## Strong endomorphism kernel property for Brouwerian algebras.(English)Zbl 1333.06025

An algebra $$A$$ has the endomorphism kernel property (EKP for short) if every congruence relation on $$A$$ different from the greatest one is the kernel of an endomorphism on $$A$$. Let $$\Theta$$ be congruence on $$A$$ and $$f$$ be an endomorphism of $$A$$. Then $$f$$ is said to be compatible with $$\Theta$$ if $$(a,b)\in\Theta$$ implies $$(f(a),f(b))\in\Theta$$. Endomorphism $$f$$ is strong if it is compatible with every congruence on $$A$$. An algebra $$A$$ has the strong endomorphism kernel property (SEKP for short) if every congruence relation on $$A$$ different from the greatest one is the kernel of a strong endomorphism on $$A$$. There are determined classes of algebras for which EKP or SEKP are preserved by finite direct products and also by a special form of infinite direct sum construction. Full characterization of finite relatively Stone algebras and L-algebras which have SEKP is given and a wide class of infinite relative Stone algebras which posses SEKP is described.

### MSC:

 06D20 Heyting algebras (lattice-theoretic aspects) 03G25 Other algebras related to logic 08A30 Subalgebras, congruence relations 08A35 Automorphisms and endomorphisms of algebraic structures 08B26 Subdirect products and subdirect irreducibility
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