Monoid-labeled transition systems

Given a ∨ -complete (semi)lattice L, we consider L-labeled transition systems as coalgebras of a functor L(−), associating with a set X the set LX of all L-fuzzy subsets. We describe simulations and bisimulations of L-coalgebras to show that L(−) weakly preserves nonempty kernel pairs iff it weakly preserves nonempty pullbacks iff L is join infinitely distributive (JID). Exchanging L for a commutative monoidM, we consider the functorM ω which associates with a set X all finite multisets containing elements of X with multiplicities m ∈ M . The corresponding functor weakly preserves nonempty pullbacks along injectives iff 0 is the only invertible element ofM, and it preserves nonempty kernel pairs iffM is refinable, in the sense that two sum representations of the same value, r1 + . . . + rm = c1 + . . . + cn, have a common refinement matrix (mi,j) whose k-th row sums to rk and whose l-th column sums to cl for any 1 ≤ k ≤ m and 1 ≤ l ≤ n.