$L$-Fuzzy Multiset Recognizer: Determinization and Minimization

This research aims to investigate the determinization and minimization problems of a new class of fuzzy recognizers with input as multisets. We begin by introducing and examining the concepts of an
$L$
-fuzzy multiset recognizer (
$L$
-
LMR
), complete, accessible, and coaccessible
$L$
-
LMR
over a quantale
$L$
. We, then characterize introduced
$L$
-
LMR
with a multiset regular language (
MRL
). Next, we investigate the method of construction of a complete
$L$
-
FMR
from a given incomplete
$L$
-
FMR
, and show that if
$K(\mathcal {R})$
is the
MRL
recognized by an incomplete
$L$
-
FMR

$\mathcal {R}$
, then there is a completion
$\mathcal {R}^{c}$
of
$\mathcal {R}$
such that
$K(\mathcal {R})=K(\mathcal {R}^{c})$
. Moreover, we also illustrate these notions with examples. Further, we introduce a new method for the construction of an equivalent deterministic
$L$
-fuzzy multiset recognizer (
DLMR
) from a given
$L$
-fuzzy multiset recognizer via
$L$
-fuzzy subsets and accessible
$L$
-fuzzy subset construction. Furthermore, we characterize introduced
DLMR
with an
$L$
-fuzzy multiset regular languages (
$L$
-
FMRL
). In the last, we introduce the concepts related to minimal
DLMR
and then we obtain a minimal
DLMR
for a given
$L$
-
FMRL
.