About a question of Gateva-Ivanova and Cameron on square-free set-theoretic solutions of the Yang-Baxter equation

Abstract

In this paper, we introduce a new sequenceN m to find a new estimation of the cardinality N m of the minimal involutive square-free solution of level m. As an application, using the first values ofN m , we improve the estimations of N m obtained by and by Lebed and Vendramin in [16]. Following the approach of the first part, in the last section we construct several new counterexamples to the Gateva-Ivanova's Conjecture.

On the indecomposable involutive set-theoretic solutions of the Yang-Baxter equation of prime-power size

“…Conversely, if (X, r) is a non-degenerate involutive solution and · the binary operation given by x · y := λ −1 x (y) for all x, y ∈ X, then (X, ·) is a non-degenerate cycle set. The existence of this bijective correspondence allows us to move the study of involutive non-degenerate solutions to non-degenerate cycle sets, as already made in [2,18,5,8,4,6,3,23,25,19].…”

Konstruksi Brace Dua Sisi Dengan Menggunakan Ring Jacobson

“…Selain itu, Agata Smoktunowicz dalam [10] dan [11] juga memberikan solusi alternatif dari sudut pandang teori himpunan. Selanjutnya, solusi dengan pendekatan teori himpunan juga diberikan oleh Castelli, et.al dalam papernya [12] dan Matsumoto & Shimizu dalam [13]. Solusi yang diperumum diberikan oleh [14] dengan menggunakan struktur aljabar baru yang disebut dengan brace.…”

Simplicity and finite primitive level of indecomposable set-theoretic solutions of the Yang-Baxter equation

“…Evidently, if X is a q-cycle set such that • and : coincide, then X is a cycle set. Cycle sets were introduced by Rump in [25] and rather investigated (see, for instance, [2,3,4,6,26,33]) for their one-to-one correspondence with left non-degenerate involutive solutions.…”