[Simon](https://forum.azimuthproject.org/discussion/comment/19999/#Comment_19999) wrote:

> But you're not using usual matrix multiplication, i.e. with the usual arithmetic operations. Is this matrix the identity with respect to the alternative multiplication using meet and join?

Whenever we have a [semi-ring or *rig*](https://en.wikipedia.org/wiki/Semiring) \\((R, +, \cdot)\\) then we can define matrix multiplication and addition.

In the special case John is thinking of here:

> \[ (\Psi\Phi)(x,z) = \bigvee_{y \in Y} \Phi(x,y) \wedge \Psi(y,z) \]

I suspect it follows that matrix multiplication is associative, even for infinite matrices, because \\(\mathbf{Bool}\\) is a [complete semiring](https://en.wikipedia.org/wiki/Semiring#Complete_and_continuous_semirings). However, I am not sure..

While I'm busy gearing up to be wrong, I also suspect every monoidal poset gives rise to a complete semiring. This might explain the \\([0-\infty]\\)-feasibility construction I was wondering about earlier.

> But you're not using usual matrix multiplication, i.e. with the usual arithmetic operations. Is this matrix the identity with respect to the alternative multiplication using meet and join?

Whenever we have a [semi-ring or *rig*](https://en.wikipedia.org/wiki/Semiring) \\((R, +, \cdot)\\) then we can define matrix multiplication and addition.

In the special case John is thinking of here:

> \[ (\Psi\Phi)(x,z) = \bigvee_{y \in Y} \Phi(x,y) \wedge \Psi(y,z) \]

I suspect it follows that matrix multiplication is associative, even for infinite matrices, because \\(\mathbf{Bool}\\) is a [complete semiring](https://en.wikipedia.org/wiki/Semiring#Complete_and_continuous_semirings). However, I am not sure..

While I'm busy gearing up to be wrong, I also suspect every monoidal poset gives rise to a complete semiring. This might explain the \\([0-\infty]\\)-feasibility construction I was wondering about earlier.