In this paper, the Boolean representation of R0-algebras are investigated. In particular, we show that directly indecomposable R0-algebras are equivalent to local R0-algebras and any nontrivial R0-algebra is representable as a weak Boolean product of local R0-algebras.

We present some results concerning elimination of quantifiers and elementary equivalence for Boolean products of real closed valuation rings and fields. We also study rings of continuous functions and rings of definable functions over real closed valuation rings under this point of view.

We characterize the elementary classes generated from a distinguished subclass closing by taking direct products and elementary equivalence. In the second part we give the same characterization in terms of atomic Boolean products. In the last part, we study the cases when the class of Boolean products is elementary but is not given by a discriminator.

We show that the class of all isomorphic images of Boolean Products of members of SR [1] is the class of all archimedean W-algebras. We obtain this result from the characterization of W-algebras which are isomorphic images of Boolean Products of CW-algebras.

A Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP generated by a universal classKof finitely subdirectly irreducible algebras such that Γa has the Fraser-Horn property. If ⟦a≠b⟧ ∩ ⟦c≠d⟧ = ∅ is definable inKandKhas a model companion ofK-simple algebras, then it is shown that ISP has a model companion. Conversely, a sufficient condition is given for ISP to have no model companion.

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLw-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and (...) that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras. (shrink)

We continue the investigation, initiated in Salibra et al. (Found Sci, 2020), of Boolean-like algebras of dimension n (nBA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document}s), algebras having n constants e1,⋯,en\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf e_1,\dots,\mathsf e_n$$\end{document}, and an (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of nBA\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document}s share many remarkable properties with the variety of Boolean algebras and with primal varieties. Putting to good use the concept of a central element, we extend the Boolean power construction to that of a semiring power and we prove two representation theorems: (i) Any pure nBA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document} is isomorphic to the algebra of n-central elements of a Boolean vector space; (ii) Any member of a variety of nBAs with one generator is isomorphic to a Boolean power of this generator. This yields a new proof of Foster’s theorem on primal varieties. (shrink)

Let ${\mathbb{BRL}}$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety ${\mathbb{V}}$ of ${\mathbb{BRL}}$ is a unary term t in the language of bounded residuated lattices such that for every ${{\bf A} \in \mathbb{V}, t^{A}}$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense (...) that there is a variety ${\mathbb{V}_{t} \subsetneq \mathbb{BRL}}$ such that a variety ${\mathbb{V} \subsetneq \mathbb{BRL}}$ admits the unary term t as a Boolean retraction term if and only if ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ . Moreover, the equation s(x) = t(x) holds in ${\mathbb{V}_{s} \cap \mathbb{V}_{t}}$ . The radical of ${{\bf A} \in \mathbb{BRL}}$ , with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ is associated a variety ${\mathbb{V}^{r}}$ formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in ${\mathbb{V}}$ . Each free algebra in such ${\mathbb{V}}$ is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety ${\mathbb{V}^{r}}$ , also with the same number of free generators.A hierarchy of subvarieties of ${\mathbb{BRL}}$ admitting Boolean retraction terms is exhibited. (shrink)

Let Γ be Mundici’s functor from the category $${\mathcal{LG}}$$ whose objects are the lattice-ordered abelian groups ( ℓ -groups for short) with a distinguished strong order unit and the morphisms are the unital homomorphisms, onto the category $${\mathcal{MV}}$$ of MV-algebras and homomorphisms. It is shown that for each strong order unit u of an ℓ -group G , the Boolean skeleton of the MV-algebra Γ ( G , u ) is isomorphic to the Boolean algebra of factor congruences (...) of G. (shrink)

In this paper we provide a categorical equivalence for the category \ of product algebras, with morphisms the homomorphisms. The equivalence is shown with respect to a category whose objects are triplets consisting of a Boolean algebra B, a cancellative hoop C and a map \ from B × C into C satisfying suitable properties. To every product algebra P, the equivalence associates the triplet consisting of the maximum boolean subalgebra B, the maximum cancellative subhoop C, (...) of P, and the restriction of the join operation to B × C. Although several equivalences are known for special subcategories of \ , up to our knowledge, this is the first equivalence theorem which involves the whole category of product algebras. The syntactic counterpart of this equivalence is a syntactic reduction of classical logic CL and of cancellative hoop logic CHL to product logic, and viceversa. (shrink)

We construct Boolean algebras with prescribed behaviour concerning depth for the free product of two Boolean algebras over a third, in ZFC using pcf; assuming squares we get results on ultraproducts. We also deal with the family of cardinalities and topological density of homomorphic images of Boolean algebras (you can translate it to topology - on the cardinalities of closed subspaces); and lastly we deal with inequalities between cardinal invariants, mainly.

The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion.Some of the main results of the paper are:• (1) If there is a maximal (...) family of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal family of size λ of pairwise almost disjoint subsets of κ+ each of size κ.• (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the factors; and a similar characterization for weak products.• (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all cardinalities of partitions of 1 of that BA.• (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras. (shrink)

The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal κ > א1, the principle □ is equivalent to the existence of a certain strong coloring c : [κ]2 → κ for which the family of fibers T is a nonspecial κ-Aronszajn tree. The theorem follows from an analysis (...) of a new characteristic function for walks on ordinals, and implies in particular that if the κ-chain condition is productive for a given regular cardinal κ > א1, then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ-chain condition is productive. (shrink)

In this paper we present an equivalence between the category of commutative regular rings and the category of Boolean-valued fields, i.e., Boolean-valued sets for which the field axioms are true. The author used this equivalence in [12] to develop a Galois theory for commutative regular rings. Here we apply the equivalence to give an alternative construction of an algebraic closure for any commutative regular ring.Boolean-valued sets were developed in 1965 by Scott and Solovay [10] to simplify independence (...) proofs in set theory. They later were applied by Takeuti [13] to obtain results on Hilbert and Banach spaces. Ellentuck [3] and Weispfenning [14] considered Boolean-valued rings which consisted of rings and associated Boolean-valued relations. To the author's knowledge, the present work is the first to employ the Boolean-valued sets of Scott and Solovay to obtain results in algebra.The idea that commutative regular rings can be studied by examining the properties of related fields is not new. For several years algebraists and logicians have investigated commutative regular rings by representing a commutative regular ring as a subdirect product of fields or as the ring of global sections of a sheaf of fields over a Boolean space. These representations depend, as does the work presented here, on the fact that the set of central idempotents of any ring with identity forms a Boolean algebra. The advantage of the Boolean-valued set approach is that the axioms of classical logic and set theory are true in the Boolean universe. Therefore, if the axioms for a field are true for a Boolean-valued set, then other properties of the set can be deduced immediately from field theory. (shrink)

Heidorf, L., Moderate families in Boolean algebras, Annals of Pure and Applied Logic 57 217–250. A subset F of a Boolean algebra B will be called moderate if no element of B splits infinitely many elements of F . Disjoint moderate sets occur in connection with a product construction that is systematically studied in this paper. In contrast to the usual full direct product, these so-called moderate products preserve many properties of their factors. This can be (...) used, for example, to construct big retractive Boolean algebras that are not embeddable into interval algebras. The paper is also concerned with those Boolean algebras whose ideals are generated by moderate sets. The results may be summarized by saying that in many respects these algebras behave like countable ones. (shrink)

In the book Cardinal Invariants on Boolean Algebras by J. Donald Monk many such cardinal functions are defined and studied. Among them several are generalizations of well known cardinal characteristics of the continuum. Alongside a long list of open problems is given. Focusing on half a dozen of those cardinal invariants some of those problems are given an answer here, which in most of the cases is a definitive one. Most of them can be divided in two groups. The (...) problems of the first group ask about the change on those cardinal functions when going from a given infinite Boolean algebra to its simple extensions, while in the second group the comparison is between a couple of given infinite Boolean algebras and their free product. (shrink)

We prove that colouring of pairs from 2 with strong properties exists. The easiest to state problem it solves is: there are two topological spaces with cellularity 1 whose product has cellularity 2; equivalently, we can speak of cellularity of Boolean algebras or of Boolean algebras satisfying the 2-c.c. whose product fails the 2-c.c. We also deal more with guessing of clubs.

Antifragility is a property from which systems are able to resist stress and furthermore benefit from it. Even though antifragile dynamics is found in various real-world complex systems where multiple subsystems interact with each other, the attribute has not been quantitatively explored yet in those complex systems which can be regarded as multilayer networks. Here we study how the multilayer structure affects the antifragility of the whole system. By comparing single-layer and multilayer Boolean networks based on our recently proposed (...) antifragility measure, we found that the multilayer structure facilitated the production of antifragile systems. Our measure and findings will be useful for various applications such as exploring properties of biological systems with multilayer structures and creating more antifragile engineered systems. (shrink)

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We (...) include examples to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

In an earlier paper the present author proved that de Finetti coherence is preserved under taking products of coherent books on two finite sets of independent events. Conversely, in this note it is proved that product is the only coherence preserving operation on coherent books. Our proof shows that the traditional definition of stochastically independent classes of events actually follows from the combination of two more basic notions: boolean algebraic independence and de Finetti coherent betting system.

An explicit categorical equivalence is defined between a proper subvariety of the class of \-algebras, as defined by Di Nola and Dvurečenskij, to be called \-algebras, and the category of semi-low \-rings. This categorical representation is done using the prime spectrum of the \-algebras, through the equivalence between \-algebras and \-groups established by Mundici, from the perspective of the Dubuc–Poveda approach, that extends the construction defined by Chang on chains. As a particular case, semi-low \-rings associated to Boolean algebras (...) are characterized. (shrink)

A Heyting algebra is supplemented if each element a has a dual pseudo-complement \, and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigate a new type of completion of Heyting algebras arising in the context of algebraic proof theory, the so-called hyper-MacNeille completion. We show that the (...) hyper-MacNeille completion of a Heyting algebra is the MacNeille completion of its centrally supplemented extension. This provides an algebraic description of the hyper-MacNeille completion of a Heyting algebra, allows development of further properties of the hyper-MacNeille completion, and provides new examples of varieties of Heyting algebras that are closed under hyper-MacNeille completions. In particular, connections between the centrally supplemented extension and Boolean products allow us to show that any finitely generated variety of Heyting algebras is closed under hyper-MacNeille completions. (shrink)

In the present paper we give a description of the free algebra over an arbitrary set of generators in the variety of nilpotent minimum algebras. Such description is given in terms of a weak Boolean product of directly indecomposable algebras over the Boolean space corresponding to the Boolean subalgebra of the free NM-algebra.

Lattice-ordered abelian groups, or abelian$$\ell $$ ℓ -groups in what follows, are categorically equivalent to two classes of 0-bounded hoops that are relevant in the realm of the equivalent algebraic semantics of many-valued logics: liftings of cancellative hoops and perfect MV-algebras. The former generate the variety of product algebras, and the latter the subvariety of MV-algebras generated by perfect MV-algebras, that we shall call $$\textsf{DLMV}$$ DLMV. In this work we focus on these two varieties and their relation to the (...) structures obtained by forgetting the falsum constant 0, i.e., product hoops and DLW-hoops. As main results, we first show a characterization of the free algebras in these two varieties as particular weak Boolean products; then, we show a construction that freely generates a product algebra from a product hoop and a DLMV-algebra from a DLW-hoop. In other words, we exhibit the free functor from the two algebraic categories of hoops to the corresponding categories of 0-bounded algebras. Finally, we use the results obtained to study projective algebras and unification problems in the two varieties (and the corresponding logics); both varieties are shown to have (strong) unitary unification type, and as a consequence they are structurally and universally complete. (shrink)

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of (...) $\mathsf {CH}$ ); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse. (shrink)

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems (...) dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces. (shrink)

We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them and introduce a new characteristic (...) of relations, ternarity, that measures their ‘complexity of relating’ and allows to refine reduction results. In particular, we refine Peirce’s controversial reduction thesis, and show that reducibility behaviour is dramatically different on finite and infinite domains. (shrink)

We present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutatively and associatively are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, (...) similar to Curry's subtraction operator, which is resituated with Linear Logic's contensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic. (shrink)

We prove that de Finetti coherence is preserved under taking products of coherent books on two sets of independent events. This establishes a desirable closure property of coherence: were it not the case it would raise a question mark over the utility of de Finetti’s notion of coherence.

In this paper we study and equationally characterize the subvarieties of BL, the variety of BL-algebras, which are generated by families of single-component BL-chains, i.e. MV-chains, Product-chain or Gödel-chains. Moreover, it is proved that they form a segment of the lattice of subvarieties of BL which is bounded by the Boolean variety and the variety generated by all single-component chains, called ŁΠG.

Mereotopology is an extension of mereology with some relations of topological nature like contact. An algebraic counterpart of mereotopology is the notion of contact algebra which is a Boolean algebra whose elements are considered to denote spatial regions, extended with a binary relation of contact between regions. Although the language of contact algebra is quite expressive to define many useful mereological relations and mereotopological relations, there are, however, some interesting mereotopological relations which are not definable in it. Such are, (...) for instance, the relation of n-ary contact, internal connectedness and some others. To overcome this disadvantage, we introduce a generalisation of contact algebra, replacing the contact with a binary relation between finite sets of regions and a region, satisfying some formal properties of Tarski consequence relation. The obtained system is called sequent algebra, considered as an algebraic counterpart of a new mereotopology. We develop the topological representation theory for sequent algebras showing in this way certain correspondence between point-free and point-based models of space. As a by-product, we show how one logical relation in nature notion, Tarski consequence relation, may have also certain spatial meaning. (shrink)

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing (...) the Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor generating p-points which are k-arrow but not \-arrow, and in a partial order of Blass producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \\). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. (shrink)

Using the Feferman-Vaught Theorem, we prove that a definable subset of a product structure must be a Boolean combination of open sets, in the product topology induced by giving each factor structure the discrete topology. We prove that for families of structures with certain properties, including families of integral domains, the pure Boolean generalized product is definable in the direct product structure. We use these results to obtain characterizations of the definable subsets of $$\prod (...) _p \mathbb {F}_p$$ —in particular, every formula is equivalent to a Boolean combination of $$\exists \forall \exists $$ formulae. (shrink)

Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to (...) computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (shrink)

Given a variety we study the existence of a class such that S1 every A can be represented as a global subdirect product with factors in and S2 every non-trivial A is globally indecomposable. We show that the following varieties (and its subvarieties) have a class satisfying properties S1 and S2: p-algebras, distributive double p-algebras of a finite range, semisimple varieties of lattice expansions such that the simple members form a universal class (bounded distributive lattices, De Morgan algebras, etc) (...) and arithmetical varieties in which the finitely subdirectly irreducible algebras form a universal class (f-rings, vector groups, Wajsberg algebras, discriminator varieties, Heyting algebras, etc). As an application we obtain results analogous to that of Nachbin saying that if every chain of prime filters of a bounded distributive lattice has at most length 1, then the lattice is Boolean. (shrink)

Dynamic algebras combine the classes of Boolean (B 0) and regular (R ; *) algebras into a single finitely axiomatized variety (B R ) resembling an R-module with scalar multiplication . The basic result is that * is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main (...) result is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of prepositional dynamic logic, and yet another notion of regular algebra. (shrink)

Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define the functor K • relating integral residuated lattices with 0 with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras. The lifting of the functor to the category of residuated lattices leads us to study (...) other adjunctions and equivalences. For example, we treat the functor C whose domain is cuRL, the category of involutive residuated lattices M whose unit is fixed by the involution and has a Boolean complement c. If we restrict to the full subcategory NRL of cuRL of those objects that have a nilpotent c, then C is an equivalence. In fact, C M is isomorphic to C e M, and C e is adjoint to, where assigns to an object A of IRL0 the product A × A 0 which is an object of NRL. (shrink)

The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to (...) provide a metamathematical interpretation of ideas sometimes considered disparate, heuristic, or simply ill-defined: the collapse of the wave function, for example; Everett's many worlds'-construal of quantum measurement; and a natural product space of contextual (nonlocal) hidden variables. (shrink)

The study of existentially closed closure algebras begins with Lipparini’s 1982 paper. After presenting new nonelementary axioms for algebraically closed and existentially closed closure algebras and showing that these nonelementary classes are different, this paper shows that the classes of finitely generic and infinitely generic closure algebras are closed under finite products and bounded Boolean powers, extends part of Hausdorff’s theory of reducible sets to existentially closed closure algebras, and shows that finitely generic and infinitely generic closure algebras are (...) elementarily inequivalent. Special properties of algebraically closed, existentially closed, finitely generic, and infinitely generic closure algebras are established along the way. (shrink)

A symmetric residuated lattice is an algebra such that is a commutative integral bounded residuated lattice and the equations x=x and =xy are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription εx=x→0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive →0)=1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider the (...) chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n=1, we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature. (shrink)

The purpose of this paper is to examine how any domain of Meinongian objects can be structured by a special kind of mereology. The basic definition of this mereology is the following: an object is part of another iff every characteristic property of the former is also a characteristic property of the latter. I will show that this kind of mereology ends up being very powerful for dealing with Meinongian objects. Mereological sums and products are not restricted in any way (...) in a domain of Meinongian objects: there is a sum and a product for any pair of Meinongian objects. With the mereological operations of sum, product and complement, and two special Meinongian objects, we can define a full boolean algebra on Meinongian objects. Moreover, this kind of mereology is atomic and extensional: an atom is a Meinongian object having just one characteristic property and two objects are identical iff the same atoms are parts of both of them. A Meinongian object can finally be defined in mereological terms as the sum of the atoms of its characteristic properties. (shrink)

Many results concerning the equivalence between a syntactic form of formulas and a model theoretic conditions are proven directly without using any form of a continuum hypothesis. In particular, it is demonstrated that any reduced product sentence is equivalent to a Horn sentence. Moreover, in any first order language without equality one now has that a reduced product sentence is equivalent to a Horn sentence and any sentence is equivalent to a Boolean combination of Horn sentences.