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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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