Multilattices are generalisations of lattices introduced by Mihail Benado. He
replaced the existence of unique lower (resp. upper) bound by the existence of
maximal lower (resp. minimal upper) bound(s). A multilattice will be called
pure if it is not a lattice. Multilattices could be endowed with a residuation,
and therefore used as set of truth-values to evaluate elements in fuzzy
setting. In this paper we exhibit the smallest pure multilattice and show that
it is a sub-multilattice of any pure multilattice. We also prove that any
bounded residuated multilattice that is not a residuated lattice has at least
seven elements. We apply the ordinal sum construction to get more examples of
residuated multilattices that are not residuated lattices. We then use these
residuated multilattices to evaluate objects and attributes in formal concept
analysis setting, and describe the structure of the set of corresponding formal
concepts. More precisely, if $\mathcal{A}_i:
=(A_i,\le_i,\top_i,\odot_i,\to_i,\bot_i)$, $i=1,2$ are two complete residuated
multilattices, $G$ and $M$ two nonempty sets and $(\varphi, \psi)$ a Galois
connection between $A_1^G$ and $A_2^M$ that is compatible with the residuation,
then we show that
\[\mathcal{C}: =\{(h,f)\in A_1^G\times A_2^M; \varphi(h)=f \text{ and }
\psi(f)=h \}\] can be endowed with a complete residuated multilattice
structure. This is a generalization of a result by Ruiz-Calvi{\~n}o and Medina
saying that if the (reduct of the) algebras […]
Graph embeddings play a significant role in the design and analysis of
parallel algorithms. It is a mapping of the topological structure of a guest
graph G into a host graph H, which is represented as a one-to-one mapping from
the vertex set of the guest graph to the vertex set of the host graph. In
multiprocessing systems the interconnection networks enhance the efficient
communication between the components in the system. Obtaining minimum
wirelength in embedding problems is significant in the designing of network and
simulating one architecture by another. In this paper, we determine the
wirelength of embedding 3-ary n-cubes into cylinders and certain trees.
Context: Petri net slicing is a technique to reduce the size of a Petri net
to ease the analysis or understanding of the original Petri net. Objective:
Presenting two new Petri net slicing algorithms to isolate those places and
transitions of a Petri net (the slice) that may contribute tokens to one or
more places given (the slicing criterion). Method: The two algorithms proposed
are formalized. The maximality of the first algorithm and the minimality of the
second algorithm are formally proven. Both algorithms, together with three
other state-of-the-art algorithms, have been implemented and integrated into a
single tool so that we have been able to carry out a fair empirical evaluation.
Results: Besides the two new Petri net slicing algorithms, a public, free, and
open-source implementation of five algorithms is reported. The results of an
empirical evaluation of the new algorithms and the slices they produce are also
presented. Conclusions: The first algorithm collects all places and transitions
that may contribute tokens (in any computation) to the slicing criterion, while
the second algorithm collects the places and transitions needed to fire the
shortest transition sequence that contributes tokens to some place in the
slicing criterion. Therefore, the net computed by the first algorithm can
reproduce any computation that contributes tokens to any place of interest. In
contrast, the second algorithm loses this possibility, but it often produces a
much more reduced subnet […]