Lattice of retracts of monounary algebras

“…The present paper can be regarded as a continuation of the authors' article [6] where several properties of the lattice of all retracts of monounary algebras were investigated. Now we focus mainly on finite retract lattices of monounary algebras.…”

Section: Introductionmentioning

confidence: 99%

“…We recall some results contained in [6]. The system of all retracts of a given monounary algebra (A, f ) is denoted R(A, f ).…”

Section: Lattice Of Retracts Of Monounary Algebrasmentioning

confidence: 99%

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On finite retract lattices of monounary algebras

Jakubíková-Studenovská

Pócs

2012

Mathematica Slovaca

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ABSTRACT. 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.

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“…For an explanation of the basic terms in universal algebra and monounary algebras, we recommend [1], [7].…”

Section: Preliminariesmentioning

confidence: 99%

The endomorphism spectrum of a monounary algebra

Jakubíková-Studenovská

Potpinková

2014

Mathematica Slovaca

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ABSTRACT. The endomorphism spectrum spec A of an algebra A is defined as the set of all positive integers, which are equal to the number of elements in an endomorphic image of A, for all endomorphisms of A. In this paper we study finite monounary algebras and their endomorphism spectrum. If a finite set S of positive integers is given, one can look for a monounary algebra A with S = spec A. We show that for countably many finite sets S, no such A exists. For some sets S, an appropriate A with spec A = S are described.For n ∈ N it is easy to find a monounary algebra A with {1, 2, . . . , n} = spec A. It will be proved that if i ∈ N, then there exists a monounary algebra A such that spec A skips i consecutive (consecutive eleven, consecutive odd, respectively) numbers.Finally, for some types of finite monounary algebras (binary and at least binary trees) A, their spectrum is shown to be complete.

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Lattice of retracts of monounary algebras

Abstract: ABSTRACT. 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.

Cited by 13 publications

References 10 publications

The number of monounary algebras

The number of monounary algebras

On finite retract lattices of monounary algebras

The endomorphism spectrum of a monounary algebra

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