## 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.

### MSC:

 06A12 Semilattices 06D15 Pseudocomplemented lattices

### Keywords:

pseudocomplemented semilattice; congruence kernel
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.