  We introduce MK-fuzzy automata over a bimonoid K which is related to the
fuzzification of the McCarthy-Kleene logic. Our automata are inspired by, and
intend to contribute to, practical applications being in development in a
project on runtime network monitoring based on predicate logic. We investigate
closure properties of the class of recognizable MK-fuzzy languages accepted by
MK-fuzzy automata as well as of deterministically recognizable MK-fuzzy
languages accepted by their deterministic counterparts. Moreover, we establish
a Nivat-like result for recognizable MK-fuzzy languages. We introduce an
MK-fuzzy MSO logic and show the expressive equivalence of a fragment of this
logic with MK-fuzzy automata, i.e., a B\"uchi type theorem.
