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Book Review: Existentially closed groups

Kenneth Hickin
1990 Bulletin of the American Mathematical Society  
After some preliminaries, Chapters 5 and 6 develop some algebraic applications of the Higman embedding theorem and its generalized version (which is deduced in the text).  ...  An existentially closed structure E for a class of algebraic systems S is a sort of "universal structure for Z-relations" in the sense that £EI and every finite set of equalities and inequalities (that  ... 
doi:10.1090/s0273-0979-1990-15943-9 fatcat:af5mkj54n5apleqqmqemmqtvaa

Page 7540 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews  
(RS-KEME; Kemerovo) Higman’s embedding theorem in a general setting and its application to existentially closed algebras. (English summary) Notre Dame J. Formal Logic 37 (1996), no. 4, 613-624.  ...  We suggest certain general conditions on K under which (1) the Higman theorem implies the generalized Higman theorem; (2) a finitely generated K-algebra A is embeddable into every existen- tially closed  ... 

On nilpotent groups of exponent p

Berthold J Maier
1989 Journal of Algebra  
If a model companion exists it is to LX what the theory of algebraically closed fields is to the class of fields.  ...  By 3.7 it remains to show that T is existentially closed in K". This follows from (II) and (III) as in 3.6.  ... 
doi:10.1016/0021-8693(89)90253-6 fatcat:mb5txr67yjbbves6supelnglwq

On countable locally described structures

Berthold J. Maier
1987 Annals of Pure and Applied Logic  
The main theorems of the general theory give characterizations for (1) the uniqueness, up to isomorphism, of the countable closed structure and (2) the existence of a countable saturated structure.  ...  In LX we study (1) closed structures also known as basically saturated structures which are algebraic analogues of existentially closed structures and (2) saturated structures which are algebraic analogues  ...  A closed structure is existentially closed. Therefore, in a closed group G in Nc + the upper and lower central series coincide [20, Theorem 1] .  ... 
doi:10.1016/0168-0072(87)90064-9 fatcat:o44ffwhcljfldbznh72ktsni2y

Decision procedure of some relevant logics: a constructive perspective

Jacques Riche
2005 Journal of Applied Non-Classical Logics  
A. Urquhart's pen and paper solution that relies on a sophisticated algebraic and geometric treatment of the problem shows the usefulness of an algebraic approach in Logic.  ...  Meyer has shown equivalent to Dickson's lemma in number theory and to his own infinite divisor lemma, henceforth, Meyer's lemma or IDP.  ...  It suffices to show that every such chain in the set A n of infinite sequences of N k has a glb. And it is the case from the proof of IDP in [RIC 98].  ... 
doi:10.3166/jancl.15.9-23 fatcat:zh7yifu5lbaonntv76fylikilu

Introduction to Sofic and Hyperlinear groups and Connes' embedding conjecture [article]

Valerio Capraro, Martino Lupini
2015 arXiv   pre-print
The aim of these notes is to present in a uniform and accessible way some cornerstone results in the study of sofic and hyperlinear groups and the Connes embedding conjecture.  ...  In this case the famous conjecture due to Connes (commonly known as the Connes embedding conjecture) that any II_1 factor can be approximated in a suitable sense by matrix algebras inspired several breakthroughs  ...  This is the content of a theorem proven in the setting of the usual first order logic by Loś in [109] . Its generalization to the logic for metric structures can be found in [13] (Theorem 5.4).  ... 
arXiv:1309.2034v6 fatcat:viw5r73kozbaplk43vzyfycwrm

Bernhard Hermann Neumann AC. 15 October 1909 -- 21 October 2002

C. E. Praeger
2010 Biographical Memoirs of Fellows of the Royal Society  
They are at the heart of G. higman's famous embedding theorem (mr0130286, 1961) showing that a finitely generated group can be embedded in a finitely presented group if and only if it is recursively  ...  In it he proved (among other things) that a finitely generated group with soluble word problem can be embedded in every algebraically closed group-one half of the macintyre-neumann Theorem. macintyre (  ...  Soc. 29, [236] [237] [238] [239] [240] [241] [242] [243] [244] [245] [246] [247] [248] an embedding theorem for algebraic systems. Proc. Lond. Math. Soc.  ... 
doi:10.1098/rsbm.2010.0002 fatcat:4n3ttuh44rai7odg7fl4wmynhi

Book Review: Fundamentals of generalized recursion theory

G. Kreisel
1985 Bulletin of the American Mathematical Society  
If Cantor's so-called generalizations of numbers are viewed as such extensions to infinite sets and well orderings and not as competing with Higher Arithmetic (say, of ideals or algebraic number fields  ...  Should I add that it is well organized, has good indices, and no misprints? I just want to say that the hand holds it well, and does not wish to let it go.  ...  (b) The original theorem of Cantor-Bendixson states that the sequence a F of derived sets of a closed set F is countable (in suitable spaces).  ... 
doi:10.1090/s0273-0979-1985-15413-8 fatcat:xq5osel6prdojeilzzfcbzhu5i

Deciding Piecewise Testable Separability for Regular Tree Languages

Jean Goubault-Larrecq, Sylvain Schmitz, Marc Herbstritt
2016 International Colloquium on Automata, Languages and Programming  
We prove a general characterisation of piecewise testable separability on languages in a well-quasiorder, in terms of ideals of the ordering.  ...  In the case of finite ranked trees ordered by homeomorphic embedding, we show using effective representations for tree ideals that it entails the decidability of piecewise testable separability when the  ...  97:13 to the growing body of algorithmic applications of downwards-closed sets and ideals of wellquasi-orders in logic and verification, e.g. in forward analysis [13, 14] , backward analysis [21] , inference  ... 
doi:10.4230/lipics.icalp.2016.97 dblp:conf/icalp/Goubault-Larrecq16 fatcat:6kzs33bamfgvlhuor72x7o4nra

The reverse mathematics of wqos and bqos [article]

Alberto Marcone
2019 arXiv   pre-print
In this paper we survey wqo and bqo theory from the reverse mathematics perspective.  ...  We consider both elementary results (such as the equivalence of different definitions of the concepts, and basic closure properties) and more advanced theorems.  ...  To prove our theorem first notice that Kruskal's theorem generalizes Higman's theorem, so that we can argue in ACA 0 .  ... 
arXiv:1707.08365v5 fatcat:ym22ayu7ibhdtfrqnf2pi5ko44

Bernhard Hermann Neumann 1909–2002

Cheryl E. Praeger
2010 Historical Records of Australian Science  
(Mike) Newman to whom I am especially indebted and without whose help and scholarship over several years this memoir could not have been written. In particular, together with L. G.  ...  I am greatly indebted to James Wiegold, Martin Taylor, and many others who have helped in various ways: Reg Allenby, Gilbert Baumslag, Keith Burns, Slava Grigorchuk, Hermann Heineken, Charles  ...  Higman's famous embedding theorem (MR0130286, 1961) showing that a finitely generated group can be embedded in a finitely presented group if and only if it is recursively presented.  ... 
doi:10.1071/hr09021 fatcat:uxo2zjtsajfafnuhiy47wfahmu

Groups with no nontrivial linear representations

A.J. Derrick
1994 Bulletin of the Australian Mathematical Society  
in C) studied by Wilson [26] et al. ( 2 ) 2 We contribute to the theory of this class of groups by showing it to be closed under perfect extension with locally soluble kernel (Theorem 2.1). (3) Applications  ...  Now, as in Theorem 2.1 below, there is an embedding PGL(V)^GL(R(PGL(V))).  ... 
doi:10.1017/s0004972700009503 fatcat:m3swsyzdlbcrjlilud5rcvul3a

More problems in rewriting [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1993 Lecture Notes in Computer Science  
It has been proved that (in A-calculus or Combinatory Logic) every recursively enumerable set of ground terms that is closed under conversion has the form {MIPM ~* Q} for some P and Q.  ...  Robertson and Seymour [Robertson and Seymour, 1982] have achieved a similar theorem for undirected graphs. However, no embedding theorem has yet been proved for directed graphs, and (consequently?)  ... 
doi:10.1007/3-540-56868-9_39 fatcat:cibthhywyrgmvcsv73tryp3v2e

Real Computational Universality: The Word Problem for a class of groups with infinite presentation [article]

Martin Ziegler, Klaus Meer
2006 arXiv   pre-print
The word problem for discrete groups is well-known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem.  ...  Most important, the free group will be generated by an uncountable set of generators with index running over certain sets of real numbers.  ...  Many of the known computability and complexity results in the BSS model are closely related to computational problems of semi-algebraic sets.  ... 
arXiv:cs/0604032v3 fatcat:jiek7okaebh6zndat5ikmfm55m

Uniform interpolation and compact congruences

Samuel J. van Gool, George Metcalfe, Constantine Tsinakis
2017 Annals of Pure and Applied Logic  
Uniform interpolation properties are defined for equational consequence in a variety of algebras and related to properties of compact congruences on first the free and then the finitely presented algebras  ...  It is also shown, following related results of Ghilardi and Zawadowski, that a combination of these properties provides a sufficient condition for the first-order theory of the variety to admit a model  ...  By Higman's embedding theorem, there exists a finitely presented group H and an embedding j : G → H. Choose a finite generating set A ⊆ H which contains jp(y) for every y ∈ y.  ... 
doi:10.1016/j.apal.2017.05.001 fatcat:lfcidbanjvcljf2lmqdpp52zsq
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