Abstract:
In studying derived objects on universal algebras, such as automorphisms, endomorphisms, congruences, subalgebras, etc., we are naturally interested in those that can be defined by the means of the universal algebras themselves (i.e., are definable in one sense or another) – in particular in what part of all relevant derived objects is constituted by these. It is proved that for any algebraic lattice L and any of its $0$-$1$-lower subsemilattices $L_0\subseteq L_1\subseteq L_2$, there exist a universal algebra $\mathcal A$ and an isomorphism $\varphi$ of the lattice $L$ onto the lattice $\mathrm{Sub}\mathcal A$ such that $\varphi(L_0)=\mathrm{OFSub}\mathcal A$, $\varphi(L_1)=\mathrm{POFSub}\mathcal A$, $\varphi(L_2)=\mathrm{FSub}\mathcal A$, and $\mathrm{PFSub}\mathcal A=\mathrm{FSub}\mathcal A$.
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