I found in Awodey's "Theory Category", second edition p34, a set of poset axioms to define Boolean algebras, whereas I don't see how they can be sufficient.


Here are the axioms:



> 
> A Boolean algebra is a poset B
>  equipped with distinguished elements
>  0, 1, binary operations $a \vee b$ of
>  "join" and $a \wedge b$ of "meet", and
>  a unary operation $\neg b$ of
>  "complementation". These are required
>  to satisfy the conditions $$0 \le a$$
>  $$a \le 1$$ $$a \le c \text{ and } b
> > \le c \text{ iff } a\vee b \le c $$
>  $$c \le a \text{ and } c \le b \text{
> > iff } c \le a \wedge b$$ $$a \le \neg
> > b \text{ iff } a \wedge b = 0$$
>  $$\neg\neg a =a$$
> 
> 
> 


I can see that from these axioms I can deduce several of the common properties of boolean algebras, so for example I can prove $\forall c, c\wedge\neg c =0$, $\forall (a,c), a\wedge c \le c$, etc. but there are some properties I cannot see how they are provable in this axiomatics.


Here one of which I tried so hard (and failed) to prove from the axioms above: $\forall (a,b), a \le a \vee b$


Could somebody guide me? I am trying to make something impossible? Is my mistake on things I think that are required to be true in a Boolean algebra and are not?



Poset axioms of Boolean algebra
As remarked by Joel, this is indeed very direct starting from $a\vee b \le a \vee b$ and using the third axiom. Many thanks to have unstucked me.


