11 research outputs found
Nuclear ranges in implicative semilattices
A nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice N A that is isomorphic to the system NA of all nuclear ranges, ordered by dual inclusion. The nuclear ranges are those closure ranges which are total subalgebras (l-ideals). Nuclei have been studied intensively in the case of complete Heyting algebras. We extend, as far as possible, results on nuclei and their ranges to the non-complete setting of implicative semilattices (whose unary meet translations have adjoints). A central tool are so-called r-morphisms, that is, residuated semilattice homomorphisms, and their adjoints, the l-morphisms. Such morphisms transport nuclear ranges and preserve implicativity. Certain completeness properties are necessary and sufficient for the existence of a least nucleus above a prenucleus or of a greatest nucleus below a weak nucleus. As in pointfree topology, of great importance for structural investigations are three specific kinds of l-ideals, called basic open, boolean and basic closed. © 2022, The Author(s)
Adjoint maps between implicative semilattices and continuity of localic maps
We study residuated homomorphisms (r-morphisms) and their adjoints, the so-called localizations (or l-morphisms), between implicative semilattices, because these objects may be characterized as semilattices whose unary meet operations have adjoints. Since left resp. right adjoint maps are the residuated resp. residual maps (having the property that preimages of principal downsets resp. upsets are again such), one may not only regard the l-morphisms as abstract continuous maps in a pointfree framework (as familiar in the complete case), but also characterize them by concrete closure-theoretical continuity properties. These concepts apply to locales (frames, complete Heyting lattices) and provide generalizations of continuous and open maps between spaces to an algebraic (not necessarily complete) pointfree setting. © 2022, The Author(s)
Relatively compatible operations in BCK-algebras and some related algebras
Let A be a BCK-algebra and f : Ak → A a function. The main goal of this paper is to give a necessary and sufficient condition for ƒ to be compatible with respect to every relative congruence of A. We extend this result in some related algebras, as for example in pocrims.Facultad de Ciencias Exacta
On subreducts of subresiduated lattices and logic
Subresiduated lattices were introduced during the decade of 1970 by Epstein
and Horn as an algebraic counterpart of some logics with strong implication
previously studied by Lewy and Hacking. These logics are examples of
subuintuitionistic logics, i.e., logics in the language of intuitionistic logic
that are defined semantically by using Kripke models, in the same way as
intuitionistic logic is defined, but without requiring of the models some of
the properties required in the intuitionistic case. Also in relation with the
study of subintuitionistic logics, Celani and Jansana get these algebras as the
elements of a subvariety of that of weak Heyting algebras.
Here, we study both the implicative and the implicative-infimum subreducts of
subresiduated lattices. Besides, we propose a calculus whose algebraic
semantics is given by these classes of algebras. Several expansions of this
calculi are also studied together to some interesting properties of them
Order algebraizable logics
AbstractThis paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic (AAL). Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: (i) it is injective, (ii) it is isotonic, (iii) it commutes with the inverse image operator of any algebraic homomorphism, and (iv) it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties (i)–(iv)
Representations and Completions for Ordered Algebraic Structures
The primary concerns of this thesis are completions and representations for various classes of
poset expansion, and a recurring theme will be that of axiomatizability. By a representation we
mean something similar to the Stone representation whereby a Boolean algebra can be homomorphically
embedded into a field of sets. So, in general we are interested in order embedding
posets into fields of sets in such a way that existing meets and joins are interpreted naturally as
set theoretic intersections and unions respectively.
Our contributions in this area are an investigation into the ostensibly second order property
of whether a poset can be order embedded into a field of sets in such a way that arbitrary meets
and/or joins are interpreted as set theoretic intersections and/or unions respectively. Among
other things we show that unlike Boolean algebras, which have such a ‘complete’ representation
if and only if they are atomic, the classes of bounded, distributive lattices and posets with
complete representations have no first order axiomatizations (though they are pseudoelementary).
We also show that the class of posets with representations preserving arbitrary joins is
pseudoelementary but not elementary (a dual result also holds).
We discuss various completions relating to the canonical extension, whose classical construction
is related to the Stone representation. We claim some new results on the structure of
classes of poset meet-completions which preserve particular sets of meets, in particular that they
form a weakly upper semimodular lattice. We make explicit the construction of \Delta_{1}-completions
using a two stage process involving meet- and join-completions.
Linking our twin topics we discuss canonicity for the representation classes we deal with,
and by building representations using a meet-completion construction as a base we show that
the class of representable ordered domain algebras is finitely axiomatizable. Our method has
the advantage of representing finite algebras over finite bases