We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ω defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with ∑ 1 b -induction over PV ω and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, (...) we apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion. (shrink)

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 ∃ 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. We show that FMBA is characterised by the disjunction of the equations ∃E = 1 and ∃E = 0. We also define a (...) weaker notion of "relatively functional" algebra, and show that every member of MBA is isomorphic to a relatively functional one. (shrink)

Let A be an algebra. We say that the functions f 1 , . . . , f m : A n → A are algebraic on A provided there is a finite system of term-equalities $${{\bigwedge t_{k}(\overline{x}, \overline{z}) = s_{k}(\overline{x}, \overline{z})}}$$ satisfying that for each $${{\overline{a} \in A^{n}}}$$, the m -tuple $${{(f_{1}(\overline{a}), \ldots , f_{m}(\overline{a}))}}$$ is the unique solution in A m to the system $${{\bigwedge t_{k}(\overline{a}, \overline{z}) = s_{k}(\overline{a}, \overline{z})}}$$. In this work we present a collection of general (...) tools for the study of algebraic functions, and apply them to obtain characterizations for algebraic functions on distributive lattices, Stone algebras, finite abelian groups and vector spaces, among other well known algebraic structures. (shrink)

A function is algebraic on an algebra if it can be implicitly defined by a system of equations on. In this note we give a semantic characterization for algebraic functions on quasiprimal algebras. This characterization is applied to obtain necessary and sufficient conditions for a quasiprimal algebra to have every one of its algebraic functions be a term function. We also apply our results to particular algebras such as finite fields and monadic algebras.

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between (...) our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis. (shrink)

This paper presents a generalization of a proposal of van Benthem’s who has shown how to provide a canonical name for any object in propositional type theory. Van Benthem’s idea is to characterize any function in the hierarchy by the Boolean values the function takes for any sequence of arguments. The recursive definition of canonical names uses only the abstraction, functional application, the identity operator and the fact that we have a name for the true and the false. (...) We show that this result can be extended to the finite theory of types by the introduction of an algebra on individuals. (shrink)

Consider any subvariety of BL-algebras generated by a single BL-chain which is the ordinal sum of the standard MV-algebra on [0, 1] and a basic hoop H. We present a geometrical characterization of elements in the finitely generated free algebra of each of these subvarieties. In this characterization there is a clear insight of the role of the regular and dense elements of the generating chain. As an application, we analyze maximal and prime filters in the free algebra.

We employ the notions of "sequential function" and "interrogation" (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using Longley's preorder-enriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results is that every realizability topos is a geometric quotient of a realizability topos on a total combinatory algebra.

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf = κ+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra equation image such that equation image, equation image for every α < η and equation image. Especially, equation image and equation image can (...) be cardinal sequences of superatomic Boolean algebras. (shrink)

The Uniformly Reflexive Structure was introduced by E. G. Wagner who showed that the theory of such structures generalized much of recursive function theory. In this paper Uniformly Reflexive Structures are constructed as factor algebras of Free nonassociative algebras. Wagner's question about the existence of a model with no computable splinter ("successor set") is answered in the affirmative by the construction of a model whose only computable sets are the finite sets and their complements. Finally, for each (...) countable Boolean algebra R of subsets of a countable set which contains the finite subsets, a model is constructed with R as its family of computable sets. (shrink)

We study ranges of algebraic functions in lattices and in algebras, such as Łukasiewicz-Moisil algebras which are obtained by extending standard lattice signatures with unary operations.We characterize algebraic functions in such lattices having intervals as their ranges and we show that in Artinian or Noetherian lattices the requirement that every algebraic function has an interval as its range implies the distributivity of the lattice.

We study ranges of algebraic functions in lattices and in algebras, such as Łukasiewicz-Moisil algebras which are obtained by extending standard lattice signatures with unary operations.We characterize algebraic functions in such lattices having intervals as their ranges and we show that in Artinian or Noetherian lattices the requirement that every algebraic function has an interval as its range implies the distributivity of the lattice.

We present a model of computation for string functions over single-sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum-Shub-Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational complexity in a BSS-like (...) manner. Relationships both to classical computability and to Friedman's concept of eds computability are established. Two kinds of nondeterminism as well as several variants of recognizability are investigated with respect to interdependencies on each other and on properties of the underlying structures. For structures of finite signatures, there are universal programs with the usual characteristics. For the general case of not necessarily finite signature, this subject will be studied in a separate, forthcoming paper. (shrink)

Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is (...) meromorphic on Kn is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting. (shrink)

In this paper we introduce the concept of d-fuzzy function which generalizes the concept of fuzzy subalgebra to a much larger class of functions in a natural way. In addition we discuss a method of fuzzification of a wide class of algebraic systems onto [0, 1] along with some consequences.

Explicit description of maps definable by formulæ of the second order intuitionistic propositional calculus is given on two classes of linear Heyting algebras—the dense ones and the ones which possess successors. As a consequence, it is shown that over these classes every formula is equivalent to a quantifier free formula in the dense case, and to a formula with quantifiers confined to the applications of the successor in the second case.

We can equivalently present this by the recursion equations f1(n) = 2n, fk+1(1) = fk(1), fk+1(n+1) = fk(fk+1(n)), where k,n ≥ 1. We define A(k,n) = fk(n).

We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the (...) representation of MBA's as powerset algebras of certain directed graphs with a set of "marked" points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly $2^{3.2^{r.} } ^{2^{{\rm{2}}^{{\rm{r - 1}}} } } $ elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r. (shrink)

The invention of symbolic algebra in the sixteenth and seventeenth centuries fundamentally changed the way we do mathematics. If we want to understand this change and appreciate its importance, we must analyze it on two levels. One concerns the compositional function of algebraic symbols as tools for representing complexity; the other concerns the referential function of algebraic symbols, which enables their use as tools for describing objects (such as polynomials), properties (such as irreducibility), relations (such as divisibility), and (...) operations (such as factorization). The reconstruction of both the compositional function and the referential function of algebraic symbols requires the use of different analytic tools and the taking of different temporal perspectives. In this chapter, we offer both: a reconstruction of the compositional function of algebraic symbols, and a reconstruction of their referential function. (shrink)

Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$. The (...) main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras. (shrink)

In this paper we consider the algebra of differential operators with polynomial coefficients acting on Clifford algebra-valued functions from both sides. We characterize the subalgebra of SO(m)-invariant differential operators, which itself contains the subalgebra of GL(m)-invariant differential operators.

A complete classification of simple function germs with respect to Lipschitz equivalence over the field of complex numbers ℂ was given by Nguyen et al. The aim of this article is to implement a classifier in terms of easy computable invariants to compute the type of the Lipschitz simple function germs without computing the normal form in the computer algebra system Singular.

This paper is devoted to the algebraization of theories in which, as in Peano arithmetic, there is a formula, Theor(x), numerating the set of theorems, and satisfying Hilbert-Bernays derivability conditions. In particular, we study the diagonalizable algebras, which are been introduced by R. Magari in [6], [7]. We prove that for every natural number n, the n-freely generated algebra $\germ{J}_{n}$ is not functionally free in the equational class of diagonalizable algebras; we also prove that the diagonalizable algebra of (...) Peano arithmetic is not an element of the equational class generated by $\{\germ{J}_{n}\}$. (shrink)

According to the algebraic approach to spacetime, a thoroughgoing dynamicism, physical fields exist without an underlying manifold. This view is usually implemented by postulating an algebraic structure (e.g., commutative ring) of scalar-valued functions, which can be interpreted as representing a scalar field, and deriving other structures from it. In this work, we point out that this leads to the unjustified primacy of an undetermined scalar field. Instead, we propose to consider algebraic structures in which all (and only) physical fields are (...) primitive. We explain how the theory of natural operations in differential geometry—the modern formalism behind classifying diffeomorphism-invariant constructions—can be used to obtain concrete implementations of this idea for any given collection of fields. For concrete examples, we illustrate how our approach applies to a number of particular physical fields, including electrodynamics coupled to a Weyl spinor. (shrink)

Let R be a real closed field, and X⊆Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\subseteq R^m}$$\end{document} semi-algebraic and 1-dimensional. We consider complete first-order theories of modules over the ring of continuous semi-algebraic functions X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\to R}$$\end{document} definable with parameters in R. As a tool we introduce -piecewise vector bundles on X and show that the category of piecewise vector bundles on X is equivalent to the category of syzygies of (...) finitely generated submodules of free modules. We give an explicit method to determine the Baur–Monk invariants of free modules in terms of pre-piecewise vector bundles. When R is a recursive real closed field this yields the decidability of the theory of free modules. Where it makes sense, we address the same questions for continuous definable functions in o-minimal expansions of a real closed field. From the free module case we are able to deduce generalisations of some results to arbitrary modules over the ring. We present a geometrically motivated quantifier elimination result down to the level of positive primitive formulae with a certain block decomposition of the matrix of coefficients. (shrink)

We introduce Pura Vida Neutrosophic Algebra, an algebraic structure consisting of neutrosophic numbers equipped with two binary operations namely addition and multiplication. The addition can be calculated sometimes with the function min and other times with the max function. The multiplication operation is the usual sum between numbers. Pura Vida Neutrosophic Algebra is an extension of both Tropical Algebra (also known as Min-Plus, or Min-Algebra) and Max-Plus Algebra (also known as Max-algebra). Tropical and Max-Plus algebras are algebraic (...) structures included in semirings and their operations can be used in matrices and vectors. Pura Vida Neutrosophic Algebra is included in Neutrosophic semirings and can be used in Neutrosophic matrices and vectors. (shrink)

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists! \mathop{\boldsymbol {\bigwedge }}\limits p = q$. For a logic L algebraized by a quasivariety $\mathcal {Q}$ we show that the AE-subclasses of $\mathcal {Q}$ correspond to certain natural expansions of L, which we call algebraic expansions. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of (...) abelian $\ell $ -groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics. (shrink)