The main part of the theory of residuated groupoids has been technical and the results apparently diverse in nature. However P. Dubreil and R. Croisot [5], using the concepts of residuated and residual maps, were able to present the basic properties of residuals in a simple unified manner. R. Croisot [3] carried on this study. In this paper we build a theory of residuated and residual maps whic...

We present the numbers of all non-isomorphic residuated lattices with up to 12 elements and a link to a database of these lattices. In addition, we explore various characteristics of these lattices such as the width, length, and various properties considered in the literature and provide the corresponding statistics. We also present algorithms for computing finite lattices and finite residuated...

A Hennessy-Milner property, relating modal equivalence and bisimulations, is defined for many-valued modal logics that combine a local semantics based on a complete MTL-chain (a linearly ordered commutative integral residuated lattice) with crisp Kripke frames. A necessary and sufficient algebraic condition is then provided for the class of image-finite models of these logics to admit the Henne...

Let B be a collection of fuzzy sets. What are the fuzzy sets which are sufficiently similar to every fuzzy set from B, i.e. ‘central’ fuzzy sets for B? Such a question naturally arises if B is large and one wishes to replace B by a single fuzzy set – the representative of B. In this paper, we develop a framework which enables us to answer this question and related ones. We use complete residuat...

We present results regarding row and column spaces of matrices whose entries are elements of residuated lattices. In particular, we define the notions of a row and column space for matrices over residuated lattices, provide connections to concept lattices and other structures associated to such matrices, and show several properties of the row and column spaces, including properties that relate ...

We develop the filter theory in triangle algebras. We define several interval valued residuated lattice-filters (IVRL-filters for short) in triangle algebras. We investigate the relationships among these types of IVRL-filters. Also, some special triangle algebras are introduced and studied in details.

In this paper, we introduce the notions of L-fuzzy quasi-uniformities and Lfuzzy interior operators in complete residuated lattices. We investigate the L-fuzzy quasiuniformities induced by L-fuzzy interior operators. We study the relationships between Lfuzzy interior operators and L-fuzzy quasi-uniformities. We give their examples. AMS Subject Classification: 03E72, 06A15, 06F07, 54F05

We study the connection between certain many-valued contexts and general geometric structures. The known one-to-one correspondence between attribute-complete many-valued contexts and complete affine ordered sets is used to extend the investigation to π-lattices and class geometries. The former are identified as a subclass of complete affine ordered sets, which exhibit a close relation to concep...

This paper considers the relations among L -fuzzy sets, rough sets and hyperring theory. Based on a complete residuated lattice, the concept of (invertible) L-fuzzy hyperideals of a hyperring is introduced and some related properties are presented. The notions of lower and upper L-fuzzy rough approximation operators with respect to an L-fuzzy hyperideal are provided and some significant propert...