The infimum is defined for boolean and by the Golden Rule:

By studying this rule we can observe several facts about the infimum. First, since it is defined in terms of and we have

{Golden Rule}

{Symmetry of and }

{Golden Rule}

i.e. is symmetric.

**Exercise**: Show that is associative.

Furthermore, is idempotent:

{Golden Rule}

{identity of ; idempotence of }

With regard to last three properties, is just like . We might very well ask, is a fixed point of as it is of ? Let’s calculate:

{Golden Rule}

{fixed point of }

{identity of }

So is the identity of .

Next we have two very useful laws of absorption/introduction:

We shall prove the first, the second follows by interchanging and .

{Golden rule}

{idempotence}

{identity of }

What about distributivity properties?

{Golden rule}

{distributivity}

{idempotence, preparing for the Golden Rule}

{Golden Rule}

{ distributes over }

{absorption}

{ distributes over }

{associativity; symmetry}

{absorption}

So distributes over . What about ? The best we can do is

{Golden rule}

{distributivity}

{symmetry, preparing for Golden rule}

{Golden rule, twice}

As a consolation we do have

.

Finally we have

{pseudo-distributivity}

{idempotence; identity of }