scholarly journals Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Francesco Catino ◽  
Ilaria Colazzo ◽  
Paola Stefanelli
Keyword(s):  
Finite Order ◽  
Algebraic Structure ◽  
Baxter Equation ◽  
Construction Technique ◽  
Semilattice Of Semigroups ◽  
Strong Semilattice ◽  
New Construction ◽  
Strong Semilattice Of Semigroups

Abstract This paper aims to introduce a construction technique of set-theoretic solutions of the Yang–Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known that braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions.

Strong Semilattices of Implicative Semigroups

2008 ◽  
Vol 15 (01) ◽  
pp. 53-62
Author(s):  
Shuk Yee Lee ◽  
K. P. Shum ◽  
Congxin Wu
Keyword(s):  
Analogous Result ◽  
Practical Method ◽  
Main Difficulty ◽  
Large Size ◽  
Semilattice Of Semigroups ◽  
Strong Semilattice ◽  
Strong Semilattice Of Semigroups

It is well known that a strong semilattice of semigroups is still a semigroup. However, it is not known whether the analogous result carries or not for a strong semilattice of implicative semigroups. The main difficulty is that the implicative operation ∗ is hard to define on the semilattice, especially we need the operation ∗ of the implicative semigroup to be compatible with the negatively ordered relation and the semigroup multiplication on the semilattice of implicative semigroups. In this paper, we provide a practical method of constructing such a strong semilattice of mplicative semigroups. Our method is particularly useful in constructing implicative semigroups of large size. A constructed example is given.

A Structure Theorem of Regular ${\cal H}^\sharp$-Cryptographs

2008 ◽  
Vol 15 (04) ◽  
pp. 653-666 ◽  
Author(s):  
Xiangzhi Kong ◽  
Zhiling Yuan ◽  
K. P. Shum
Keyword(s):  
Structure Theorem ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Abundant Semigroups ◽  
Completely Regular Semigroups ◽  
Semilattice Of Semigroups ◽  
Strong Semilattice ◽  
Green Relations ◽  
Strong Semilattice Of Semigroups

A new set of generalized Green relations is given in studying the [Formula: see text]-abundant semigroups. By using the generalized strong semilattice of semigroups recently developed by the authors, we show that an [Formula: see text]-abundant semigroup is a regular [Formula: see text]-cryptograph if and only if it is an [Formula: see text]-strong semilattice of completely [Formula: see text]-simple semigroups. This result not only extends the well known result of Petrich and Reilly from the class of completely regular semigroups to the class of semiabundant semigroups, but also generalizes a well known result of Fountain on superabundant semigroups from the class of abundant semigroups to the class of semiabundant semigroups.

scholarly journals The strong nil-cleanness of semigroup rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1491-1500
Author(s):  
Yingdan Ji
Keyword(s):  
Inverse Semigroup ◽  
Simple Semigroup ◽  
Semigroup Ring ◽  
Semigroup Rings ◽  
Locally Inverse Semigroup ◽  
Semilattice Of Semigroups ◽  
Strong Semilattice ◽  
Alpha Beta ◽  
Strong Semilattice Of Semigroups

Abstract In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M={ {\mathcal M} }^{0}(G;I,\text{Λ};P) , we show that the contracted semigroup ring {R}_{0}{[}M] is strongly nil-clean if and only if either |I|=1 or |\text{Λ}|=1 , and R{[}G] is strongly nil-clean; as a corollary, we characterize the strong nil-cleanness of locally inverse semigroup rings. Moreover, let S={[}Y;{S}_{\alpha },{\varphi }_{\alpha ,\beta }] be a strong semilattice of semigroups, then we prove that R{[}S] is strongly nil-clean if and only if R{[}{S}_{\alpha }] is strongly nil-clean for each \alpha \in Y .

New Construction Technique of Soft Soil Foundation Consolidated by Plastic Drainage Slab

2012 ◽  
Vol 446-449 ◽  
pp. 2673-2680
Author(s):  
Song Lei Tao ◽  
Zhi Fang Zhao ◽  
Zi He Zhang ◽  
Yuan Hong Yu ◽  
Xing Ge Lian
Keyword(s):  
Soft Soil ◽  
Construction Technique ◽  
New Construction ◽  
Soil Foundation ◽  
Soft Soil Foundation

scholarly journals Results of R&D on a new construction technique for W/ScFi Calorimeters

2012 ◽  
Vol 404 ◽  
pp. 012023 ◽  
Author(s):  
O D Tsai ◽  
L E Dunkelberger ◽  
C A Gagliardi ◽  
S Heppelmann ◽  
H Z Huang ◽  
...  
Keyword(s):  
Construction Technique ◽  
New Construction

scholarly journals Quantum Spectral Curve of γ-Twisted 𝒩 = 4 SYM Theory and Fishnet CFT

2018 ◽  
Vol 30 (07) ◽  
pp. 1840010 ◽  
Author(s):  
Vladimir Kazakov
Keyword(s):  
Algebraic Structure ◽  
Spectral Curve ◽  
Scaling Limit ◽  
Integrable Model ◽  
Quantization Condition ◽  
Spin Chains ◽  
Hasse Diagram ◽  
Baxter Equation ◽  
Heisenberg Spin Chain ◽  
Wheel Graphs

We review the quantum spectral curve (QSC) formalism for the spectrum of anomalous dimensions of [Formula: see text] SYM, including its [Formula: see text]-deformation. Leaving aside its derivation, we concentrate on the formulation of the “final product” in its most general form: a minimal set of assumptions about the algebraic structure and the analyticity of the [Formula: see text]-system — the full system of Baxter [Formula: see text]-functions of the underlying integrable model. The algebraic structure of the [Formula: see text]-system is entirely based on (super)symmetry of the model and is efficiently described by Wronskian formulas for [Formula: see text]-functions organized into the Hasse diagram. When supplemented with analyticity conditions on [Formula: see text]-functions, it fixes completely the set of physical solutions for the spectrum of an integrable model. First, we demonstrate the spectral equations on the example of [Formula: see text] and [Formula: see text] Heisenberg (super)spin chains. Supersymmetry [Formula: see text] occurs as a simple “rotation” of the Hasse diagram for a [Formula: see text] system. Then we apply this method to the spectral problem of [Formula: see text]/CFT4-duality, describing the QSC formalism. The main difference with the spin chains consists in more complicated analyticity constraints on [Formula: see text]-functions which involve an infinitely branching Riemann surface and a set of Riemann–Hilbert conditions. As an example of application of QSC, we consider a special double scaling limit of [Formula: see text]-twisted [Formula: see text] SYM, combining weak coupling and strong imaginary twist. This leads to a new type of non-unitary CFT dominated by particular integrable, and often computable, 4D fishnet Feynman graphs. For the simplest of such models — the bi-scalar theory — the QSC degenerates into the [Formula: see text]-system for integrable non-compact Heisenberg spin chain with conformal, [Formula: see text] symmetry. We describe the QSC derivation of Baxter equation and the quantization condition for particular fishnet graphs — wheel graphs, and review numerical and analytic results for them.

Sign in / Sign up

Export Citation Format

Share Document