Semimonoidal Join Semilattice

A semimonoidal join semilattice is a weaker form of Quantale, where we only require the underlying poset be a Join Semilattice.

Explicitly, $(X,\vee)$ is a semimonoidal join semilattice if

• it is equipped iwth an associative operation $\otimes:X\to X\to X$, such that

• $\otimes$ obeys left and right distributivity laws

  	$\displaystyle x\otimes(y\vee z)$	$\displaystyle=(x\otimes y)\vee(x\otimes z)$
  	$\displaystyle(x\vee y)\otimes z$	$\displaystyle=(x\otimes z)\vee(y\otimes z)$

The canonical example a of semimonoidal join semilattices is the Min Tropical Semiring.