scholarly journals Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 972
Author(s):  
Steven Duplij
Keyword(s):  
Special Kind ◽  
Braid Groups ◽  
Matrix Group ◽  
Baxter Equation ◽  
Regular Semigroups ◽  
Von Neumann ◽  
R Matrix ◽  
Artin Braid Group ◽  
Higher Dimensional ◽  
Von Neumann Regular

In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.

V-regular semigroups

1981 ◽  
Vol 88 (3-4) ◽  
pp. 275-291 ◽  
Author(s):  
K. S. S. Nambooripad ◽  
F. Pastijn
Keyword(s):  
Regular Semigroup ◽  
Transformation Semigroup ◽  
Inverse Semigroups ◽  
Full Transformation ◽  
Regular Semigroups ◽  
Von Neumann ◽  
Biordered Set ◽  
Von Neumann Regular ◽  
Strongly Regular

SynopsisA regular semigroup S is called V-regular if for any elements a, b ∈ S and any inverse (ab)′ of ab, there exists an inverse a′ of a and an inverse b′ of b such that (ab)′ = b′a′. A characterization of a V-regular semigroup is given in terms of its partial band of idempotents. The strongly V-regular semigroups form a subclass of the class of V-regular semigroups which may be characterized in terms of their biordered set of idempotents. It is shown that the class of strongly V-regular semigroups comprises the elementary rectangular bands of inverse semigroups (including the completely simple semigroups), a special class of orthodox semigroups (including the inverse semigroups), the strongly regular Baer semigroups (including the semigroups that are the multiplicative semigroup of a von Neumann regular ring), the full transformation semigroup on a set, and the semigroup of all partial transformations on a set.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

On (Strongly) Gorenstein Von Neumann Regular Rings

2011 ◽  
Vol 39 (9) ◽  
pp. 3242-3252 ◽  
Author(s):  
Najib Mahdou ◽  
Mohammed Tamekkante ◽  
Siamak Yassemi
Keyword(s):  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

1988 ◽  
Vol 53 (4) ◽  
pp. 1177-1187
Author(s):  
W. A. MacCaull
Keyword(s):  
Intuitionistic Logic ◽  
Main Idea ◽  
Rational Functions ◽  
Completeness Theorem ◽  
Point Of View ◽  
Polynomial Rings ◽  
Von Neumann ◽  
Ordered Fields ◽  
Von Neumann Regular ◽  
Theoretic Point

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

scholarly journals RINGS OF DEFINABLE SCALARS OF VERMA MODULES

2007 ◽  
Vol 06 (05) ◽  
pp. 779-787 ◽  
Author(s):  
SONIA L'INNOCENTE ◽  
MIKE PREST
Keyword(s):  
Lie Algebra ◽  
Model Theory ◽  
Regular Ring ◽  
Verma Module ◽  
Closed Field ◽  
Verma Modules ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Canonical Map ◽  
Von Neumann Regular

Let M be a Verma module over the Lie algebra, sl 2(k), of trace zero 2 × 2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U( sl 2(k)) to RM is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules.

scholarly journals Degenerate Sklyanin algebras

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Andrey Smirnov
Keyword(s):  
Difference Operators ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Rational Solutions ◽  
R Matrix ◽  
Gauge Transformations ◽  
Sklyanin Algebras ◽  
Sklyanin Algebra ◽  
The Difference

AbstractNew trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.

scholarly journals Normal Subgroups of Classical Groups over von Neumann Regular Rings

1994 ◽  
Vol 169 (3) ◽  
pp. 863-873
Author(s):  
F.A. Arlinghaus ◽  
L.N. Vaserstein ◽  
H. You
Keyword(s):  
Classical Groups ◽  
Normal Subgroups ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings
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