The type monoid of a Boolean inverse semigroup (BIS) $S$ is the universal monoid of the partial semigroup consisting of the quotient of S, endowed with its orthogonal addition, by its GreenÕs relation $\mathcal{D}$. It bears strong analogies with such different contexts as abstract measure theory (DobbertinÕs $V$-measures), lattice theory (the dimension monoid), ring theory (nonstable $K$-theory). A related analogy is that BISs, endowed with binary operations suitably defined from the multiplication and orthogonal addition, form a congruence-permutable variety [of universal algebras].

The type monoid of a BIS is a conical refinement monoid, and further, due to Dobbertin's results on $V$-measures, the converse holds in the countable case. On the other hand, there are counterexamples, due to the author, in cardinality $\aleph_2$. By using a small fragment of the dimension monoid in lattice theory, one can see that the positive cone of any abelian lattice-ordered group is isomorphic to the type monoid of a BIS. We survey some further results of that sort, and we relate the type monoid of $S$ and the nonstable $K_0$-theory of a $K$-algebra

denoted by $K(S)$, for a unital ring $K$, observing in particular that even for $K$ a field, the canonical map between those monoids can fail to be an isomorphism.