The theory of boolean algebras with an additional binary operation

This paper deals with Boolean algebras supplied with an additional binary operation, calledB-algebras for short.The aim of the paper is to generalize some theorems concerning topological Boolean algebras to more comprehensive classes ofB-algebras, to formulate fundamental properties ofB-algebras, and to find more important relationships of these algebras to other known algebras.The paper consists of two parts. At the beginning of the first one, several subclasses ofB-algebras are distinguished, and then, their basic properties, connections between them as well as certain relationships with other algebras, are investigated. In particular, it is shown that the class of Boolean algebras together with an arbitrary unary operation is polynomially equivalent to the class ofB ₁-algebras.The second part of the paper is concerned with the theory of filters and congruences inB-algebras.     

Functional Monadic Bounded Algebras

The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ${\exists} reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions.     

MS-Algebras Whose e-Ideals are Kernel Ideals

Studia Logica - We consider, in the context of an MS-algebra L, the ideals I of L that are kernels of L. We characterize two kinds of de Morgan algebras: the class Boolean algebras and the...     

Persistence and Atomic Generation for Varieties of Boolean Algebras with Operators

A variety V of Boolean algebras with operators is singleton-persistent if it contains a complex algebra whenever it contains the subalgebra generated by the singletons. V is atom-canonical if it contains the complex algebra of the atom structure of any of the atomic members of V.This paper explores relationships between these "persistence" properties and questions of whether V is generated by its complex algebras or its atomic members, or is closed under canonical embedding algebras or completions. It also develops a general theory of when operations involving complex algebras lead to the construction of elementary classes of relational structures.     

A Comprehensive Picture of the Decidability of Mereological Theories

The signature of the formal language of mereology contains only one binary predicate which stands for the relation “being a part of” and it has been strongly suggested that such a predicate must at least define a partial ordering. Mereological theories owe their origin to Le?niewski. However, some more recent authors, such as Simons as well as Casati and Varzi, have reformulated mereology in a way most logicians today are familiar with. It turns out that any theory which can be formed by using the reformulated mereological axioms or axiom schemas is in a sense a subtheory of the elementary theory of Boolean algebras or of the theory of infinite atomic Boolean algebras. It is known that the theory of partial orderings is undecidable while the elementary theory of Boolean algebras and the theory of infinite atomic Boolean algebras are decidable. In this paper, I will look into the behaviors in terms of decidability of those mereological theories located in between. More precisely, I will give a comprehensive picture of the said issue by offering solutions to the open problems which I have raised in some of my papers published previously.     

Plain Semi-Post algebras as a poset-based generalization of post algebras and their representability

Semi-Post algebras of any type T being a poset have been introduced and investigated in [CR87a], [CR87b]. Plain Semi-Post algebras are in this paper singled out among semi-Post algebras because of their simplicity, greatest similarity with Post algebras as well as their importance in logics for approximation reasoning ([Ra87a], [Ra87b], [RaEp87]). They are pseudo-Boolean algebras generated in a sense by corresponding Boolean algebras and a poset T. Every element has a unique descending representation by means of elements in a corresponding Boolean algebra and primitive Post constants which form a poset T. An axiomatization and another characterization, subalgebras, homomorphisms, congruences determined by special filters and a representability theory of these algebras, connected with that for Boolean algebras, are the subject of this paper.To the memory of Jerzy SupeckiResearch reported here has been supported by Polish Government Grant CPBP 01.01     

Averaging the truth-value in Łukasiewicz logic

Chang's MV algebras are the algebras of the infinite-valued sentential calculus of ukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of average degree of truth of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.     

Categoricity Spectra for Polymodal Algebras

We investigate effective categoricity for polymodal algebras (i.e., Boolean algebras with distinguished modalities). We prove that the class of polymodal algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. In particular, this implies that every categoricity spectrum is the categoricity spectrum of a polymodal algebra.     

Sahlqvist's theorem for boolean algebras with operators with an application to cylindric algebras

For an arbitrary similarity type of Boolean Algebras with Operators we define a class ofSahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities arecanonical, that is, their validity is preserved under taking canonical embedding algebras. Taken together, these properties imply that results about a Sahlqvist variety V van be obtained by reasoning in the elementary class of canonical structures of algebras in V.We give an example of this strategy in the variety of Cylindric Algebras: we show that an important identity calledHenkin's equation is equivalent to a simpler identity that uses only one variable. We give a conceptually simple proof by showing that the first-order correspondents of these two equations are equivalent over the class of cylindric atom structures.Presented byIstván Németi     

The First-Order Theories of Dedekind Algebras

A Dedekind Algebra is an ordered pair (B,h) where B is a non-empty set and h is an injective unary function on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called configurations of the Dedekind algebra. There are N₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on omega called its configuration signature. The configuration signature of a Dedekind algebra counts the number of configurations in the decomposition of the algebra in each isomorphism type.The configuration signature of a Dedekind algebra encodes the structure of that algebra in the sense that two Dedekind algebras are isomorphic iff their configuration signatures are identical. Configuration signatures are used to establish various results in the first-order model theory of Dedekind algebras. These include categoricity results for the first-order theories of Dedekind algebras and existence and uniqueness results for homogeneous, universal and saturated Dedekind algebras. Fundamental to these results is a condition on configuration signatures that is necessary and sufficient for elementary equivalence.     

On Complete Representations of Reducts of Polyadic Algebras

Following research initiated by Tarski, Craig and Németi, and futher pursued by Sain and others, we show that for certain subsets G of ^ω ω, atomic countable G polyadic algebras are completely representable. G polyadic algebras are obtained by restricting the similarity type and axiomatization of ω-dimensional polyadic algebras to finite quantifiers and substitutions in G. This contrasts the cases of cylindric and relation algebras. Presented by Robert Goldblatt     

A Note on Matrices for Systems of nonsense-logics

We construct a class K of algebras which are matrices of the logical system Z introduced in [4]. It is shown that algebras belonging to the class K are decomposable into disjoint subalgebras which are Boolean algebras.To the memory of Jerzy SupeckiTranslated From the Polish by Jan Zygmunt.     

Q-ultrafilters and normal ultrafilters in B-algebras

The first part of the paper deals with some subclasses of B-algebras and their applications to the semantics of SCI _B , the Boolean strengthening of the sentential calculus with identity (SCI). In the second part a generalization of the McKinsey-Tarski construction of well-connected topological Boolean, algebras to the class of B-algebras is given.     

Partial Boolean algebras in a broader sense

The article deals with compatible families of Boolean algebras. We define the notion of a partial Boolean algebra in a broader sense (PBA(bs)) and then we show that there is a mutual correspondence between PBA(bs) and compatible families of Boolean algebras (Theorem (1.8)). We examine in detail the interdependence between PBA(bs) and the following classes: partial Boolean algebras in the sense of Kochen and Specker (§ 2), ortholattices (§ 3, § 5), and orthomodular posets (§ 4), respectively.     

The countable homogeneous universal model of B 2

We give a detailed account of the Algebraically Closed and Existentially Closed members of the second Lee class B ₂ of distributive p-algebras, culminating in an explicit construction of the countable homogeneous universal model of B ₂. The axioms of Schmid [7], [8] for the AC and EC members of B ₂ are reduced to what we prove to be an irredundant set of axioms. The central tools used in this study are the strong duality of Clark and Davey [3] for B ₂ and the method of Clark [2] for constructing AC and EC algebras using a strong duality. Applied to B ₂, this method transfers the entire discussion into an equivalent dual category X ₂ of Boolean spaces which carry a pair of tightly interacting orderings. The doubly ordered spaces of X ₂ prove to be much more readily constructed and analyzed than the corresponding algebras in B ₂.     

Concerning axiomatizability of the quasivariety generated by a finite Heyting or topological Boolean algebra

In classes of algebras such as lattices, groups, and rings, there are finite algebras which individually generate quasivarieties which are not finitely axiomatizable (see [2], [3], [8]). We show here that this kind of algebras also exist in Heyting algebras as well as in topological Boolean algebras. Moreover, we show that the lattice join of two finitely axiomatizable quasivarieties, each generated by a finite Heyting or topological Boolean algebra, respectively, need not be finitely axiomatizable. Finally, we solve problem 4 asked in Rautenberg [10].     

Grishin Algebras and Cover Systems for Classical Bilinear Logic

Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (??propositions??) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, by a map that preserves all existing joins and meets. This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions.     

Homogeneous and Universal Dedekind Algebras

A Dedekind algebra is an order pair (B, h) where B is a non-empty set and h is a similarity transformation on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type which occur in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. It is shown that configuration signatures can be used to characterize the homogeneous, universal and homogeneous-universal Dedekind algebras. This characterization is used to prove various results about these subclasses of Dedekind algebras.     

An Admissible Semantics for Propositionally Quantified Relevant Logics

The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p-instantiations of A. It is also shown that without the admissibility qualification many of the systems considered are semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap’s system E of entailment, extended by the mingle axiom and the Ackermann constant t. The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras.     

Non-standard Stochastics with a First Order Algebraization

Internal sets and the Boolean algebras of the collection of the internal sets are of central importance in non-standard analysis. Boolean algebras are the algebraization of propositional logic while the logic applied in non-standard analysis (in non-standard stochastics) is the first order or the higher order logic (type theory). We present here a first order logic algebraization for the collection of internal sets rather than the Boolean one. Further, we define an unusual probability on this algebraization.