Weak representability of \(M\)-semilattices (Q1272219)

It was shown by \textit{G. Grätzer} that for an algebra \(A\) its congruence lattice \(\operatorname {Con} (A)\) and endomorphism monoid \(\operatorname {End} (A)\) are not independent. Let \(M\) be a monoid with unit element \(e\) and let \(L\) be a complete \(\wedge \)-semilattice with 0 and 1. An action of \(M\) on \(L\) is a mapping \(\Phi:M\times L\to L\), \(\Phi(\mu,\theta)=\mu\cdot\theta\), s.t. {1.} \(e\cdot \theta =\theta \) {2.} \(\mu \cdot 1=1\) {3.} \(\mu \cdot (\nu \cdot \theta)=(\mu \nu)\cdot \theta \) {4.} \(\mu \cdot (\wedge \theta _i)=\wedge (\mu \cdot \theta _i)\) \noindent A triple \(\langle L,M,\Phi \rangle \) is called an \(M\)-semilattice. A homomorphism from \(\langle L_1,M_1,\Phi _1\rangle \) to \(\langle L_2,M_2,\Phi _2\rangle \) is a pair \((\omega ,\iota)\) where \(\omega \: L_1\to L_2\) is a 0,1-preserving complete \(\wedge \)-homomorphism, \(\iota \: M_1\to M_2\) is an isomorphism and \(\omega \big (\Phi _1(\mu ,\theta)\big)= \Phi _2\big (\iota (\mu),\omega (\theta)\big).\) We say that \(\langle L,M,\Phi \rangle \) is weakly representable if there exists an algebra \(A\) and an embedding of \(\langle L,M,\Phi \rangle \) into \(\langle \operatorname {Con} (A), \operatorname {End} (A),\Phi _1\rangle ,\) where for \(\mu \in \operatorname {End} (A)\) and \(\theta \in \operatorname {Con} (A)\) \[ \Phi _1(\mu ,\theta)=\Big \{\langle a,b\rangle \in A^2\: \big \langle \mu (a),\mu (b)\big \rangle \in \theta \Big \}\in \operatorname {Con} (A). \] \textit{L. A. Skornyakov} [Colloq. Math. Soc. J. Bolyai 43, 469-496 (1986; Zbl 0597.08003)] discovered necessary condition for weak representability of \(M\)-semilattices. The author shows that this condition is also sufficient.