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...

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...

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.

We determine a general formula to compute the number of saturated chains in Dyck lattices, and we apply it to find the number of saturated chains of length 2 and 3. We also compute what we call the Hasse index (of order 2 and 3) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2 and 3) and the cardinality of the underlying poset.

The operation ‘·’, often called fusion is distributive over join. In finite residuated lattices, fusion and join determine residuation uniquely, although residuation cannot be defined equationally from other operations. The class R of residuated lattices is a variety. It is arithmetical, has CEP, and is generated by its finite members (cf. [7]). It is also congruence 1-regular, i.e., for any co...

Alexandrov topologies are the topologies induced by relations. This paper addresses the properties of Alexandrov topologies as the extensions of strong topologies and strong cotopologies in complete residuated lattices. With the concepts of Zhang’s completeness, the notions are discussed as extensions of interior and closure operators in a sense as Pawlak’s the rough set theory. It is shown tha...

The Mamdani model [1] of Fuzzy Systems is the earliest and the most widely studied type of Fuzzy Systems. In this work, Residuated Implication (R-implication) operators have been explored for rule reduction in Mamdani-Type Fuzzy Systems with lossless inferencing.