@MISC{Gumm_monoid-labeledtransition, author = {H. Peter Gumm and Tobias Schröder}, title = {Monoid-Labeled Transition Systems}, year = {} }

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Abstract

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 L X of all L-fuzzy subsets. We describe simulations and bisimulations of L-coalgebras to show that L (-) weakly preserves nonempty kernel pairs i# it weakly preserves nonempty pullbacks i# L is join infinitely distributive (JID). Exchanging L for a commutative monoid M, we consider the functor M (-) # 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 i# 0 is the only invertible element of M, and it preserves nonempty kernel pairs i# M is refinable, in the sense that two sum representations of the same value, r 1 + . . . + r m = c 1 + . . . + c n , have a common refinement matrix (m i,j ) whose k-th row sums to r k and whose l-th column sums to c l for any 1 # k # m and 1 # l # n.