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07 February 2025
 
  » arxiv » 2301.01643

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Idempotent set-theoretical solutions of the pentagon equation
Marzia Mazzotta ;
Date 4 Jan 2023
AbstractA set-theoretical solution of the pentagon equation on a non-empty set $X$ is a function $s:X imes X o X imes X$ satisfying the relation $s_{23}, s_{13}, s_{12}=s_{12}, s_{23}$, with $s_{12}=s imes ,id_X$, $s_{23}=id_X imes , s$ and $s_{13}=(id_X imes , au)s_{12}(id_X imes , au)$, where $ au:X imes X o X imes X$ is the flip map given by $ au(x,y)=(y,x)$, for all $x,yin X$. Writing a solution as $s(x,y)=(xy , heta_x(y))$, where $ heta_x: X o X$ is a map, for every $xin X$, one has that $X$ is a semigroup. In this paper, we study idempotent solutions, i.e., $s^2=s$, by showing that the idempotents of $X$ have a key role in such an investigation. In particular, we describe all such solutions on monoids having central idempotents. Moreover, we focus on idempotent solutions defined on monoids for which the map $ heta_1$ is a monoid homomorphism, by showing that they have to be derived considering the kernel congruence of the map $ heta_1$.
Source arXiv, 2301.01643
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