Joins of subalgebras and normals in 0-regular varieties
Let \(\mathcal V\) be a variety of algebras with a nullary operation \(0\). If for each congruence \(\theta\) on each algebra \(A\in\mathcal V\), the \(\theta\)-class containing \(0\) determines \(\theta\), then \(\mathcal V\) is called \textit{\(0\)-regular}. If additionally, for each algebra \(A\in\mathcal V\), the singleton \(\{0\}\) is a subalgebra of \(A\), then \(\mathcal V\) is called \textit{\(0\)-normal}. Evidently, a \(0\)-regular variety is \(0\)-normal if and only if for any congruence \(\theta\) on any \(A\in\mathcal V\), the \(\theta\)-class containing \(0\) is a subalgebra of \(A\). Examples of \(0\)-normal varieties are groups or rings or -- more generally -- multioperator groups. If \(\mathcal V\) is a \(0\)-regular variety, then for any \(A\in\mathcal V\) the congruence classes containing \(0\) are called its \textit{normals}. If \(I\) is a normal of \(A\) and \(a\in A\), then \(aI\) denotes the \(\theta\)-class containing \(a\), where \(\theta\) is the congruence corresponding to \(I\). Evidently, the lattice \((\mathcal I(A),\cap,+)\) of normals of \(A\) is isomorphic to the congruence lattice of \(A\). A \(0\)-normal variety \(\mathcal V\) is called \textit{join invariant} if for any \(A\in\mathcal V\) the set \(\mathcal I(A)\) forms a sublattice of the lattice \((\mathcal S(A),\cap,\vee)\) of subuniverses of \(A\), i.e., if \(I+J=I\vee J\) for all \(I,J\in\mathcal I(A)\). Let \(\mathcal V\) be a \(0\)-normal variety and \(A\in\mathcal V\). Two congruences \(\rho\), \(\theta\) on \(A\) are said to \textit{permute at \(0\)} if for any \(a\in A\), \((a,0)\in\rho\circ\theta\) implies \((a,0)\in\theta\circ\rho\) and vice versa. The variety \(\mathcal V\) is called \textit{ideal determined} if for any \(A\in\mathcal V\) any two congruences on \(A\) permute at \(0\) (for this notion, see in particular [\textit{H. P. Gumm} and \textit{A. Ursini}, ibid. 19, 45--54 (1984; Zbl 0547.08001)]). The variety is called \textit{\(0\)-coherent} if for any \(A\in\mathcal V\), \(I\in\mathcal I(A)\) and \(S\in\mathcal S(A)\), \(S\vee I=S\cdot I\), where \(S\cdot I=\bigcup\{sI\mid s\in S\}\). Again, groups and rings are examples for \(0\)-coherent varieties. The main results of the paper under consideration are:{\parindent=8mm \begin{itemize}\item[(i)] If \(\mathcal V\) is a \(0\)-normal variety, \(A\in\mathcal V\) and \(I,J\in\mathcal S(A)\), then \(I\vee J\subseteq I\cdot J\subseteq I+J\). \item[(ii)] Every \(0\)-coherent variety is join invariant and congruence permutable. \item[(iii)] Every join invariant variety is ideal determined. \end{itemize}} An example is given for which both inclusions in (i) are proper. Furthermore, it is shown: The variety of implication algebras is ideal determined, but not join invariant. The variety of distributive \text{EQ}-lattices is join invariant, but not \(0\)-coherent. The authors provide an example of a variety which is join invariant but not congruence permutable, whereas it is still open whether a congruence permutable \(0\)-normal variety must be join invariant. Last but not least, let us note that a Mal'cev condition for join invariance is established.
scientific article; zbMATH DE number 6494312