## 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.

### MSC:

 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) 08A35 Automorphisms and endomorphisms of algebraic structures
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### References:

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