Skew lattices are the most successful generalization of lattices to the noncommutative case to date. Roughly speaking, each skew lattice can be seen as a lattice of rectangular bands. A coset decomposition can be given to each pair of comparable maximal rectangular bands. The internal structure of skew lattices is revealed by their coset structure. In the present paper we study the coset struct...

The present paper has been an attempt to investigate the general fuzzy automata on the basis of complete residuated lattice-valued ($L$-GFAs). The study has been chiefly inspired from the work by Mockor cite{15, 16, 17}. Regarding this, the categorical issue of $L$-GFAs has been studied in more details. The main issues addressed in this research include: (1) investigating the relationship betwe...

In this paper, we characterize residuated lattices for which the topological space of prime ideals is a Noetherian space. The notion i-Noetherian lattice introduced and related properties are investigated. We proved that iff every ideal principal. Moreover, show has spectrum it i-Noetherian.

We show that a lattice-ordered field (not necessarily commutative) is totally ordered if and only if each square is positive, answering a generalized question of Conrad and Dauns [6] in the affirmative. As a consequence, any lattice-ordered skew-field in [5] is totally ordered. Furthermore , we note that each lattice order determined by a pre-positive cone P on a skew-filed F is linearly ordere...

The first part of this paper provides a comprehensive and self-contained account of the interrelationships between algebraic properties of varieties and properties of their free algebras and equational consequence relations. In particular, proofs are given of known equivalences between the amalgamation property and the Robinson property, the congruence extension property and the extension prope...

We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x∨¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To pr...

The notion of a state is an analogue of a probability measure and was first introduced by Kôpka and Chovanec for MV-algebras and by Riec̆an for BLalgebras. The states have also been studied for different types of non-commutative fuzzy structures such as pseudo-MV algebras, pseudo-BL algebras, bounded R`monoids, residuated lattices and pseudo-BCK semilattices. In this paper we investigate the sta...

Nonassociative Lambek Calculus (NL) is a pure logic of residuation, involving one binary operation (product) and its two residual operations defined on a poset [26]. Generalized Lambek Calculus GL involves a finite number of basic operations (with an arbitrary number of arguments) and their residual operations [7]. In this paper we study a further generalization of GL which admits operations wh...