scholarly journals Limits of Quantum B-Algebras

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3184
Author(s):  
Aiping Gan ◽  
Aziz Muzammal ◽  
Yichuan Yang
Keyword(s):  
Propositional Logic ◽  
Binary Operation ◽  
Baxter Equation ◽  
Inverse Limits ◽  
True Statement ◽  
Direct Limits

Every set with a binary operation satisfying a true statement of propositional logic corresponds to a solution of the quantum Yang-Baxter equation. Quantum B-algebras and L-algebras are closely related to Yang-Baxter equation theory. In this paper, we study the categories with quantum B-algebras with morphisms of exact ones or spectral ones. We guarantee the existences of both direct limits and inverse limits.

Quasi-linear Cycle Sets and the Retraction Problem for Set-theoretic Solutions of the Quantum Yang-Baxter Equation

2016 ◽  
Vol 23 (01) ◽  
pp. 149-166 ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Abelian Group ◽  
Algebraic System ◽  
Binary Operation ◽  
Group Structure ◽  
Baxter Equation ◽  
Proper Extension ◽  
Group Structures ◽  
New Evidence

Cycle sets were introduced to reduce non-degenerate unitary Yang-Baxter maps to an algebraic system with a single binary operation. Every finite cycle set extends uniquely to a finite cycle set with a compatible abelian group structure. Etingof et al. introduced affine Yang-Baxter maps. These are equivalent to cycle sets with a specific abelian group structure. Abelian group structures have also been essential to get partial results for the still unsolved retraction problem. We introduce two new classes of cycle sets with an underlying abelian group structure and show that they can be transformed into each other while keeping the group structure fixed. This leads to a proper extension of the retractibility conjecture and new evidence for its truth.

scholarly journals Equilibrium states on higher-rank Toeplitz non-commutative solenoids

2019 ◽  
Vol 40 (11) ◽  
pp. 2881-2912 ◽  
Author(s):  
ZAHRA AFSAR ◽  
ASTRID AN HUEF ◽  
IAIN RAEBURN ◽  
AIDAN SIMS
Keyword(s):  
Continuous Functions ◽  
Equilibrium States ◽  
Probability Measures ◽  
Inverse Limits ◽  
Inverse Temperature ◽  
Higher Dimensional ◽  
Direct Limits ◽  
Higher Rank ◽  
Natural Dynamics

We consider a family of higher-dimensional non-commutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz non-commutative solenoids are direct limits of the Toeplitz extensions of non-commutative tori. We consider natural dynamics on these Toeplitz algebras, and we compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrized by the probability measures on an (ordinary) solenoid.

SEMIDIRECT PRODUCTS IN ALGEBRAIC LOGIC AND SOLUTIONS OF THE QUANTUM YANG–BAXTER EQUATION

2008 ◽  
Vol 07 (04) ◽  
pp. 471-490 ◽  
Author(s):  
WOLFGANG RUMP
Keyword(s):  
Semidirect Product ◽  
Binary Operation ◽  
Algebraic Logic ◽  
Type Theorem ◽  
Combinatorial Theory ◽  
Baxter Equation ◽  
Semidirect Products ◽  
Special Classes

A semidirect product is introduced for cycloids, i.e. sets with a binary operation satisfying (x · y) · (x · z) = (y · x) · (y · z). Special classes of cycloids arise in the combinatorial theory of the quantum Yang–Baxter equation, and in algebraic logic. In the first instance, semidirect products can be used to construct new solutions of the quantum Yang–Baxter equation, while in algebraic logic, they lead to a characterization of L-algebras satisfying a general Glivenko type theorem.

On GL2(R) Where R is a Boolean Ring

1972 ◽  
Vol 15 (2) ◽  
pp. 263-275 ◽  
Author(s):  
Joseph G. Rosenstein
Keyword(s):  
Symmetric Group ◽  
Negative Answer ◽  
Inverse Limits ◽  
Boolean Ring ◽  
Invertible Matrix ◽  
Boolean Rings ◽  
Direct Sum ◽  
Euclidean Domain ◽  
Direct Limits ◽  
Invertible Matrices

In this paper we characterize the 2 × 2 invertible matrices over a Boolean ring, and, using this characterization, show that every invertible matrix has order dividing 6. This suggests that GL2 of a Boolean ring is built up out of copies of the symmetric group S3. Indeed, if B is a finite Boolean ring, then GL2(B) turns out to be a direct sum of copies of S3. If B is infinite, then GL2(B) is more difficult to calculate; we present here descriptions of GL2(B) for the "extreme" cases of countable Boolean rings—namely, the Boolean ring which is generated by its atoms and the atomless Boolean ring. The former provides a negative answer to the question of whether the functor GL2(⋅) preserves inverse limits; the latter is a corollary of a theorem which states that, under certain circumstances, GL2(⋅) preserves direct limits. It turns out, in addition, that every invertible matrix is a product of elementary ones, as is the case for matrices over a Euclidean domain.

Hilbert-style proof Calculus for Propositional Logic in ABC notation

2019 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Propositional Logic ◽  
Inference Rule ◽  
Modus Ponens ◽  
Axiomatic System ◽  
First Contact

All nine axioms and a single inference rule of logic (Modus Ponens) within the Hilbert axiomatic system are presented using capital letters (ABC) in order to familiarize the beginner student in hers/his first contact with the topic.

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