Abstract: | Given a
-complete (semi)lattice
, we consider
-labeled transition systems as coalgebras of a functor
(−), associating with a set X the set
X of all
-fuzzy subsets. We describe simulations and bisimulations of
-coalgebras to show that L(−) weakly preserves nonempty kernel pairs iff it weakly preserves nonempty pullbacks iff L is join infinitely distributive (JID).Exchanging
for a commutative monoid
, we consider the functor
(−)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
, and it preserves nonempty kernel pairs iff
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. |