commutative unary algebra

On Congruence Lattices of Direct Sums of Strongly Connected Commutative Unary Algebras

Authors: Kartashova A. V.
A union of mutually disjoint unary algebras is called their direct sum. A unary algebra is said to be strongly connected if it is generated by its arbitrary element. In the present paper we investigate congruence lattices of the class of all algebras with finitely many operations whose every connected component is strongly connected. We give a necessary and sufficient condition for an algebra from this class to have a distributive congruence lattice (Theorem 1). Besides, all distributive congruence lattices of algebras from the above class are discribed (Theorem 2).

On Conditions for Distributivity or Modularity of Congruence Lattices of Commutative Unary Algebras

Authors: Kartashov V. K., Kartashova A. V.
The paper is devoted to the problem of describing unary algebras whose congruence lattices have a given property. By now this problem has been solved for algebras with one unary operation. In the paper it is shown that this problem is much more difficult for arbitrary commutative unary algebras. We give some necessary conditions for such lattices to be distributive or modular. Besides, it is proved here that a lattice of all subsets of a set is isomorphic to the congruence lattice of a suitable connected commutative unary algebra.