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19 pages, 378 KiB

Open AccessArticle

Partial Residuated Implications Induced by Partial Triangular Norms and Partial Residuated Lattices

by Xiaohong Zhang, Nan Sheng and Rajab Ali Borzooei

Axioms 2023, 12(1), 63; https://doi.org/10.3390/axioms12010063 - 6 Jan 2023

Cited by 13 | Viewed by 2054

This paper reveals some relations between fuzzy logic and quantum logic on partial residuated implications (PRIs) induced by partial t-norms as well as proposes partial residuated monoids (PRMs) and partial residuated lattices (PRLs) by defining partial adjoint pairs. First of all, we introduce the connection between lattice effect algebra and partial t-norms according to the concept of partial t-norms given by Borzooei, together with the proof that partial operation in any commutative quasiresiduated lattice is partial t-norm. Then, we offer the general form of PRI and the definition of partial fuzzy implication (PFI), give the condition that partial residuated implication is a fuzzy implication, and prove that each PRI is a PFI. Next, we propose PRLs, study their basic characteristics, discuss the correspondence between PRLs and lattice effect algebras (LEAs), and point out the relationship between LEAs and residuated partial algebras. In addition, like the definition of partial t-norms, we provide the notions of partial triangular conorms (partial t-conorms) and corresponding partial co-residuated lattices (PcRLs). Lastly, based on partial residuated lattices, we define well partial residuated lattices (wPRLs), study the filter of well partial residuated lattices, and then construct quotient structure of PRMs. Full article

(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)

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36 pages, 494 KiB

Open AccessReview

Polynomial Automorphisms, Deformation Quantization and Some Applications on Noncommutative Algebras

by Wenchao Zhang, Roman Yavich, Alexei Belov-Kanel, Farrokh Razavinia, Andrey Elishev and Jietai Yu

Mathematics 2022, 10(22), 4214; https://doi.org/10.3390/math10224214 - 11 Nov 2022

Cited by 1 | Viewed by 2015

Abstract

This paper surveys results concerning the quantization approach to the Jacobian Conjecture and related topics on noncommutative algebras. We start with a brief review of the paper and its motivations. The first section deals with the approximation by tame automorphisms and the Belov–Kontsevich Conjecture. The second section provides quantization proof of Bergman’s centralizer theorem which has not been revisited for almost 50 years and formulates several related centralizer problems. In the third section, we investigate a free algebra analogue of a classical theorem of Białynicki-Birula’s theorem and give a noncommutative version of this famous theorem. Additionally, we consider positive-root torus actions and obtain the linearity property analogous to the Białynicki-Birula theorem. In the last sections, we introduce Feigin’s homomorphisms and we see how they help us in proving our main and fundamental theorems on screening operators and in the construction of our lattice

W_{n}

-algebras associated with

{sl}_{n}

, which is by far the simplest known approach concerning constructing such algebras until now. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

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