We study L-categories of lattice-valued convergence spaces. Suchcategories are obtained by fuzzifying" the axioms of a lattice-valued convergencespace. We give a natural example, study initial constructions andfunction spaces. Further we look into some L-subcategories. Finally we usethis approach to quantify how close certain lattice-valued convergence spacesare to being lattice-valued topologi...

The amalgamation property (AP) is of particular interest in the study of residuated lattices due to its relationship with various syntactic interpolation properties of substructural logics. There are no examples to date of non-commutative varieties of residuated lattices that satisfy the AP. The variety SemRL of semilinear residuated lattices is a natural candidate for enjoying this property, s...

In this paper, we present triangle algebras: residuated lattices equipped with two modal, or approximation, operators and with a third angular point u, different from 0 (false) and 1 (true), intuitively denoting ignorance about a formula’s truth value. We prove that these constructs, which bear a close relationship to several other algebraic structures including rough approximation spaces, prov...

Hahn’s famous structure theorem states that totally ordered Abelian groups can be embedded in the lexicographic product of real groups. Our main theorem extends this structural description to order-dense, commutative, group-like residuated chains, which has only finitely many idempotents. It is achieved via the so-called partial-lexicographic product construction (to be introduced here) using t...

In this paper, it is shown that the category of stratified $L$-generalized convergence spaces is monoidal closed if the underlying truth-value table $L$ is a complete residuated lattice. In particular, if the underlying truth-value table $L$ is a complete Heyting Algebra, the Cartesian closedness of the category is recaptured by our result.

Right-residuated binars and right-divisible residuated binars are defined as precursors of generalized hoops, followed by some results and open problems about these partially ordered algebras. Next we show that all complete homomorphic images of a complete residuated lattice A can be constructed easily on certain definable subsets of A. Applying these observations to the algebras of Hajek’s Bas...

Introduction and summary. In the theory of non-commutative rings certain distinguished subrings, one-sided and two-sided ideals, play the important roles. Ideals combine under crosscut, union and multiplication and hence are an instance of a lattice over which a non-commutative multiplication is defined.f The investigation of such lattices was begun by W. Krull (Krull [3]) who discussed decompo...

We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into near and far sets and an integer threshold t, a threshold-coloring of the graph is an assignment of integers to the vertices so that endpoints of near edges differ by t or less, while endpoints of far edges differ by more than t. We study threshold-coloring of tilings of th...

Exact querying and retrieving relevant data from a database is a difficult task. We present an approach for flexibly answering algebraic queries using an extension of Codd’s relational model with ordinal ranks based on residuated lattices and similarities on attribute domains.