A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that (...) emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)

Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic (...) methods will also be used to establish the embedding of subintuitionistic logics into the corresponding modal logics. Finally, proof-theoretic embeddings will be interpreted as a reduction of classes of word problems. (shrink)

We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine (...) Medvedev’s logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces). In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open elements of this space. This connection yields a new topological semantics for inquisitive logic. (shrink)

This paper studies the topological duality between diagonalizable algebras and bi-topological spaces. In particular, the correspondence between algebraic properties of a diagonalizable algebra and topological properties of its dual space is investigated. Since the main example of a diagonalizable algebra is the Lindenbaum algebra of an r.e. theory extending Peano Arithmetic, endowed with an operator defined by means of the provability predicate of the theory, this duality gives the possibility to study arithmetical properties of theories from a topological point (...) of view. We find topological characterization of $\Sigma_1$-sound theories and of sentences that are $\Sigma_1$-conservative over such a theory. (shrink)

In this paper, we consider a collection of filters of a BL-algebra A. We use the concept of congruence relation with respect to filters to construct a uniformity which induces a topology on A. We study the properties of this topology regarding different filters.

Let M be an MV-algebra and ΩM be the set of all σ -valuations from M into the MV-unit interval. This paper focuses on the characterization of MV-algebras using σ -valuations of MV-algebras and proves that a σ -complete MV-algebra is σ -regular, which means that a ≤ b if and only if v ≤ v for any v ∈ ΩM. Then one can introduce in a natural way a fuzzy topology δ on ΩM. The representation theorem forMV-algebras is (...) established by means of fuzzy topology. Some properties of fuzzy topology δ and its cut topology U are investigated. (shrink)

We study the flow of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable f‐generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We shall give a description of f‐generic types of trigonalizable algebraic groups, and prove that every f‐generic type is almost periodic.

We consider semi‐algebraic sets and properties of these sets that are expressible by sentences in first‐order logic over the reals. We are interested in first‐order properties that are invariant under topological transformations of the ambient space. Two semi‐algebraic sets are called topologically elementarily equivalent if they cannot be distinguished by such topological first‐order sentences. So far, only semi‐algebraic sets in one and two‐dimensional space have been considered in this context. Our contribution is a natural characterisation of topological (...) elementary equivalence of regular closed semi‐algebraic sets in three‐dimensional space, extending a known characterisation for the two‐dimensional case. Our characterisation is based on the local topological behaviour of semi‐algebraic sets and the key observation that topologically elementarily equivalent sets can be transformed into each other by means of geometric transformations, each of them mapping a set to a first‐order indistinguishable one. (shrink)

We investigate topological properties of subsets S of the real plane, expressed by first-order logic sentences in the language of the reals augmented with a binary relation symbol for S. Two sets are called topologically elementary equivalent if they have the same such first-order topological properties. The contribution of this paper is a natural and effective characterization of topological elementary equivalence of closed semi-algebraic sets.

Post algebras of order + as a semantic foundation for +-valued predicate calculi were examined in [5]. In this paper Post spaces of order + being a modification of Post spaces of order n2 (cf. Traczyk [8], Dwinger [1], Rasiowa [6]) are introduced and Post fields of order + are defined. A representation theorem for Post algebras of order + as Post fields of sets is proved. Moreover necessary and sufficient conditions for the existence of representations preserving a given set (...) of infinite joins and infinite meets are established and applied to Lindenbaum-Tarski algebras of elementary theories based on +-valued predicate calculi in order to obtain a topological characterization of open theories. (shrink)

Lawvere hyperdoctrines give categorical algebraic semantics for intuitionistic predicate logic. Here we extend the hyperdoctrinal semantics to a broad variety of substructural predicate logics over the Typed Full Lambek Calculus, verifying their completeness with respect to the extended hyperdoctrinal semantics. This yields uniform hyperdoctrinal completeness results for numerous logics such as different types of relevant predicate logics and beyond, which are new results on their own; i.e., we give uniform categorical semantics for a broad variety of non-classical predicate logics. (...) And we introduce an analogue of Lawvere–Tierney topology and cotopology in the hyperdoctrinal setting, which gives a unifying perspective on different logical translations, in particular allowing for a uniform treatment of Girard’s exponential translation between linear and intuitionistic logics and of Kolmogorov’s double negation translation between intuitionistic and classical logics. In the hyerdoctrinal conception, type theories are categories, logics over type theories are functors, and logical translations between them, then, are natural transformations, in particular Lawvere–Tierney topologies and cotopologies on hyperdoctrines. The view of logical translations as hyperdoctrinal Lawvere–Tierney topologies and cotopologies has not been elucidated before, and may be seen as a novel contribution of the present work. From a broader perspective, this work may be regarded as taking first steps towards interplay between algebraic and categorical logics; it is, technically, a combination of substructural algebraic logic and hyperdoctrinal categorical logic, as the hyperdoctrinal completeness theorem is shown via the integration of the Lindenbaum–Tarski algebra construction with the syntactic category construction. As such this work lays a foundation for further interactions between algebraic and categorical logics. (shrink)

For the modal logic $\textsf {S4}^{C}_{I}$, we identify the class of completable $\textsf {S4}^{C}_{I}$-algebras and prove for them a Stone-type representation theorem. As a consequence, we obtain strong algebraic and topological completeness of the logic $\textsf {S4}^{C}_{I}$ in the case of local semantic consequence relations. In addition, we consider an extension of the logic $\textsf {S4}^{C}_{I}$ with certain infinitary derivations and establish the corresponding strong completeness results for the enriched system in the case of global semantic consequence relations.

For any subset $Z \subseteq {\mathbb {Q}}$, consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$, we show that if Z is not thin in ${\mathbb {Q}}$, then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here, thin and meager both mean (...) “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $-definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials. (shrink)

A sequent calculus S for the variety tqBa of all topological quasi-Boolean algebras is established. Using a construction of syntactic finite algebraic model, the finite model property of S is shown, and thus the decidability of S is obtained. We also introduce two non-distributive variants of topological quasi-Boolean algebras. For the variety TDM5 of all topological De Morgan lattices with the axiom 5, we establish a sequent calculus S5 and prove that the cut elimination holds for it. Consequently the (...) decidability of S5 is established. Furthermore, this proof-theoretic method is applied to some subvarieties of TDM5. (shrink)

The aim of this paper is to show that every topological space gives rise to a wealth of topological models of the modal logic S4.1. The construction of these models is based on the fact that every space defines a Boolean closure algebra (to be called a McKinsey algebra) that neatly reflects the structure of the modal system S4.1. It is shown that the class of topological models based on McKinsey algebras contains a canonical model that can be used to (...) prove a completeness theorem for S4.1. Further, it is shown that the McKinsey algebra MKX of a space X endoewed with an alpha-topologiy satisfies Esakia's GRZ axiom. (shrink)

In this work we provide a new topological representation for implication algebras in such a way that its one-point compactification is the topological space given in [1]. Some applications are given thereof.

Starting from Poincaré’s assignment of an algebraic object to a topological manifold, namely the fundamental group, this article introduces the concept of categories and their language of arrows that has, since their mid-20th-century inception, altered how large areas of mathematics, from algebra to abstract logic and computer programming, are conceptualized. The assignment of the fundamental group is an example of a functor, an arrow construction central to the notion of a category. The exposition of category theory’s arrows, which operate (...) at three distinct but deeply interconnected levels, is framed by a comparison with the language and outlook of set theory founded on the concept of membership; sets and their theorization having provided, famously through the Bourbaki initiative, the basic ontological and epistemological vocabulary for defining and handling all mathematical entities. The comparison with sets emphasizes how categories offer a form of diagrammatic argument and thought against set theory’s fidelity to syntax-based proofs; how categories invert set theory’s priority of objects and their attributes over relations by making the relations of an object to others of its kin primary; and how categories replace identity, that is, equality, between objects, by the weaker notion of isomorphism, restricting equality to identity between arrows. The article concludes with a return to topology and some remarks about the question of its possible use in articulating/characterizing cultural dynamics. (shrink)

In 2015, tense n × m-valued Lukasiewicz–Moisil algebras were introduced by A. V. Figallo and G. Pelaitay as an generalization of tense n-valued Łukasiewicz–Moisil algebras. In this paper we continue the study of tense LMn×m-algebras. More precisely, we determine a Priestley-style duality for these algebras. This duality enables us not only to describe the tense LMn×m-congruences on a tense LMn×m-algebra, but also to characterize the simple and subdirectly irreducible tense LMn×m-algebras.

In this note, we give a representation of distributive Ockham algebras via natural hom-functors. In order to do this, we describe two different structures (one algebraic, and the other order-topological) on the set of subsets of the natural numbers. The topological duality previously obtained by A. Urquhart is used throughout.

Giovanni Sambin has recently introduced the notion of an overlap algebra in order to give a constructive counterpart to a complete Boolean algebra. We propose a new notion of regular open subset within the framework of intuitionistic, predicative topology and we use it to give a representation theorem for overlap algebras. In particular we show that there exists a duality between the category of set-based overlap algebras and a particular category of topologies in which all open subsets are regular.

We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem of an ultraproduct of systems from K. In this paper we obtain a (...) similar characterization of approximability of a locally compact system A by systems from K using the language of nonstandard analysis. In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula φ holds in A if and only if all precise enough approximations of φ hold in all precise enough approximations of A. We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be non-commutative). Finite approximations of the field R can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings. (shrink)

Algebraic/topological descriptions of living processes are indispensable to the understanding of both biological and cognitive functions. This paper presents a fundamental algebraic description of living/cognitive processes and exposes its inherent ambiguity. Since ambiguity is forbidden to computation, no computational description can lend insight to inherently ambiguous processes. The impredicativity of these models is not a flaw, but is, rather, their strength. It enables us to reason with ambiguous mathematical representations of ambiguous natural processes. The noncomputability of these structures (...) means computerized simulacra of them are uninformative of their key properties. This leads to the question of how we should reason about them. That question is answered in this paper by presenting an example of such reasoning, the demonstration of a topological strategy for understanding how the fundamental structure can form itself from within itself. (shrink)

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]. (shrink)

Using nonstandard methods, we generalize the notion of an algebraic primitive element to that of an hyperalgebraic primitive element, and show that under mild restrictions, such elements can be found infinitesimally close to any given element of a topological field.

A Hilbert algebra with supremum is a Hilbert algebra where the associated order is a join-semilattice. This class of algebras is a variety and was studied in Celani and Montangie . In this paper we shall introduce and study the variety of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras, which are Hilbert algebras with supremum endowed with a modal operator $${\Diamond}$$ ◊ . We give a topological representation for these algebras using the topological spectral-like representation for Hilbert algebras with supremum given in (...) Celani and Montangie . We will consider some particular varieties of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras. These varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic $${\mathbf{IntK}_{\Diamond}}$$ IntK ◊ . We also determine the congruences of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras in terms of certain closed subsets of the associated space, and in terms of a particular class of deductive systems. These results enable us to characterize the simple and subdirectly irreducible $${H_{\Diamond}^{\vee }}$$ H ◊ ∨ -algebras. (shrink)

We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.

Homologous operational localization processes are effectuated in terms of generalized topological covering systems on structures of physical events. We study localization systems of quantum events' structures by means of Gtothendieck topologies on the base category of Boolean events' algebras. We show that a quantum events algebra is represented by means of a Grothendieck sheaf-theoretic fibred structure, with respect to the global partial order of quantum events' fibres over the base category of local Boolean frames.

As is well known, any preorder R on a set U induces an Alexandrov topology on U. In some interesting cases related to data mining an Alexandrov topology can be transformed into different types of logico-algebraic models. In some cases, (pre)topological operators provided by Pointless Topology may define a topological space on U even if R is not a preorder. If this is the case, then we call R a crypto-preorder. The paper studies the conditions under (...) which a relation R is a crypto-preorder and how to transform a crypto-preorder into a proper preorder. (shrink)

We prove a topological completeness theorem for the modal logic $$\textsf{GLP}$$ GLP containing operators $$\{\langle \xi \rangle :\xi \in \textsf{Ord}\}$$ { ⟨ ξ ⟩ : ξ ∈ Ord } intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence $$\phi $$ ϕ consistent with $$\textsf{GLP}$$ GLP can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the (...) finitely many modalities that occur in $$\phi $$ ϕ. (shrink)

A sequence x=xn:nω of elements of a complete Boolean algebra converges to a priori if lim infx=lim supx=b. The sequential topology τs on is the maximal topology on such that x→b implies x→τsb, where →τs denotes the convergence in the space — the a posteriori convergence. These two forms of convergence, as well as the properties of the sequential topology related to forcing, are investigated. So, the a posteriori convergence is described in terms of killing of tall (...) ideals on ω, and it is shown that the a posteriori convergence is equivalent to the a priori convergence iff forcing by does not produce new reals. A property of c.B.a.’s, satisfying -cc -cc and providing an explicit definition of the a posteriori convergence, is isolated. Finally, it is shown that, for an arbitrary c.B.a. , the space is sequentially compact iff the algebra has the property and does not produce independent reals by forcing, and that implies P is the unique sequentially compact c.B.a. in the class of Suslin forcing notions. (shrink)

Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not expressible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In fact, many (...) formal topologies can be presented in a predicative way by an inductive generation and thus their properties can be proved inductively. We show however that some natural complete Heyting algebra cannot be inductively defined. (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)

We associate a canonical unital involutive quantale to a topological groupoid. When the groupoid is also étale, this association is compatible with but independent from the theory of localic étale groupoids and their quantales [9] of P. Resende. As a motivating example, we describe the connection between the quantale and the C*-algebra that both classify Penrose tilings, which was left as an open problem in [5].

At the turn of the 21st century, topology, the mathematical study of spatial properties that remain the same under the continuous deformation of objects, has come to invest all fields of aesthetics and culture. In particular, the algebraic topology of continuity has added to the digital realm of binary information, the on and off states of 0s and 1s, an invariant property, which now governs the relation between different forms of data. As this invariant function of continual (...) transformation has entered the field of automated computation, the culture of binary digits has shifted towards a new level of calculation derived from the introduction of temporal quantities into finite sets of algorithmic instructions and parameters. This new level of topological computation, it will be argued, defines new operative procedures of control, constantly adding axioms at the limit of calculation through an invariant function that establishes a smooth or uninterrupted connectivity between distinct data. The establishment of a continual function between distinct forms of data is based on homeomorphism or topological isomorphism between data objects, of which parametricism, as the new global style for architecture and design, is a perfect example. (shrink)