English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The Yang–Baxter and pentagon equations are two well-known equations of Mathematical Physic. If S is a set, a map $$s:S\times S\rightarrow S\times S$$ s : S × S → S × S is said to be a set-theoretical solution of the quantum Yang–Baxter equation if $$\begin{aligned} s_{23}\, s_{13}\, s_{12} = s_{12}\, s_{13}\, s_{23}, \end{aligned}$$ s 23 s 13 s 12 = s 12 s 13 s 23 , where $$s_{12}=s\times {{\,\mathrm{id}\,}}_S$$ s 12 = s × id S , $$s_{23}={{\,\mathrm{id}\,}}_S\times s$$ s 23 = id S × s , and $$s_{13}=({{\,\mathrm{id}\,}}_S\times \tau )\,s_{12}\,({{\,\mathrm{id}\,}}_S\times \tau )$$ s 13 = ( id S × τ ) s 12 ( id S × τ ) and $$\tau $$ τ is the flip map, i.e., the map on $$S\times S$$ S × S given by $$\tau (x,y)=(y,x)$$ τ ( x , y ) = ( y , x ) . Instead, s is called a set-theoretical solution of the pentagon equation if $$\begin{aligned} s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}. \end{aligned}$$ s 23 s 13 s 12 = s 12 s 23 . The main aim of this work is to display how solutions of the pentagon equation turn out to be a useful tool to obtain new solutions of the Yang–Baxter equation. Specifically, we present a new construction of solutions of the Yang–Baxter equation involving two specific solutions of the pentagon equation. To this end, we provide a method to obtain solutions of the pentagon equation on the matched product of two semigroups, that is a semigroup including the classical Zappa product.

Set-theoretical solutions of the Yang–Baxter and pentagon equations on semigroups