$\newcommand{\angles}[1]{\langle { #1 } \rangle}$ I did the following exercise, can you tell me if I got it right? Thanks:
EXERCISE 31(G): Consider $\angles{ [0,1) , \leq } \otimes^s \angles{ [0,1] , \leq }$, $\angles{ [0,1) , \leq } \otimes^l \angles{ [0,1] , \leq }$, and $\angles{ [0,1) , \leq } \otimes^a \angles{ [0,1] , \leq }$. Find all minimal, minimum, maximal and maximum elements of these p.o.'s.
where we have the simple product
One can meaningfully define three kinds of products of p.o.'s. The simple product $\angles{ A , \preceq_A } \otimes^s \angles{ B , \preceq_B }$ (or shorthand $A \otimes^s B$) is the p.o. $\angles{ A \times B , \preceq_s }$, where the relation $\preceq_s$ is given by $\angles{ a , b } \preceq_s \angles{ c , d } \quad\text{iff}\quad a \preceq_A b \text{ and } b \preceq_B d.$
The lexicographic product or lexicographic order $\angles{ A , \preceq_A } \otimes^l \angles{ B , \preceq_B }$ (or shorthand $A \otimes^l A$) is the p.o. $\angles{ A \times B , \preceq_l }$, where the relation $\preceq_l$ is given by $ \angles{ a , b } \preceq_l \angles{ c , d } \;\;\text{iff}\;\; \begin{cases} a \prec_A c \quad\text{or}\\ a = c \text{ and } b \preceq_B d. \end{cases}$
and the antilexicographic order
The antilexicographic product or antilexicographic order $\angles{ A , \preceq_A } \otimes^a \angles{ B , \preceq_B }$ (or shorthand $A \otimes^a A$) is the p.o. $\angles{ A \times B , \preceq_a }$, where the relation $\preceq_a$ is given by $ \angles{ a , b } \preceq_a \angles{ c , d } \;\;\text{iff}\;\; \begin{cases} b \prec_B d \quad\text{or}\\ b = d \text{ and } a \preceq_A c. \end{cases}$
(Quoted text -- and linked images -- are from W. Just and M. Weese, Discovering Modern Set Theory, vol.1, p.25.)
For the simple product I got:
minimum = $(0,0)$ and hence only one minimal element. No maximum but maximal elements $\{ (a,1) \mid a \in [0,1) \}$.
For the lexicographic order I got:
minimum = $(0,0)$ and hence only one minimal element. No maximal elements and no maximum.
For the antilexicographic order I got:
minimum = $(0,0)$ and hence only one minimal element. No maximal elements and no maximum.
Thanks for your help!