A characterization of congruence kernels in pseudocomplemented semilattices. (English) Zbl 1042.06002

The author considers the question which (nonempty) subsets \(I\) of a pseudocomplemented semilattice (for short: a PCS) \({\mathcal{S}}=\langle S;\wedge,^*,0\rangle\) are congruence kernels, that is, occur as the congruence class of \(0\) for some PCS-congruence on \(\mathcal{S}\). He shows that this is the case iff (i) \(x\in I\) and \(a\in S\) jointly imply that \(x\wedge a\in I\) and (ii) \((x^*\wedge y^*)^*\in I\) whenever \(x,y\in I\).
Reviewer’s remark: Let \(\text{Sk} ({\mathcal{S}})=\{x\in S;\;x = x^{**}\}\) be the skeleton of \(\mathcal{S}\) which is a sub-PCS of \(\mathcal{S}\) and a Boolean algebra under the operations \(^*\), \(\wedge\) and \(x\oplus y=(x^*\wedge y^*)^*\). Then the conditions stated above just say that \(I\) is a congruence kernel of \(\mathcal{S}\) iff \(I\) is the inverse image of some (Boolean) ideal \(I_B\subseteq \text{Sk} (\mathcal{S}) \) under the Glivenko homomorphism \(\gamma:{\mathcal{S}}\rightarrow \text{Sk}({\mathcal{S}})\) given by \(\gamma(x)=x^{**}\); so PCS congruence kernels occur as most natural preimages of Boolean congruence kernels.


06A12 Semilattices
06D15 Pseudocomplemented lattices
Full Text: EuDML


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