Monoid-labeled transition systems

Abstract Given a V -complete (semi)lattice L , we consider L -labeled transition systems as coalgebras of a functor L (−), associating with a set X the set L X 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 monoid M , we consider the functor M (−)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 of M , and it preserves nonempty kernel pairs iff M is refinable, in the sense that two sum representations of the same value, r1 + … + rm = c1 + … + cn, have a common refinement matrix (m(i, 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.