A Cayley theorem for distributive lattices (Q1047092)

For every set \(M\), let \({\mathcal F}(M)\) denote the algebra \((M^{M^2},\diamond,\ast)\) of type (2,2) defined by \((f\diamond g)(x,y)=f(g(x,y),y)\) and \((f\ast g)(x,y)=f(x,g(x,y))\). For every lattice \((L,\vee,\wedge)\) and every \(a\in L\), let \(f_a\) denote the mapping \((x,y)\mapsto (a\vee x)\wedge y\) from \(L^2\) to \(L\) and \(\varphi\) the mapping \(a\mapsto f_a\) from \(L\) to \(L^{L^2}\). If \({\mathcal L}=(L,\vee,\wedge)\) is a distributive lattice, then \({\mathcal F}(L)\) is a distributive lattice and \(\varphi\) is an embedding of \(\mathcal L\) into \({\mathcal F}(L)\).