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

Chajda, Ivan

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 41, 35-37 (2002).

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.

Reviewer: Jürg Schmid (Bern)

MSC:

06A12	Semilattices
06D15	Pseudocomplemented lattices

Keywords:

pseudocomplemented semilattice; congruence kernel

References:

[1]	Agliano P., Ursini A.: On subtractive varieties III: From ideals to congruences. Algebra Universalis 37 (1997), 296-333. · Zbl 0906.08005
[2]	Chajda I., Länger H.: Ideals in locally regular and permutable at 0 varieties. Contributions to General Algebra 13 (2001), 63-72. · Zbl 0989.08003
[3]	Frink O.: Pseudo-complements in semi-lattices. Duke Math. J. 29 (1962), 505-514. · Zbl 0114.01602