The finitary independence of epimorphism and embedding relations (Q1922119)

The main result of this paper is the following: Theorem. For any discriminator variety \({\mathfrak M}\) with a finitely generated simple rich algebra with a one-element subalgebra, the epimorphism and embedding relations on the family of countable \({\mathfrak M}\)-algebras are finitary independent. Where the epimorphism and embedding relations are finitary independent on the class \({\mathcal K}\) of universal algebras if for any finite set \(A\) with two quasiorders \(\leq_1\) and \(\leq_2\) on \(A\), the twice quasiordered set \(\langle A; \leq_1, \leq_2\rangle\) is an isomorphic embedding in the double skeleton \(\langle {\mathcal I} {\mathcal K}; \leq,\ll \rangle\) of the class \({\mathcal K}\). The algebra \({\mathfrak A}\) is rich if there exists a disjunctive embedding of the tree of all finite corteges consisting of 0 and 1 \((S_1\leq S_2\) iff the cortege \(S_2\) is an initial interval of the cortege \(S_1)\) into the semilattice of sets of solutions of all systems of equations on \({\mathfrak A}\).