Some properties of retract lattices of monounary algebras (original) (raw)

On finite retract lattices of monounary algebras

Mathematica Slovaca, 2012

For a monounary algebra (A, f ) we denote R ∅ (A, f ) the system of all retracts (together with the empty set) of (A, f ) ordered by inclusion. This system forms a lattice. We prove that if (A, f ) is a connected monounary algebra and R ∅ (A, f ) is finite, then this lattice contains no diamond. Next distributivity of R ∅ (A, f ) is studied. We find a representation of a certain class of finite distributive lattices as retract lattices of monounary algebras. c 2012 Mathematical Institute Slovak Academy of Sciences 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 08A60. K e y w o r d s: monounary algebra, retract, lattice of retracts.

Lattice of retracts of monounary algebras

Mathematica Slovaca, 2011

We investigate lattices of retracts of monounary algebras. Semimodularity and concepts related to semimodularity (M-symmetry and Mac Lane's condition) are dealt with. Further, we give a description of all connected monounary algebras with modular retract lattice. c 2011 Mathematical Institute Slovak Academy of Sciences 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 08A60. K e y w o r d s: monounary algebra, retract, lattice of retracts.

Retract irreducibility of connected monounary algebras II

Czechoslovak Mathematical Journal

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Congruence lattices of connected monounary algebras

Algebra Universalis, 2020

The system of all congruences of an algebra (A, F) forms a lattice, denoted {{\\,\\mathrm{Con}\\,}}(A, F).Further,thesystemofallcongruencelatticesofallalgebraswiththebasesetAformsalattice. Further, the system of all congruence lattices of all algebras with the base set A forms a lattice.Further,thesystemofallcongruencelatticesofallalgebraswiththebasesetAformsalattice\\mathcal {E}_A.Wedealwithmeet−irreducibilityin. We deal with meet-irreducibility in.Wedealwithmeet−irreducibilityin\\mathcal {E}_AforagivenfinitesetA.Allmeet−irreducibleelementsoffor a given finite set A. All meet-irreducible elements offoragivenfinitesetA.Allmeet−irreducibleelementsof\\mathcal {E}_Aarecongruencelatticesofmonounaryalgebras.Sometypesofmeet−irreduciblecongruencelatticeswerealreadydescribed.Inthecasewhenamonounaryalgebra(A,f)isconnected,weprovenecessaryandsufficientconditionunderwhichare congruence lattices of monounary algebras. Some types of meet-irreducible congruence lattices were already described. In the case when a monounary algebra (A, f) is connected, we prove necessary and sufficient condition under whicharecongruencelatticesofmonounaryalgebras.Sometypesofmeet−irreduciblecongruencelatticeswerealreadydescribed.Inthecasewhenamonounaryalgebra(A,f)isconnected,weprovenecessaryandsufficientconditionunderwhich{{\\,\\mathrm{Con}\\,}}(A, f)isisis\\wedge -irreducible.

DR-irreducibility of connected monounary algebras

Czechoslovak Mathematical Journal, 2000

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Retract irreducibility of monounary algebras

1999

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Unoriented graphs of monounary algebras

Discrete Mathematics, 2000

A problem proposed by G. Birkho concerns the relation between ÿnite lattices and unoriented graphs. In the present paper we investigate an analogous problem concerning the relations between monounary algebras and unoriented graphs. To each monounary algebra A we assign in a natural way an unoriented graph G(A) without loops and multiple edges. We describe all monounary algebras B such that G(A) and G(B) are isomorphic. Further, we characterize all monounary algebras A having the property that whenever A1 is a monounary algebra whose unoriented graph G(A1) is isomorphic to G(A); then A1 is isomorphic to A.