308 research outputs found
Eilenberg theorems for many-sorted formations
A theorem of Eilenberg establishes that there exists a bijection between the
set of all varieties of regular languages and the set of all varieties of
finite monoids. In this article after defining, for a fixed set of sorts
and a fixed -sorted signature , the concepts of formation of
congruences with respect to and of formation of -algebras, we
prove that the algebraic lattices of all -congruence formations and of
all -algebra formations are isomorphic, which is an Eilenberg's type
theorem. Moreover, under a suitable condition on the free -algebras and
after defining the concepts of formation of congruences of finite index with
respect to , of formation of finite -algebras, and of formation
of regular languages with respect to , we prove that the algebraic
lattices of all -finite index congruence formations, of all
-finite algebra formations, and of all -regular language
formations are isomorphic, which is also an Eilenberg's type theorem.Comment: 46 page
Relational Galois connections between transitive fuzzy digraphs
Fuzzy-directed graphs are often chosen as the data structure to model and implement solutions to several problems in the applied sciences. Galois connections have also shown to be useful both in theoretical and in practical problems. In this paper, the notion of relational Galois connection is extended to be applied between transitive fuzzy directed graphs. In this framework, the components of the connection are crisp relations satisfying certain reasonable properties given in terms of the so-called full powering
On tractability and congruence distributivity
Constraint languages that arise from finite algebras have recently been the
object of study, especially in connection with the Dichotomy Conjecture of
Feder and Vardi. An important class of algebras are those that generate
congruence distributive varieties and included among this class are lattices,
and more generally, those algebras that have near-unanimity term operations. An
algebra will generate a congruence distributive variety if and only if it has a
sequence of ternary term operations, called Jonsson terms, that satisfy certain
equations.
We prove that constraint languages consisting of relations that are invariant
under a short sequence of Jonsson terms are tractable by showing that such
languages have bounded relational width
Linear representations of regular rings and complemented modular lattices with involution
Faithful representations of regular -rings and modular complemented
lattices with involution within orthosymmetric sesquilinear spaces are studied
within the framework of Universal Algebra. In particular, the correspondence
between classes of spaces and classes of representables is analyzed; for a
class of spaces which is closed under ultraproducts and non-degenerate finite
dimensional subspaces, the latter are shown to be closed under complemented
[regular] subalgebras, homomorphic images, and ultraproducts and being
generated by those members which are associated with finite dimensional spaces.
Under natural restrictions, this is refined to a --correspondence between
the two types of classes
- …