The Algebraic Counterpart of the Wagner Hierarchy

2008 ◽  
pp. 100-109
Author(s):  
Jérémie Cabessa ◽  
Jacques Duparc
Keyword(s):  
Algebraic Counterpart

A completeness theorem for higher order logics

2000 ◽  
Vol 65 (2) ◽  
pp. 857-884 ◽  
Author(s):  
Gábor Sági
Keyword(s):  
Set Theory ◽  
Representation Theorem ◽  
Completeness Theorem ◽  
Higher Order ◽  
Algebraic Solution ◽  
Algebraic Proof ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Strong Representation ◽  
Algebraic Counterpart

AbstractHere we investigate the classes of representable directed cylindric algebras of dimension α introduced by Németi [12]. can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, “purely cylindric algebraic” proof for the following theorems of Németi: (i) is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain a strong representation theorem for if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories.

scholarly journals On an interpretation of second order quantification in first order intuitionistic propositional logic

1992 ◽  
Vol 57 (1) ◽  
pp. 33-52 ◽  
Author(s):  
Andrew M. Pitts
Keyword(s):  
Propositional Logic ◽  
Heyting Algebra ◽  
Propositional Calculus ◽  
Interpolation Theorem ◽  
Second Order ◽  
Heyting Algebras ◽  
Intuitionistic Propositional Logic ◽  
First Order ◽  
Intuitionistic Propositional Calculus ◽  
Algebraic Counterpart

AbstractWe prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, ϕ, built up from propositional variables (p, q, r, …) and falsity (⊥) using conjunction (∧), disjunction (∨) and implication (→). Write ⊢ϕ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula ϕ there exists a formula Apϕ (effectively computable from ϕ), containing only variables not equal to p which occur in ϕ, and such that for all formulas ψ not involving p, ⊢ψ → Apϕ if and only if ⊢ψ → ϕ. Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions.An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra.

scholarly journals Logarithmic Abelian Varieties

2008 ◽  
Vol 189 ◽  
pp. 63-138 ◽  
Author(s):  
Takeshi Kajiwara ◽  
Kazuya Kato ◽  
Chikara Nakayama
Keyword(s):  
Abelian Varieties ◽  
Algebraic Theory ◽  
Analytic Theory ◽  
Previous Part ◽  
Algebraic Counterpart

AbstractWe develop the algebraic theory of log abelian varieties. This is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I ([6]), where we developed the analytic theory of log abelian varieties.

Nonfinitizability of classes of representable cylindric algebras

1969 ◽  
Vol 34 (3) ◽  
pp. 331-343 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Predicate Logic ◽  
Infinite Dimension ◽  
Fundamental Result ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Locally Finite ◽  
First Order ◽  
Alfred Tarski ◽  
Sentential Logic ◽  
Algebraic Counterpart

Cylindric algebras were introduced by Alfred Tarski about 1952 to provide an algebraic analysis of (first-order) predicate logic. With each cylindric algebra one can, in fact, associate a certain, in general infinitary, predicate logic; for locally finite cylindric algebras of infinite dimension the associated predicate logics are finitary. As with Boolean algebras and sentential logic, the algebraic counterpart of completeness is representability. Tarski proved the fundamental result that every locally finite cylindric algebra of infinite dimension is representable.

SUBHIERARCHIES OF THE SECOND LEVEL IN THE STRAUBING–THÉRIEN HIERARCHY

2011 ◽  
Vol 21 (07) ◽  
pp. 1195-1215
Author(s):  
ONDŘEJ KLÍMA ◽  
LIBOR POLÁK
Keyword(s):  
Wide Class ◽  
Nonnegative Integer ◽  
Positive Variety ◽  
Polynomial Operators ◽  
Algebraic Counterpart

In a recent paper we assigned to each positive variety [Formula: see text] and each nonnegative integer k the class of all finite unions of finite intersections or Boolean combinations of the languages of the form L0a1L1a2…aℓLℓ, where ℓ ≤ k, a1,…,aℓ are letters and L0,…,Lℓ are in the variety [Formula: see text]. For these polynomial operators on a wide class of varieties we gave a certain algebraic counterpart in terms of identities satisfied by syntactic (ordered) monoids of languages considered. Here we apply our constructions to particular examples of varieties of languages obtaining four hierarchies of (positive) varieties. Two of them have the 3/2 level of the Straubing–Thérien hierarchy as their limits, and two others tend to the level two of this hierarchy. We concentrate here on the existence of finite bases of identities for corresponding pseudovarieties of (ordered) monoids and we are looking for inclusions among those varieties.

scholarly journals MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY

2016 ◽  
Vol 4 ◽  
Author(s):  
RENZO CAVALIERI ◽  
SIMON HAMPE ◽  
HANNAH MARKWIG ◽  
DHRUV RANGANATHAN
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Tropical Geometry ◽  
Stable Curves ◽  
Tropical Curves ◽  
Graphic Matroid ◽  
Fibre Products ◽  
Algebraic Counterpart

We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$-stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

scholarly journals IRREDUCIBLE QUASIORDERS OF MONOUNARY ALGEBRAS

2012 ◽  
Vol 93 (3) ◽  
pp. 259-276 ◽  
Author(s):  
DANICA JAKUBÍKOVÁ-STUDENOVSKÁ ◽  
REINHARD PÖSCHEL ◽  
SÁNDOR RADELECZKI
Keyword(s):  
Partial Order ◽  
Unary Operation ◽  
Natural Order ◽  
Monounary Algebra ◽  
Natural Partial Order ◽  
Ordered Structure ◽  
Subdirectly Irreducible ◽  
Irreducible Elements ◽  
Algebraic Counterpart

AbstractRooted monounary algebras can be considered as an algebraic counterpart of directed rooted trees. We work towards a characterization of the lattice of compatible quasiorders by describing its join- and meet-irreducible elements. We introduce the limit $\cB _\infty $ of all $d$-dimensional Boolean cubes $\Two ^d$ as a monounary algebra; then the natural order on $\Two ^d$ is meet-irreducible. Our main result is that any completely meet-irreducible quasiorder of a rooted algebra is a homomorphic preimage of the natural partial order (or its inverse) of a suitable subalgebra of $\cB _\infty $. For a partial order, it is known that complete meet-irreducibility means that the corresponding partially ordered structure is subdirectly irreducible. For a rooted monounary algebra it is shown that this property implies that the unary operation has finitely many nontrivial kernel classes and its graph is a binary tree.

ON GROUPS OF TYPE $\mathcal{L}^{-1}$

2010 ◽  
Vol 21 (06) ◽  
pp. 727-736
Author(s):  
JANG HYUN JO
Keyword(s):  
Finite Group ◽  
Algebraic Property ◽  
Cohomological Dimension ◽  
Finite Dimensional ◽  
Free Actions ◽  
Virtual Cohomological Dimension ◽  
Algebraic Counterpart

We show that every finite group G has a set of cohomological elements satisfying ceratin algebraic property [Formula: see text] which can be regarded as a generalized notion of an algebraic counterpart to the topological phenomenon of free actions on finite dimensional homotopy spheres. We extend this result to a certain class of groups which contains groups of finite virtual cohomological dimension.

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