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.

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...

Bosbach states represent a way of probabilisticly evaluating the formulas from various (commutative or non-commutative) many-valued logics. They are defined on the algebras corresponding to these logics with values in [0, 1]. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure of [0, 1], in this paper we introduce Bosbach sta...

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.

My primary research interest is universal algebra. My thesis is about selfdistributive groupoids and their connection to other algebraic structures, such as groups and weakly associative loops. I am also interested ordered structures (residuated lattices in particular) and in the theory of quasigroups and loops. I am open to new interesting directions of research. A summary of my results follow...

In this paper, we will show that the variety of residuated lattices is generated by finite simple residuated lattices. The “simplicity” part of the proof is based on Grǐsin’s idea from [5], whereas the “finiteness” part employs a kind of algebraic filtration argument. Since the set of formulas valid in all residuated lattices is equal to the set of formulas provable in the propositional logic F...