On subalgebra lattices of universal algebras

Abstract:

If $Ad$ is a universal algebra, $S(A)$ is the lattice of all subalgebras of $A$. If $B \subseteq A \times A$, $B^\ast$ is $\{ (x,y) : (y,x) \in B \}$. Theorem. Let $L_1$, $L_2$, $L_3$ be algebraic lattices such that $|L_1|$, $|L_2| > 1$. Let $\alpha _i$ be an involutive automorphism of $L_i$, $i = 1$, $2$. Then there are two universal algebras $A_1$, $A_2$ of the same similarity type, having the properties: (a) there are lattice isomorphisms $\beta _i$ of $L_i$ onto $S(A_i \times A_i)$, $i = 1$, $2$, and ${\beta _3}$ of ${L_3}$ onto $S(A_1 \times A_2)$; (b) $(l \alpha _i) \beta _i = (l \beta _i)^\ast$, $l \in L_i$, $i = 1$, $2$.