Polymorphism-Homogeneous Monounary Algebras

Farkasová, Zuzana; Jakubíková-Studenovská, Danica

doi:10.1515/ms-2015-0028

Cited by 5 publications

(6 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Polymorphism-homogeneous monounary algebras were characterized by Z. Farkasová and D. Jakubíková-Studenovská in [6] using Proposition 2.2 and the description of homomorphism-homogeneous monounary algebras obtained by É. Jungábel and D. Mašulović [13]. As an illustration of the results of Section 3, we present a simple self-contained proof, which relies on the following technical lemma about quantifier elimination in monounary algebras.…”

Section: Monounary Algebras a Monounary Algebra Is An Algebramentioning

confidence: 92%

“…If A is k-polymorphism-homogeneous for every natural number k, then we say that A is polymorphism-homogeneous [15]. These two notions are linked by the following result, which was proved for relational structures by C. Pech and M. Pech [15] and for algebraic structures by Z. Farkasová and D. Jakubíková-Studenovská [6], but the same proof works for arbitrary first-order structures.…”

Section: Polymorphism-homogeneitymentioning

confidence: 92%

See 1 more Smart Citation

Polymorphism-homogeneity and universal algebraic geometry

Tóth¹,

Waldhauser²

2020

Preprint

View full text Add to dashboard Cite

We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is polymorphism-homogeneous. We show that polymorphism-homogeneity is also equivalent to the property that algebraic sets (i.e., solution sets of systems of equations) are exactly those sets of tuples that are closed under the centralizer clone of the algebra. Furthermore, we prove that the aforementioned properties hold if and only if the algebra is injective in the category of its finite subpowers. We also consider two additional conditions: a stronger variant for polymorphism-homogeneity and for injectivity, and we describe explicitly the finite semilattices, lattices, Abelian groups and monounary algebras satisfying any one of these three conditions. Key words and phrases. Polymorphism-homogeneity, algebraic set, universal algebraic geometry, solution set of a system of equations, quantifier elimination, injective algebra.1 Note that the word variety has a different meaning in universal algebra: a variety is an equationally definable class of algebras, or, equivalently, a class of algebras that is closed under homomorphic images, subalgebras and direct products.

show abstract

Section: Monounary Algebras a Monounary Algebra Is An Algebramentioning

confidence: 92%

Section: Polymorphism-homogeneitymentioning

confidence: 92%

Polymorphism-homogeneity and universal algebraic geometry

Tóth¹,

Waldhauser²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…If A is k-polymorphismhomogeneous for every natural number k, then we say that A is polymorphism-homogeneous [15]. These two notions are linked by the following result, which was proved for relational structures by C. Pech and M. Pech [15] and for algebraic structures by Z. Farkasová and D. Jakubíková-Studenovská [6], but the same proof works for arbitrary first-order structures.…”

Section: Polymorphism-homogeneitymentioning

confidence: 92%

“…Proposition 2.3 ( [6,15]). A first-order structure A is polymorphism-homogeneous if and only if A k is homomorphism-homogeneous for all natural numbers k.…”

Section: Polymorphism-homogeneitymentioning

confidence: 99%

Polymorphism-homogeneity and universal algebraic geometry

Tóth¹,

Waldhauser²

2022

Discrete Mathematics &Amp; Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…A rather large series of further algebraic notions is based on endomorphisms. Some properties of monounary algebras connected with the notion of homomorphism were studied by several authors ( [5], [10], [3], [11], [12]). …”

Section: Introductionmentioning

confidence: 99%

The endomorphism spectrum of a monounary algebra

Jakubíková-Studenovská

Potpinková

2014

Mathematica Slovaca

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Polymorphism-Homogeneous Monounary Algebras

Cited by 5 publications

References 16 publications

Polymorphism-homogeneity and universal algebraic geometry

Polymorphism-homogeneity and universal algebraic geometry

Polymorphism-homogeneity and universal algebraic geometry

The endomorphism spectrum of a monounary algebra

Contact Info

Product

Resources

About