0 \lor \ldots \lor h_{n-1}$$
However, a bit later in the book, George Grätzer states the following to prove another result (page 34) :
$I$ is an ideal generated by subset $C$ of $L$. Let us take $t \in I$. From previous lemma, we can affirm that $\exists c \in C, t \leq c$.
I do not understand how the previous lemma makes it possible to do such a statement.
If anybody is able to tell me where I am wrong, would be really grateful.
Thanks in advance :)
There is nothing wrong.
In page 34, $C$ is not just any subset of the lattice; it is a convex sub-lattice.
Thus, if $t \leq c_1 \vee \cdots \vee c_k$, for some $c_1, \ldots, c_k \in C$, then $c \in C$, where $c = c_1 \vee \cdots \vee c_k$; thus $t \leq c \in C$.