Implicative Semilattices Research Articles

Intuitionistic Modal Algebras

Recent research on algebraic models of quasi-Nelson logic has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a nucleus. Among these various algebraic structures, for which we employ the umbrella term intuitionistic modal algebras, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of weak implicative semilattices, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.

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 mathrm{N }A that is isomorphic to the system {mathcal {N}}A 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.

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.

Diego’s Theorem for nuclear implicative semilattices

We prove that the variety of nuclear implicative semilattices is locally finite, thus generalizing Diego’s Theorem. The key ingredients of our proof include the coloring technique and construction of universal models from modal logic. For this we develop duality theory for finite nuclear implicative semilattices, generalizing Köhler duality. We prove that our main result remains true for bounded nuclear implicative semilattices, give an alternative proof of Diego’s Theorem, and provide an explicit description of the free cyclic nuclear implicative semilattice.

On the free frontal implicative semilattice extension of a frontal Hilbert algebra

In this paper, we define a functor which is left adjoint to the forgetful functor from the category of frontal implicative semilattices to that of frontal Hilbert algebras.

Variations of the free implicative semilattice extension of a Hilbert algebra

Fil: Castiglioni, Jose Luis. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

On Kalman’s functor for bounded hemi-implicative semilattices and hemi-implicative lattices

Hemi-implicative semilattices (lattices), originally defined under the name of weak implicative semilattices (lattices), were introduced by the second author of the present paper. A hemi-implicative semilattice is an algebra $(H,\wedge,\rightarrow,1)$ of type $(2,2,0)$ such that $(H,\wedge)$ is a meet semilattice, $1$ is the greatest element with respect to the order, $a\rightarrow a = 1$ for every $a\in H$ and for every $a$, $b$, $c\in H$, if $a\leq b\rightarrow c$ then $a\wedge b \leq c$. A bounded hemi-implicative semilattice is an algebra $(H,\wedge,\rightarrow,0,1)$ of type $(2,2,0,0)$ such that $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice and $0$ is the first element with respect to the order. A hemi-implicative lattice is an algebra $(H,\wedge,\vee,\rightarrow,0,1)$ of type $(2,2,2,0,0)$ such that $(H,\wedge,\vee,0,1)$ is a bounded distributive lattice and the reduct algebra $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice. In this paper we introduce an equivalence for the categories of bounded hemi-implicative semilattices and hemi-implicative lattices, respectively, which is motivated by an old construction due J. Kalman that relates bounded distributive lattices and Kleene algebras.

Spectral-like duality for distributive Hilbert algebras with infimum

Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and $${\wedge}$$ -semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.

Locally Finite Reducts of Heyting Algebras and Canonical Formulas

The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics (si-logics for short). The ∨-free reducts of Heyting algebras give rise to the (∧,→)-canonical formulas that we studied in an earlier work. Here we introduce the (∧,∨)-canonical formulas, which are obtained from the study of the →-free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by (∧,∨)-canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas. One of the main ingredients of these formulas is a designated subset D of pairs of elements of a finite subdirectly irreducible Heyting algebra A. When D=A2, we show that the (∧,∨)-canonical formula of A is equivalent to the Jankov formula of A. On the other hand, when D=∅, the (∧,∨)-canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics.

On congruences in weak implicative semi-lattices

In this paper we characterize the congruences in classes of algebras that properly include varieties of interest for the logic. These algebras are obtained by weakening the main features of Heyting algebras $$(H, \wedge , \vee , \rightarrow ,0, 1)$$(H,ź,ź,ź,0,1) but retaining most of their algebraic consequences. The residuation property $$z\le x\rightarrow y$$z≤xźy if and only if $$x \wedge z \le y$$xźz≤y is replaced by the condition: if $$z\le x\rightarrow y$$z≤xźy then $$x \wedge z \le y$$xźz≤y. Some of further algebraic conditions of Heyting algebras are required.

Hilbert algebras with supremum

In this paper, we will study the class of Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice. First, we will give a simplified topological duality for Hilbert algebras using sober topological spaces with a basis of open-compact sets satisfying an additional condition. Next, we will extend this duality to Hilbert algebras with supremum. We shall prove that the ordered set of all ideals of a Hilbert algebra with supremum has a lattice structure. We will also see that in this lattice, it is possible to define an implication, but the resulting structure is neither a Heyting algebra nor an implicative semilattice. Finally, we will give a dual description of the lattice of ideals of a Hilbert algebra with supremum.

On the free implicative semilattice extension of a Hilbert algebra

AbstractHilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.

Esakia Style Duality for Implicative Semilattices

We develop a new duality for implicative semilattices, generalizing Esakia duality for Heyting algebras. Our duality is a restricted version of generalized Priestley duality for distributive semilattices, and provides an improvement of Vrancken-Mawet and Celani dualities. We also show that Heyting algebra homomorphisms can be characterized by means of special partial functions between Esakia spaces. On the one hand, this yields a new duality for Heyting algebras, which is an alternative to Esakia duality. On the other hand, it provides a natural generalization of Kohler’s partial functions between finite posets to the infinite case.

Generalized Priestley Quasi-Orders

We introduce generalized Priestley quasi-orders and show that subalgebras of bounded distributive meet-semilattices are dually characterized by means of generalized Priestley quasi-orders. This generalizes the well-known characteriza- tion of subalgebras of bounded distributive lattices by means of Priestley quasi- orders (Adams, Algebra Univers 3:216-228, 1973; Cignoli et al., Order 8(3):299- 315, 1991; Schmid, Order 19(1):11-34, 2002). We also introduce Vietoris families and prove that homomorphic images of bounded distributive meet-semilattices are dually characterized by Vietoris families. We show that this generalizes the well- known characterization (Priestley, Proc Lond Math Soc 24(3):507-530, 1972 )o f homomorphic images of a bounded distributive lattice by means of closed subsets of its Priestley space. We also show how to modify the notions of generalized Priestley quasi-order and Vietoris family to obtain the dual characterizations of subalgebras and homomorphic images of bounded implicative semilattices, which generalize the well-known dual characterizations of subalgebras and homomorphic images of Heyting algebras (Esakia, Sov Math Dokl 15:147-151, 1974).

Pure Hilbert Algebras with Infimum

In [6], iH-algebras were introduced in order to indicate an equational version of the class of Hilbert algebras where each pair of elements has infimum. These authors also proved that this variety hasthe class of Curry's implicative semilattices ([9]) as a proper subvariety. On the other hand, in [4] a special class of Hilbert algebras associated with ordered sets, which they called order algebras, were investigated. These algebras were also studied in [1] under the name of pure Hilbert algebras.Bearing in mind the above results, in this paper we introduce the notion of pure Hilbert algebras with infimum (or ipH-algebras, for short). Furthermore, we characterize the lattice of ipH-congruences and we determine the subdirectly irreducible ipH-algebras. Besides, we prove that subdirectly irreducible ipH-algebras are alsosubdirectly irreducible iH-algebras.

Hilbert algebras as implicative partial semilattices

Abstract The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.

Representation of Hilbert algebras and implicative semilattices

Abstract In this paper we shall give a topological representation for Hilbert algebras that extend the topological representation given by A. Diego in [4]. For implicative semilattices this representation gives a full duality. We shall also consider the representation for Boolean ring.

On filters of implicative semilattices

On filters of implicative semilattices

Homomorphisms of implicative semigroups

Implicative semigroups and Brouwerian semigroups were studied by W. C. Nemitz and T. S. Blyth, respectively. In this paper, following the ideas of Nemitz and Blyth, we introduce the notion of negatively partially ordered implicative semigroups and studied the homomorphisms between these semigroups. Some results of Nemitz on implicative semilattices are generalized and amplified to implicative semigroups.

Pseudocomplemented and Implicative Semilattices

Let L be a semilattice and let a ∊ L. We refer the reader to Definitions 2.2, 2.4, 2.5 and 2.12 below for the terminology. If L is a-implicative, let Ca be the set of a-closed elements of L, and let Da be the filter of a-dense elements of L. Then Ca is a Boolean algebra. If a = 0, then C0 and D0 are the usual closed algebra and dense filter of L. If L is a-admissible and f : Ca × Da → Da is the corresponding admissible map, we can form a quotient semilattice Ca × D0f. In case a = 0, Murty and Rao [4] have shown that C0 × D0/f is isomorphic to L, and hence that C0 × D0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C0 × D0/f is isomorphic to L, and that C0 × D0/f is also implicative.