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A note on normal varieties of monounary algebras

Ivan Chajda, Helmut Länger (2002)

Czechoslovak Mathematical Journal

A variety is called normal if no laws of the form s = t are valid in it where s is a variable and t is not a variable. Let L denote the lattice of all varieties of monounary algebras ( A , f ) and let V be a non-trivial non-normal element of L . Then V is of the form M o d ( f n ( x ) = x ) with some n > 0 . It is shown that the smallest normal variety containing V is contained in H S C ( M o d ( f m n ( x ) = x ) ) for every m > 1 where C denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of L consisting of all normal elements of...

Algebraic approach to locally finite trees with one end

Bohdan Zelinka (2003)

Mathematica Bohemica

Let T be an infinite locally finite tree. We say that T has exactly one end, if in T any two one-way infinite paths have a common rest (infinite subpath). The paper describes the structure of such trees and tries to formalize it by algebraic means, namely by means of acyclic monounary algebras or tree semilattices. In these algebraic structures the homomorpisms and direct products are considered and investigated with the aim of showing, whether they give algebras with the required properties. At...

Atomary tolerances on finite algebras

Bohdan Zelinka (1996)

Mathematica Bohemica

A tolerance on an algebra is defined similarly to a congruence, only the requirement of transitivity is omitted. The paper studies a special type of tolerance, namely atomary tolerances. They exist on every finite algebra.

Cardinalities of lattices of topologies of unars and some related topics

Anna Kartashova (2001)

Discussiones Mathematicae - General Algebra and Applications

In this paper we find cardinalities of lattices of topologies of uncountable unars and show that the lattice of topologies of a unar cannor be countably infinite. It is proved that under some finiteness conditions the lattice of topologies of a unar is finite. Furthermore, the relations between the lattice of topologies of an arbitrary unar and its congruence lattice are established.

Cardinality of retracts of monounary algebras

Danica Jakubíková-Studenovská, Jozef Pócs (2008)

Czechoslovak Mathematical Journal

For an uncountable monounary algebra ( A , f ) with cardinality κ it is proved that ( A , f ) has exactly 2 κ retracts. The case when ( A , f ) is countable is also dealt with.

Congruence lattices of intransitive G-Sets and flat M-Sets

Steve Seif (2013)

Commentationes Mathematicae Universitatis Carolinae

An M-Set is a unary algebra 〈 X , M 〉 whose set M of operations is a monoid of transformations of X ; 〈 X , M 〉 is a G-Set if M is a group. A lattice L is said to be represented by an M-Set 〈 X , M 〉 if the congruence lattice of 〈 X , M 〉 is isomorphic to L . Given an algebraic lattice L , an invariant Π ( L ) is introduced here. Π ( L ) provides substantial information about properties common to all representations of L by intransitive G-Sets. Π ( L ) is a sublattice of L (possibly isomorphic to the trivial lattice), a Π -product lattice. A Π -product...

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