33 research outputs found
Residuated structures and orthomodular lattices
The variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., ℓ-groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated ℓ-groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated ℓ-groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated ℓ-groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices
Enriched Topology and Asymmetry
Mathematically modeling the question of how to satisfactorily compare, in many-valued ways, both bitstrings and the predicates which they might satisfy-a surprisingly intricate question when the conjunction of predicates need not be commutative-applies notions of enriched categories and enriched functors. Particularly relevant is the notion of a set enriched by a po-groupoid, which turns out to be a many-valued preordered set, along with enriched functors extended as to be variable-basis . This positions us to model the above question by constructing the notion of topological systems enriched by many-valued preorders, systems whose associated extent spaces motivate the notion of topological spaces enriched by many-valued preorders, spaces which are non-commutative when the underlying lattice-theoretic base is equipped with a non-commutative (semi-)tensor product. Of special interest are crisp and many-valued specialization preorders generated by many-valued topological spaces, orders having these consequences for many-valued spaces: they characterize the well-established L-T0 separation axiom, define the L-T1(1) separation axiom-logically equivalent under appropriate lattice-theoretic conditions to the L-T1 axiom of T. Kubiak, and define an apparently new L-T1(2) separation axiom. Along with the consequences of such ideas for many-valued spectra, these orders show that asymmetry has a home in many-valued topology comparable in at least some respects to its home in traditional topology
Involutive Commutative Residuated Lattice without Unit: Logics and Decidability
We investigate involutive commutative residuated lattices without unit, which
are commutative residuated lattice-ordered semigroups enriched with a unary
involutive negation operator. The logic of this structure is discussed and the
Genzten-style sequent calculus of it is presented. Moreover, we prove the
decidability of this logic.Comment: 16 page
Algebraic structures from quantum and fuzzy logics
This thesis concerns the wide research area of logic. In particular, the first
part is devoted to analyze different kinds of relational systems (orthogonal
and residuated), by investigating the properties of the algebras associated
to them. The second part is focused on algebras of logic, in particular, the
relationship between prominent quantum and fuzzy structures with certain
semirings is proved. The last chapter concerns an application of group theory
to some well known mathematical puzzles
Why most papers on filters are really trivial (including this one)
The aim of this note is to show that many papers on various kinds of filters
(and related concepts) in (subreducts of) residuated structures are in fact
easy consequences of more general results that have been known for a long time