Pseudocomplemented Semilattices Research Articles

Algebraic structures formalizing the logic with unsharp implication and negation

Abstract It is well-known that intuitionistic logics can be formalized by means of Heyting algebras, i.e. relatively pseudocomplemented semilattices. Within such algebras the logical connectives implication and conjunction are formalized as the relative pseudocomplement and the semilattice operation meet, respectively. If the Heyting algebra has a bottom element $0$, then the relative pseudocomplement with respect to $0$ is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with $0$ satisfying only the Ascending Chain Condition (these assumptions are trivially satisfied in finite meet-semilattices) and introduce the operators formalizing the connectives negation $x^{0}$ and implication $x\rightarrow y$ as the set of all maximal elements $z$ satisfying $x\wedge z=0$ and as the set of all maximal elements $z$ satisfying $x\wedge z\leq y$, respectively. Such a negation and implication is ‘unsharp’ since it assigns to one entry $x$ or to two entries $x$ and $y$ belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. We present several examples. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.

On a new construction of pseudocomplemented semilattices

On a new construction of pseudocomplemented semilattices

$$\delta $$-ideals of a pseudocomplemented semilattice

In this paper we study $$\delta $$ -ideals of a pseudocomplemented semilattice. We give some characterizations of $$\delta $$ -ideals. We also prove Stone type separation theorem for $$\delta $$ -ideals of a pseudocomplemented semilattice. We show that the set of $$\delta $$ -ideals of a pseudocomplemented semilattice form a distributive pseudocomplemented lattice. We also prove that the set of principal $$\delta $$ -ideals form a Boolean algebra.

Lorenzen's Proof of Consistency for Elementary Number Theory

We present a manuscript of Paul Lorenzen that provides a proof of consistency for elementary number theory as an application of the construction of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also compare this proof to Gentzen's and Novikov's, and provide a translation of the manuscript.

Fuzzy Ideals and Fuzzy Filters of Pseudocomplemented Semilattices

In this paper, we introduce the concept of kernel fuzzy ideals and ⁎-fuzzy filters of a pseudocomplemented semilattice and investigate some of their properties. We observe that every fuzzy ideal cannot be a kernel of a ⁎-fuzzy congruence and we give necessary and sufficient conditions for a fuzzy ideal to be a kernel of a ⁎-fuzzy congruence. On the other hand, we show that every fuzzy filter is the cokernel of a ⁎-fuzzy congruence. Finally, we prove that the class of ⁎-fuzzy filters forms a complete lattice that is isomorphic to the lattice of kernel fuzzy ideals.

Correction to: The model companion of the class of pseudocomplemented semilattices is finitely axiomatizable

In the paper “The model companion of the class of pseudocomplemented semilattices is finitely axiomatizable” a list of nine axioms, which are claimed to axiomatize the class of existentially closed pseudocomplemented semilattices, is given. After its publication an error in a key lemma was found by the reviewer of a subsequent paper. The error was rectified by strengthening one of these nine axioms and then proving that the resulting nine axioms do axiomatize this class, thereby proving the statement of the title. Here we explain the error, outline the correction and refer the reader to arXiv.org for a full revision of the published paper.

The countable existentially closed pseudocomplemented semilattice

As the class \(\mathcal {PCSL}\) of pseudocomplemented semilattices is a universal Horn class generated by a single finite structure it has a \(\aleph _0\)-categorical model companion \(\mathcal {PCSL}^*\). As \(\mathcal {PCSL}\) is inductive the models of \(\mathcal {PCSL}^*\) are exactly the existentially closed models of \(\mathcal {PCSL}\). We will construct the unique existentially closed countable model of \(\mathcal {PCSL}\) as a direct limit of algebraically closed pseudocomplemented semilattices.

Free pseudocomplemented semilattices: a new approach

An intrinsic characterization of free pseudocomplemented semilattices is presented. The second principal result gives a method for constructing free pseudocomplemented semilattices.

The model companion of the class of pseudocomplemented semilattices is finitely axiomatizable

It is shown that the class $${\mathcal{PCSL}^{ec}}$$ of existentially closed pseudocomplemented semilattices is finitely axiomatizable by appropriately extending a finite axiomatization of the class $${\mathcal{PCSL}^{ac}}$$ of algebraically closed pseudocomplemented semilattices. Because $${\mathcal{PCSL}^{ec}}$$ coincides with the model companion of the class $${\mathcal{PCSL}}$$ of pseudocomplemented semilattices, this answers the question asked by Albert and Burris in a paper in 1986: “Does the class of pseudocomplemented semilattices have a finitely axiomatizable model companion?"

NEUTRAL OPERATORS AND TRIPLE REPRESENTATION

The basic idea of triple representation of a structure is to associate two simpler structures and a connecting map. Triple representation was first introduced by Chen and Gratzer to Stone Lattices (2), (3). The concept was gen- eralised to distributive pseudo-complemented lattices and then to distributive pseudo-complemented semilattices by Katrinak. The triple representation for modular Pseudo Complemented semilattice was suceeded by W.H.Cornish(4). Now the triple representation was obtained for a supermodular semilattice and thereby the supermodular semilattices are characterised.

The class of algebraically closed p-semilattices is finitely axiomatizable

We prove our title, and thereby establish the base for a positive solution of Albert and Burris’ problem on the finite axiomatizability of the model companion of the class of all pseudocomplemented semilattices.

PROPER HYPERSUBSTITUTIONS OF VARIETIES OF PSEUDO-COMPLEMENTED SEMILATTICES

We determine all V-proper hypersubstitutions for the variety of pseudo-complemented semilattices. Moreover, for any monoid [Formula: see text] of hypersubstitutions we will give the greatest M-solid variety of pseudo-complemented semilattices.

Principal congruences on pseudocomplemented semilattices

We give a description of the principal congruences on pseudocomplemented semilattices, clarifying previous work of Sankappanavar. The proof of the main result exploits the theory of binary discriminator varieties.

Properties of relatively pseudocomplemented directoids

The concept of a relatively pseudocomplemented directoid was introduced recently by the first author. It was shown that the class of relatively pseudocomplemented directoids forms a variety whose axiom system contains seven identities. The aim of this paper is three-fold. First we show that these identities are not independent and their independent subset is presented. Second, we modify the adjointness property known for relatively pseudocomplemented semilattices in the way which is suitable for relatively pseudocomplemented directoids. Hence, they can also be considered as residuated structures in a rather modified version. We also get two important congruence properties, namely congruence distributivity and $3$-permutability valid in the variety $\mathcal {V}$ of relatively pseudocomplemented directoids. Then we show some basic results connected with subdirect irreducibility in $\mathcal {V}$. Finally, we show another way how to introduce pseudocomplementation on directoids via relative pseudocomplementation.

Lee classes for pseudocomplemented semilattices, revisited

We give syntactic and semantic characterisations of the Lee classes of the varieties of pseudocomplemented semilattices and pseudocomplemented distributive lattices. The main result clarifies earlier work of K. B. Lee and of the second author.

Free products of pseudocomplemented semilattices – revisited

We analyze the structure of free products of pseudocomplemented semilattices in terms of their skeletons and Glivenko classes by giving a rather explicit construction, complementing the description given by Katriňák and Heleyová in [6].

DISTRIBUTIVE PSEUDOCOMPLEMENTED SEMILATTICES

In this paper we will extend the topological duality given in [1] to the class of distributive pseudocomplemented semilattices and to the class of distributive Stone semilattices.

Finiteness conditions and distributive laws for Boolean algebras

AbstractWe compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Pseudocomplemented semilattices are finite-to-finite relatively universal

It is shown that the category of directed graphs is isomorphic to a subcategory of the variety S of all pseudocomplemented semilattices which contains all homomorphisms whose images do not lie in the subvariety B of all Boolean pseudocomplemented semilattices. Moreover, the functor exhibiting the isomorphism may be chosen such that each finite directed graph is assigned a finite pseudocomplemented semilattice. That is to say, it is shown that the variety S of all pseudocomplemented semilattices is finite-to-finite B-relatively universal. This illustrates the complexity of the endomorphism monoids of pseudocomplemented semilattices since it follows immediately that, for any monoid M, there exists a proper class of non-isomorphic pseudocomplemented semilattices such that, for each member S, the endomorphisms of S which do not have an image contained in the skeleton of S form a submonoid of the endomorphism monoid of S which is isomorphic to M.

Subdirectly irreducible sectionally pseudocomplemented semilattices

Sectionally pseudocomplemented semilattices are an extension of relatively pseudocomplemented semilattices—they are meet-semilattices with a greatest element such that every section, i.e., every principal filter, is a pseudocomplemented semilattice. In the paper, we give a simple equational characterization of sectionally pseudocomplemented semilattices and then investigate mainly their congruence kernels which leads to a characterization of subdirectly irreducible sectionally pseudocomplemented semilattices.