Idempotent set-theoretical solutions of the pentagon equation

Abstract

AbstractA set-theoretical solution of the pentagon equation on a non-empty set X is a function $$s:X\times X\rightarrow X\times X$$ s : X × X → X × X satisfying the relation $$s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}$$ s 23 s 13 s 12 = s 12 s 23 , with $$s_{12}=s\times \,{{\,\textrm{id}\,}}_X$$ s 12 = s × id X , $$s_{23}={{\,\textrm{id}\,}}_X \times \, s$$ s 23 = id X × s and $$s_{13}=({{\,\textrm{id}\,}}_X\times \, \tau )s_{12}({{\,\textrm{id}\,}}_X\times \,\tau )$$ s 13 = ( id X × τ ) s 12 ( id X × τ ) , where $$\tau :X\times X\rightarrow X\times X$$ τ : X × X → X × X is the flip map given by $$\tau (x,y)=(y,x)$$ τ ( x , y ) = ( y , x ) , for all $$x,y\in X$$ x , y ∈ X . Writing a solution as $$s(x,y)=(xy,\theta _x(y))$$ s ( x , y ) = ( x y , θ x ( y ) ) , where $$\theta _x: X \rightarrow X$$ θ x : X → X is a map, for every $$x\in X$$ x ∈ X , one has that X is a semigroup. In this paper, we study idempotent solutions, i.e., $$s^2=s$$ s 2 = s , by showing that the idempotents of X have a crucial 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 $$\theta _1$$ θ 1 is a monoid homomorphism.