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**The strong endomorphism kernel property in Ockham algebras.**
*(English)*
Zbl 1148.06005

An endomorphism of an algebra \(\mathcal A\) is said to be strong if it is compatible with every congruence on \(\mathcal A\), and \(\mathcal A\) is said to have the strong endomorphism kernel property (SEKP in this review) if every congruence on \(\mathcal A\), different from the universal congruence, is the kernel of a strong endomorphism on \(\mathcal A\).

It is proved that an algebra with SEKP has at most one maximal congruence, hence a finite such algebra is directly indecomposable. The only finite distributive lattice with SEKP is the two-element chain. A semisimple algebra has SEKP if and only if it is simple.

Then the paper focuses on Ockham algebras, which are bounded distributive lattices \(L\) endowed with a dual endomorphism \(f\). The Berman classes \({\mathbf K}_{pq}\) of Ockham algebras are defined by the condition \(f^q=f^{2p+q}\), where \(p\geq1\) and \(q\geq0\). An Ockham algebra in \({\mathbf K}_{11}\) has SEKP if and only if it is simple.

The authors study SEKP mostly in terms of the dual space \({\mathcal X}=(X,g)\) of an Ockham algebra. Let \(C_g(X)\) be the set of the closed subsets \(Q\subseteq X\) such that \(g(Q)\subseteq Q\), and let \(\Gamma({\mathcal X})\) be the set of the endomorphisms \(\alpha\) of \(\mathcal X\) such that \(\alpha(Q)\subseteq Q\) for every \(Q\in C_g(X)\). An Ockham algebra has SEKP if and only if for every nonempty subset \(Q\in C_g(X)\) there exists \(\alpha\in\Gamma({\mathcal X})\) such that \(\alpha(X)=Q\).

The last two sections deal with \({\mathbf K}_{11}\)-algebras and its subclass \({\mathbf {MS}}\) of De Morgan-Stone algebras. The structure of the dual space of an \({\mathbf {MS}}\)-algebra with SEKP is described in terms of 1-point compactifications of discrete spaces.

It is proved that an algebra with SEKP has at most one maximal congruence, hence a finite such algebra is directly indecomposable. The only finite distributive lattice with SEKP is the two-element chain. A semisimple algebra has SEKP if and only if it is simple.

Then the paper focuses on Ockham algebras, which are bounded distributive lattices \(L\) endowed with a dual endomorphism \(f\). The Berman classes \({\mathbf K}_{pq}\) of Ockham algebras are defined by the condition \(f^q=f^{2p+q}\), where \(p\geq1\) and \(q\geq0\). An Ockham algebra in \({\mathbf K}_{11}\) has SEKP if and only if it is simple.

The authors study SEKP mostly in terms of the dual space \({\mathcal X}=(X,g)\) of an Ockham algebra. Let \(C_g(X)\) be the set of the closed subsets \(Q\subseteq X\) such that \(g(Q)\subseteq Q\), and let \(\Gamma({\mathcal X})\) be the set of the endomorphisms \(\alpha\) of \(\mathcal X\) such that \(\alpha(Q)\subseteq Q\) for every \(Q\in C_g(X)\). An Ockham algebra has SEKP if and only if for every nonempty subset \(Q\in C_g(X)\) there exists \(\alpha\in\Gamma({\mathcal X})\) such that \(\alpha(X)=Q\).

The last two sections deal with \({\mathbf K}_{11}\)-algebras and its subclass \({\mathbf {MS}}\) of De Morgan-Stone algebras. The structure of the dual space of an \({\mathbf {MS}}\)-algebra with SEKP is described in terms of 1-point compactifications of discrete spaces.

Reviewer: Sergiu Rudeanu (Bucureşti)

### MSC:

06D30 | De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) |

08A35 | Automorphisms and endomorphisms of algebraic structures |

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\textit{T. S. Blyth} and \textit{H. J. Silva}, Commun. Algebra 36, No. 5, 1682--1694 (2008; Zbl 1148.06005)

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### References:

[1] | Blyth T. S., Lattices and Ordered Algebraic Structures (2005) · Zbl 1073.06001 |

[2] | Blyth T. S., Ockham Algebras (1994) |

[3] | DOI: 10.1081/AGB-120037216 · Zbl 1060.06018 |

[4] | Burris S., A Course in Universal Algebra 78 (1981) · Zbl 0478.08001 |

[5] | Davey B. A., Introduction to Lattices and Order., 2. ed. (2002) · Zbl 1002.06001 |

[6] | McKenzie R. N., Algebras, Lattices, Varieties 1 (1987) |

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