36 research outputs found
Quasivarieties of Wajsberg hoops
In this paper we deal with quasivarieties of residuated structures which form the equivalent algebraic semantics of a positive frag-
ment of some substructural logic. Our focus is mainly on varieties and quasivarieties of Wajsberg hoops, which are the equivalent
algebraic semantics of the positive fragment of Łukasiewicz many-valued logic. In particular we study the lattice of subquasivari-
eties of Wajsberg hoops and we describe completely all the subvarieties of Wajsberg hoops that are primitive. Though the treatment
is mostly algebraic in nature, there are obvious connections with the underlying logic
Introducing Boolean Semilattices
We present and discuss a variety of Boolean algebras with operators that is closely related to the variety generated by all complex algebras of semilattices. We consider the problem of finding a generating set for the variety, representation questions, and axiomatizability. Several interesting subvarieties are presented. We contrast our results with those obtained for a number of other varieties generated by complex algebras of groupoids
Idempotent residuated structures : some category equivalences and their applications
This paper concerns residuated lattice-ordered idempotent commutative
monoids that are subdirect products of chains. An algebra of this
kind is a generalized Sugihara monoid (GSM) if it is generated by the lower
bounds of the monoid identity; it is a Sugihara monoid if it has a compatible
involution :. Our main theorem establishes a category equivalence
between GSMs and relative Stone algebras with a nucleus (i.e., a closure
operator preserving the lattice operations). An analogous result is obtained
for Sugihara monoids. Among other applications, it is shown that Sugihara
monoids are strongly amalgamable, and that the relevance logic RMt has
the projective Beth de nability property for deduction.http://www.ams.org//journals/tran/hb201
Structural completeness in quasivarieties
In this paper we study various forms of (hereditary) structural completeness
for quasivarieties of algebras, using mostly algebraic techniques. More
specifically we study relative weakly projective algebras and the way they
interact with structural completeness in quasivarieties. These ideas are then
applied to the study of -structural completeness and -primitivity,
through an algebraic generalization of Prucnal's substitution. Finally we study
in depth dual i-discriminator quasivarieties in which a particular instance of
Prucnal's substitution is used to prove that if each fundamental operation
commutes with the i-discriminator, then it is primitive
Inconsistency lemmas: an algebraic approach
Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona, Curs: 2019-2020, Tutor: Ramon Jansana i Tommaso MoraschiniThe purpose of this work is to investigate what is the algebraic counterpart of the inconsistency lemma. This question already has an answer in [29]. We have only put together in this monograph all the results and concepts that are necessary to understand the whole building
Inconsistency lemmas in algebraic logic
In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly.
We prove that, when a (finitary) deductive system is algebraized by a variety K, then has an inconsistency
lemma—in the abstract sense—iff every algebra in K has a dually pseudo-complemented join semilattice of
compact congruences. In this case, the following are shown to be equivalent: (1) has a classical inconsistency
lemma; (2) has a greatest compact theory and K is filtral, i.e., semisimple with EDPC; (3) the compact
congruences of any algebra in K form a Boolean lattice; (4) the compact congruences of any A ∈ K constitute
a Boolean sublattice of the full congruence lattice of A. These results extend to quasivarieties and relative
congruences. Except for (2), they extend even to protoalgebraic logics, with deductive filters in the role of
congruences. A protoalgebraic system with a classical inconsistency lemma always has a deduction-detachment
theorem (DDT), while a system with a DDT and a greatest compact theory has an inconsistency lemma. The
converses are false.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-3870hb201