4 research outputs found
Hyperstates of Involutive MTL-Algebras that Satisfy
States of MV-algebras have been the object of intensive study and attempts of
generalizations. The aim of this contribution is to provide a preliminary
investigation for states of prelinear semihoops and hyperstates of algebras in
the variety generated by perfect and involutive MTL-algebras (IBP0-algebras for
short). Grounding on a recent result showing that IBP0-algebras can be
constructed from a Boolean algebra, a prelinear semihoop and a suitably defined
operator between them, our first investigation on states of prelinear semihoops
will support and justify the notion of hyperstate for IBP0- algebras and will
actually show that each such map can be represented by a probability measure on
its Boolean skeleton, and a state on a suitably defined abelian l-group.Comment: 12 page
Equivalences between subcategories of MTL-algebras via Boolean algebras and prelinear semihoops
This article studies the class of strongly perfect MTL-algebras, i.e. MTL-algebras having an involutive co-radical, and the variety they generate, namely SBP0. Once these structures will be introduced, we will first establish categorical equivalences for several of their relevant proper subvarieties by employing a generalized notion of triplets whose main components are a Boolean algebra and a prelinear semihoop. When triplets are further expanded by a suitable operation between their semihoop reducts, we define a category of quadruples that are equivalent to the whole category of SBP0-algebras. Finally, we will provide an explicit representation of SBP0-algebras in terms of (weak) Boolean products
Categories of Residuated Lattices
We present dual variants of two algebraic constructions of certain classes of residuated lattices: The Galatos-Raftery construction of Sugihara monoids and their bounded expansions, and the Aguzzoli-Flaminio-Ugolini quadruples construction of srDL-algebras. Our dual presentation of these constructions is facilitated by both new algebraic results, and new duality-theoretic tools. On the algebraic front, we provide a complete description of implications among nontrivial distribution properties in the context of lattice-ordered structures equipped with a residuated binary operation. We also offer some new results about forbidden configurations in lattices endowed with an order-reversing involution. On the duality-theoretic front, we present new results on extended Priestley duality in which the ternary relation dualizing a residuated multiplication may be viewed as the graph of a partial function. We also present a new Esakia-like duality for Sugihara monoids in the spirit of Dunn\u27s binary Kripke-style semantics for the relevance logic R-mingle
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group