123 research outputs found
Infinite combinatorial issues raised by lifting problems in universal algebra
The critical point between varieties A and B of algebras is defined as the
least cardinality of the semilattice of compact congruences of a member of A
but of no member of B, if it exists. The study of critical points gives rise to
a whole array of problems, often involving lifting problems of either diagrams
or objects, with respect to functors. These, in turn, involve problems that
belong to infinite combinatorics. We survey some of the combinatorial problems
and results thus encountered. The corresponding problematic is articulated
around the notion of a k-ladder (for proving that a critical point is large),
large free set theorems and the classical notation (k,r,l){\to}m (for proving
that a critical point is small). In the middle, we find l-lifters of posets and
the relation (k, < l){\to}P, for infinite cardinals k and l and a poset P.Comment: 22 pages. Order, to appea
Some order dualities in logic, games and choices
We first present the concept of duality appearing in order theory, i.e. the notions of dual isomorphism and of Galois connection. Then, we describe two fundamental dualities, the duality extension/intention associated with a binary relation between two sets, and the duality between implicational systems and closure systems. Finally, we present two "concrete" dualities occuring in social choice and in choice functions theories.antiexchange closure operator, Galois connection, implicational system, path-independent choice function, simple game.
On the existence of right adjoints for surjective mappings between fuzzy structures0
En este trabajo los autores continĂşan su estudio de la caracterizaciĂłn de la existencia de adjunciones (conexiones de Galois isĂłtonas) cuyo codominio no está dotado de estructura en principio. En este artĂculo se considera el caso difuso en el que se tiene un orden difuso R definido en un conjunto A y una aplicaciĂłn sobreyectiva f:A-> B compatible respecto de dos relaciones de similaridad definidas en el dominio A y en el condominio B, respectivamente. Concretamente, el problema es encontrar un orden difuso S en B y una aplicaciĂłn g:B-> A compatible tambiĂ©n con las correspondientes similaridades definidas en A y en B, de tal forma que el par (f,g) constituya un adjunciĂłn
Fuzzy closure structures as formal concepts
Galois connections seem to be ubiquitous in mathematics. They have been used to model solutions for both pure and application-oriented problems. Throughout the paper, the general framework is a complete fuzzy lattice over a complete residuated lattice. The existence of three fuzzy Galois connections (two antitone and one isotone) between three specific ordered sets is proved in this paper. The most interesting part is that fuzzy closure systems, fuzzy closure operators and strong fuzzy closure relations are formal concepts of these fuzzy Galois connections.This research is partially supported by the State Agency of Research (AEI), the Spanish Ministry of Science, Innovation and Universities (MCIU), the European Social Fund (FEDER), the Junta de AndalucĂa (JA), and the Universidad de Málaga (UMA) through the FPU19/01467 (MCIU) internship and the research projects with reference PGC2018-095869-B-I00, TIN2017-89023-P, PID2021-127870OB-I00 (MCIU/AEI/FEDER, UE) and UMA18-FEDERJA-001 (JA/ UMA/ FEDER, UE).
Funding for open access charge: Universidad de Málaga / CBU
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