A small infinitely-ended 2-knot group

AbstractThe motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ⊑ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ⊔ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) of abstract $$(\le , \circ )$$ ( ≤ , ∘ ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( ≤ , ∘ ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( ≤ , ∘ ) structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representation property holds for finite structures.

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A correspondence between a class of monoids and self-similar group actions II

International Journal of Algebra and Computation ◽

10.1142/s0218196715500137 ◽

2015 ◽

Vol 25 (04) ◽

pp. 633-668

Author(s):

Mark V. Lawson ◽

Alistair R. Wallis

Keyword(s):

Group Actions ◽

Similar Group ◽

Universal Group ◽

Length Functions ◽

Group Of Units ◽

Hnn Extensions ◽

Hnn Extension ◽

Self Similar ◽

Cancellative Monoids

The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa–Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych. In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be significant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.

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Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2018.0761 ◽

2019 ◽

Vol 475 (2223) ◽

pp. 20180761 ◽

Cited By ~ 1

Author(s):

Matteo Petrera ◽

Jennifer Smirin ◽

Yuri B. Suris

Keyword(s):

Vector Field ◽

Base Point ◽

Hamiltonian Vector ◽

Geometric Condition ◽

Hamiltonian Vector Field ◽

Cubic Curves ◽

Birational Map ◽

Base Points ◽

Finite Base ◽

Pass Through

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic canonical Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic canonical Hamiltonian vector field.

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On the profinite topology on solvable groups

Journal of Group Theory ◽

10.1515/jgth-2016-0048 ◽

2017 ◽

Vol 20 (4) ◽

Author(s):

Khadijeh Alibabaei

Keyword(s):

Abelian Group ◽

Wreath Product ◽

Solvable Group ◽

Solvable Groups ◽

Polycyclic Group ◽

Finitely Generated ◽

Metabelian Groups ◽

Finitely Generated Group ◽

Hnn Extension ◽

Profinite Topology

AbstractWe show that the wreath product of a finitely generated abelian group with a polycyclic group is a LERF group. This theorem yields as a corollary that finitely generated free metabelian groups are LERF, a result due to Coulbois. We also show that a free solvable group of class 3 and rank at least 2 does not contain a strictly ascending HNN-extension of a finitely generated group. Since such groups are known not to be LERF, this settles, in the negative, a question of J. O. Button.

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Equational properties of fixed-point operations in cartesian categories: An overview

Mathematical Structures in Computer Science ◽

10.1017/s0960129518000361 ◽

2019 ◽

Vol 29 (06) ◽

pp. 909-925

Author(s):

Z Ésik

Keyword(s):

Fixed Point ◽

Equational Theory ◽

Continuous Functions ◽

Equivalence Classes ◽

Finitely Based ◽

Additive Structure ◽

Complete Lattices ◽

Finite Base ◽

Language Equivalence ◽

Context Free

AbstractSeveral fixed-point models share the equational properties of iteration theories, or iteration categories, which are cartesian categories equipped with a fixed point or dagger operation subject to certain axioms. After discussing some of the basic models, we provide equational bases for iteration categories and offer an analysis of the axioms. Although iteration categories have no finite base for their identities, there exist finitely based implicational theories that capture their equational theory. We exhibit several such systems. Then we enrich iteration categories with an additive structure and exhibit interesting cases where the interaction between the iteration category structure and the additive structure can be captured by a finite number of identities. This includes the iteration category of monotonic or continuous functions over complete lattices equipped with the least fixed-point operation and the binary supremum operation as addition, the categories of simulation, bisimulation, or language equivalence classes of processes, context-free languages, and others. Finally, we exhibit a finite equational system involving residuals, which is sound and complete for monotonic or continuous functions over complete lattices in the sense that it proves all of their identities involving the operations and constants of cartesian categories, the least fixed-point operation and binary supremum, but not involving residuals.

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The Infinite Commitment of Finite Minds

Canadian Journal of Philosophy ◽

10.1080/00455091.1984.10716380 ◽

1984 ◽

Vol 14 (2) ◽

pp. 235-255 ◽

Cited By ~ 1

Author(s):

Timothy Williamson

Keyword(s):

Finite Axiomatization ◽

Semantic Constraint ◽

Theory Of Meaning ◽

Finite Base ◽

Finite Minds

Chomsky (in syntax) and Davidson (in semantics) have made much of the constraint that speakers’ competence must have a finite base. This base is often supposed to mean a finite axiomatization of beliefs. Section I shows why this is plausible. Section II shows why it is wrong. Section III shows why the semantic constraint is thereby trivialized.'Finite minds cannot have infinitely many beliefs’ has been taken for a useful truism. A theory of meaning, say, for a language may, for each of its infinitely many sentences, attribute to competent speakers knowledge, in some sense, and so in a corresponding sense belief, about the meaning (or some substitute for it) of that sentence. Our truism seems to force this paradox to immediate and constructive resolution: as finite minds we have a finite view on our language that recognizably entails an infinity of propositions about it.

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On the Existence of a Finite Base for Complete Trace Equivalence over BPA with Interrupt

BRICS Report Series ◽

10.7146/brics.v14i5.21928 ◽

2007 ◽

Vol 14 (5) ◽

Author(s):

Luca Aceto ◽

Silvio Capobianco ◽

Anna Ingólfsdóttir

Keyword(s):

Process Algebra ◽

Basic Process ◽

Single Action ◽

Finite Axiomatization ◽

Finite Base ◽

Special Case ◽

Trace Equivalence

We study Basic Process Algebra with interrupt modulo complete trace equivalence. We show that, unlike in the setting of the more demanding bisimilarity, a ground complete finite axiomatization exists. We explicitly give such an axiomatization, and extend it to a finite complete one in the special case when a single action is present.

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DUALITY FOR COHOMOLOGY OF CURVES WITH COEFFICIENTS IN ABELIAN VARIETIES

Nagoya Mathematical Journal ◽

10.1017/nmj.2018.46 ◽

2018 ◽

Vol 240 ◽

pp. 42-149 ◽

Cited By ~ 1

Author(s):

TAKASHI SUZUKI

Keyword(s):

Function Field ◽

Positive Characteristic ◽

Abelian Varieties ◽

Base Field ◽

Tate Pairing ◽

Global Function ◽

Crystalline Cohomology ◽

Local Duality ◽

Height Pairing ◽

Finite Base

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

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