Fixed-points and algebras with infinitely long expressions: Part II. μ-clones of regular algebras

1979 ◽  
Vol 2 (1) ◽  
pp. 317-335
Author(s):  
Jerzy Tiuryn
Keyword(s):  
Fixed Points ◽  
Regular Algebra ◽  
Arbitrary Signature ◽  
Regular Algebras ◽  
Main Notion

This paper is a continuation of Tiuryn [16]. The main notion presented in the latter paper is the notion of a regular algebra. As it was proved in Tiuryn [16] for an arbitrary signature Σ the algebra of regular Σ-trees is an initial regular algebra. This means that there are naturally defined “polynomials” for regular algebras, which are determined by infinitely long expressions. The aim of this paper is to investigate this phenomenon.

scholarly journals A Complete Classification of 3-dimensional Quadratic AS-regular Algebras of Type EC

2020 ◽  
pp. 1-19
Author(s):  
Masaki Matsuno
Keyword(s):  
Algebraic Geometry ◽  
Complete Classification ◽  
Complete List ◽  
Graded Algebra ◽  
Regular Algebra ◽  
3 Dimensional ◽  
Noncommutative Algebraic Geometry ◽  
Regular Algebras ◽  
Point Scheme

Abstract Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$ -dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb {P}^{2}$ . In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$ -dimensional quadratic AS-regular algebras that cannot be written as a twist of a Calabi–Yau AS-regular algebra by a graded algebra automorphism.

scholarly journals Algunos tipos especiales de determinantes en extensiones PBW torcidas graduadas

2021 ◽  
Vol 39 (1) ◽  
Author(s):  
Héctor Suárez ◽  
Duban Cáceres ◽  
Armando Reyes
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Regular Algebra ◽  
Noncommutative Algebraic Geometry ◽  
Noncommutative Differential Geometry ◽  
Regular Algebras ◽  
Skew Pbw Extensions ◽  
Finitely Presented

In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over a finitely presented Koszul Auslander-regular algebra has trivial homological determinant. For A = σ(R)<x1, x2> a graded skew PBW extension over a connected algebra R, we compute its P-determinant and the inverse of σ. In the particular case of quasi-commutative skew PBW extensions over Koszul Artin-Schelter regular algebras, we show explicitly the connection between the Nakayama automorphism of the ring of coefficients and the extension. Finally, we give conditions to guarantee that A is Calabi-Yau. We provide illustrative examples of the theory concerning algebras of interest in noncommutative algebraic geometry and noncommutative differential geometry.

scholarly journals AS-REGULARITY OF GEOMETRIC ALGEBRAS OF PLANE CUBIC CURVES

2021 ◽  
pp. 1-25
Author(s):  
AYAKO ITABA ◽  
MASAKI MATSUNO
Keyword(s):  
Elliptic Curves ◽  
Three Dimensional ◽  
Preceding Paper ◽  
Regular Algebra ◽  
Noncommutative Algebraic Geometry ◽  
Defining Relations ◽  
Morita Equivalent ◽  
Cubic Curve ◽  
Regular Algebras ◽  
Geometric Algebras

Abstract In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

scholarly journals GRADED MORITA EQUIVALENCES FOR GEOMETRIC AS-REGULAR ALGEBRAS

2012 ◽  
Vol 55 (2) ◽  
pp. 241-257 ◽  
Author(s):  
IZURU MORI ◽  
KENTA UEYAMA
Keyword(s):  
Algebraic Geometry ◽  
Three Dimensional ◽  
Koszul Algebras ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Regular Algebra ◽  
Morita Equivalent ◽  
Regular Algebras

AbstractClassification of AS-regular algebras is one of the major projects in non-commutative algebraic geometry. In this paper, we will study when given AS-regular algebras are graded Morita equivalent. In particular, for every geometric AS-regular algebra A, we define another graded algebra A, and show that if two geometric AS-regular algebras A and A' are graded Morita equivalent, then A and A' are isomorphic as graded algebras. We also show that the converse holds in many three-dimensional cases. As applications, we apply our results to Frobenius Koszul algebras and Beilinson algebras.

Fixed-points and algebras with infinitely long expressions: Part I. Regular algebras

1979 ◽  
Vol 2 (1) ◽  
pp. 103-127
Author(s):  
Jerzy Tiuryn
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Upper Bound ◽  
The Other ◽  
Intrinsic Interest ◽  
Other Hand ◽  
Initial Algebra ◽  
Regular Algebras ◽  
Context Free

This paper introduces a class of algebras (called the class of regular algebras), in which the algebra of regular trees (unfoldments of monadic program schemes) is an initial algebra. This means that we have for the above-mentioned class “semantics” of monadic program schemes. We show how to treat, in a unified way, such concepts as: monadic and recursive monadic program schemes, regular and context-free languages. On the other hand, the investigation of the properties of regular algebras may be of intrinsic interest, in particular this leads to a very nice generalization of the notion of a polynomial in an algebra. These “new” polynomials, in general, are determined by infinitely long expressions, and existence of such polynomials in the class of regular algebras is closely connected with the property that every finite tuple of algebraic mappings has a least fixed-point which is obtainable as a least upper bound of a denumerable chain of “approximations”.

On $\alpha$-nonexpansive mapping and fixed-points in Banach spaces

2018 ◽  
Vol 2018 (-) ◽  
Author(s):  
Prondanai Kaskasem ◽  
Chakkrid Klin-eam ◽  
Suthep Suantai
Keyword(s):  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Banach Spaces

Fixed Points of Sequences of Mappings

2018 ◽  
Vol 2018 (-) ◽  
Author(s):  
C. Ganesa Moorthy ◽  
S. Iruthaya Raj
Keyword(s):  
Fixed Points
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