Help calculating relative pseudo-complements in a Heyting algebra/lattice

Question

I'm trying to work some examples of relative pseudo-complements in lattices, to make sure I understand them. I wonder if anybody could check my examples, and tell me if I'm correct or if I've misunderstood.

I'm looking at the definition of relative pseudo-complement in Rasiowa/Sikorski, The Mathematics of Metamathematics, p. 54. It says that an element $c$ is the pseudo-complement of $a$ relative to $b$ (denoted $a \Rightarrow b$), if $c$ is the greatest element such that $a \wedge c \leq b$.

I think I grasp why it is that if $a \leq b$, then $a \Rightarrow b$ is always 1. I'm now trying to understand $a \Rightarrow b$ when $a \not \leq b$.

Example 1.

Suppose I have this lattice:

  1
 / \
a   b
 \ /
  0

I want to figure out which element $x$ is $b \Rightarrow 0$, i.e., which element $x$ is the pseudo complement of $b$ relative to $0$.

If I follow Rasiowa/Skiorski's definition, I find this by putting together the set $\{ x : b \wedge x \leq 0\}$, and then I take the greatest element from that.

Here, I see $\{ x : b \wedge x \leq 0\}$ = $\{ 0, a \}$, since $b \wedge 0 = 0 \leq 0$, and $b \wedge a = 0 \leq 0$. Then, the greatest of $\{ 0, a \}$ is $a$, so I conclude that the pseudo complement of $b$ relative to $0$ is $a$, i.e., that $b \Rightarrow 0 = a$.

Did I do that correctly?

Example 2a.

Suppose I have this lattice (a powerset lattice on three elements $a$, $b$, $c$):

    abc
  /  |  \
 ab  ac  bc
 | x   x |
 a   b   c
  \  |  /
     0  

I want to find $c \Rightarrow 0$.

I follow the same procedure: I find $\{ x : c \wedge x \leq 0 \}$, which is $\{ 0, b, a, ab\}$. Then I take the greatest element from that, which is $ab$. So I conclude that $c \Rightarrow 0 = ab$.

Did I do that correctly?

Example 2b. By a similar argument to 2a, the pseudo-complement of $c$ relative to $b$ is $ab$ as well, i.e., $c \Rightarrow b = ab$.

Am I understanding this correctly?

Answer 1

Thanks to @amrsa's comment above, I'll post the answer.

The answers I calculated above in the original post are correct.

@amrsa also noted that the two examples of lattices I used are Boolean algebras. In these, $x \Rightarrow 0$ always equals the complement of $x$ (sometimes written $x'$).

In example (1) above, $b \Rightarrow 0$ will thus be the complement of $b$. If we consider the lattice in example (1) to be the powerset of $\{a, b\}$, then the complement of $\{b\}$ is $\{a\}$, which is indeed the pseudo-complement of $\{b\}$ with respect to $\varnothing$.

In example (2a) above, $\{c\} \Rightarrow \varnothing$ will be the complement of $\{c\}$, which is $\{a, b\}$, and that is the pseudo-complement of $\{c\}$ with respect to $\varnothing$.

In example (2b), there is the principle that in Boolean algebras, $x \Rightarrow y$ is the same as $x' \lor y$, i.e., the complement of $x$, or $y$. In the lattice from (2b), the complement of $\{c\}$ is $\{a, b\}$, which is the pseudo-complement of $\{c\}$ relevant to $\{b\}$.