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Start Over Creator Pócs, Jozef

1. Cardinality of retracts of monounary algebras

Creator:
Jakubíková-Studenovská, Danica and Pócs, Jozef
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
monounary algebra, retract, and cardinality
Language:
English
Description:
For an uncountable monounary algebra $(A,f)$ with cardinality $\kappa $ it is proved that $(A,f)$ has exactly $2^{\kappa }$ retracts. The case when $(A,f)$ is countable is also dealt with.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. On ⋆-associated comonotone functions

Creator:
Hutník, Ondrej and Pócs, Jozef
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
comonotone functions, binary operation, ⋆-associatedness, and Sugeno integral
Language:
English
Description:
We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper \cite{BK}. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to +-associatedness of functions (as proved by Boczek and Kaluszka), but also to their ⋆-associatedness with ⋆ being an arbitrary strictly monotone and right-continuous binary operation. The second open problem deals with an existence of a pair of binary operations for which the generalized upper and lower Sugeno integrals coincide. Using a fairly elementary observation we show that there are many such operations, for instance binary operations generated by infima and suprema preserving functions.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. Test elements and the Retract Theorem for monounary algebras

Creator:
Jakubíková-Studenovská, Danica and Pócs, Jozef
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
monounary algebra, retract, and test element
Language:
English
Description:
The term “Retract Theorem” has been applied in literature in connection with group theory. In the present paper we prove that the Retract Theorem is valid (i) for each finite structure, and (ii) for each monounary algebra. On the other hand, we show that this theorem fails to be valid, in general, for algebras of the form $\mathcal{A}=(A,F)$, where each $f\in F$ is unary and $\operatorname{card}F >1$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public