36 research outputs found
On locally finite varieties of Heyting algebras
For every , we construct a variety of Heyting algebras,
whose -generated free algebra is finite but whose -generated free
algebra is infinite
On prevarieties of logic
It is proved that every prevariety of algebras is categorically equivalent to
a "prevariety of logic", i.e., to the equivalent algebraic semantics of some
sentential deductive system. This allows us to show that no nontrivial equation
in the language "meet, join, and relational product" holds in the congruence
lattices of all members of every variety of logic, and that being a
(pre)variety of logic is not a categorical property
Epimorphism surjectivity in varieties of Heyting algebras
It was shown recently that epimorphisms need not be surjective in a variety K
of Heyting algebras, but only one counter-example was exhibited in the
literature until now. Here, a continuum of such examples is identified, viz.
the variety generated by the Rieger-Nishimura lattice, and all of its (locally
finite) subvarieties that contain the original counter-example K. It is known
that, whenever a variety of Heyting algebras has finite depth, then it has
surjective epimorphisms. In contrast, we show that for every integer n greater
or equal than 2, the variety of all Heyting algebras of width at most n has a
non-surjective epimorphism. Within the so-called Kuznetsov-Gerciu variety
(i.e., the variety generated by finite linear sums of one-generated Heyting
algebras), we describe exactly the subvarieties that have surjective
epimorphisms. This yields new positive examples, and an alternative proof of
epimorphism surjectivity for all varieties of Goedel algebras. The results
settle natural questions about Beth-style definability for a range of
intermediate logics
Logics of left variable inclusion and PÅ‚onka sums of matrices
The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic ⊢. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic ⊢ is related to the construction of Płonka sums of the matrix models of ⊢. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate them in the Leibniz hierarchy
Bi-intermediate logics of trees and co-trees
A bi-Heyting algebra validates the G\"odel-Dummett axiom iff the poset of its prime filters is a disjoint union of co-trees (i.e.,
order duals of trees). Bi-Heyting algebras of this kind are called bi-G\"odel
algebras and form a variety that algebraizes the extension
- of bi-intuitionistic logic axiomatized by the
G\"odel-Dummett axiom. In this paper we initiate the study of the lattice
- of extensions of
-.
We develop the methods of Jankov-style formulas for bi-G\"odel algebras and
use them to prove that there are exactly continuum many extensions of
-. We also show that all these extensions can be
uniformly axiomatized by canonical formulas. Our main result is a
characterization of the locally tabular extensions of
-. We introduce a sequence of co-trees, called the
finite combs, and show that a logic in - is locally
tabular iff it contains at least one of the Jankov formulas associated with the
finite combs. It follows that there exists the greatest non-locally tabular
extension of - and consequently, a unique pre-locally
tabular extension of -. These results contrast with
the case of the intermediate logic axiomatized by the G\"odel-Dummett axiom,
which is known to have only countably many extensions, all of which are locally
tabular