An endomorphism h of a group G is said to be strong whenever for every congruenceθon G,(x,y)∈θimplies(h(x),h(y))∈θfor every x,y∈G.A group G is said to have the strong endomorphism kernel property if every congru...An endomorphism h of a group G is said to be strong whenever for every congruenceθon G,(x,y)∈θimplies(h(x),h(y))∈θfor every x,y∈G.A group G is said to have the strong endomorphism kernel property if every congruence on G is the kernel of a strong endomorphism.In this note,we study the strong endomorphism kernel property in the class of Abelian groups.In particular,we show that a finite Abelian group has the strong endomorphism kernel property if and only if it is cyclic.展开更多
Here we introduce a subclass of the class of Ockham algebras （L; f） for which L satisfies the property that for every x ∈ L, there exists n 〉 0 such that fn（x） and f^n＋1（x） are complementary. We characterize ...Here we introduce a subclass of the class of Ockham algebras （L; f） for which L satisfies the property that for every x ∈ L, there exists n 〉 0 such that fn（x） and f^n＋1（x） are complementary. We characterize the structure of the lattice of congruences on such an algebra （L; f）. We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.展开更多
基金The authors would like to express their appreciation to the referee for helpful comments and suggestions.
文摘An endomorphism h of a group G is said to be strong whenever for every congruenceθon G,(x,y)∈θimplies(h(x),h(y))∈θfor every x,y∈G.A group G is said to have the strong endomorphism kernel property if every congruence on G is the kernel of a strong endomorphism.In this note,we study the strong endomorphism kernel property in the class of Abelian groups.In particular,we show that a finite Abelian group has the strong endomorphism kernel property if and only if it is cyclic.
文摘Here we introduce a subclass of the class of Ockham algebras （L; f） for which L satisfies the property that for every x ∈ L, there exists n 〉 0 such that fn（x） and f^n＋1（x） are complementary. We characterize the structure of the lattice of congruences on such an algebra （L; f）. We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.
