**171**

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**Semigroup Structure of Sets of Solutions to Equation X^s = X^m**

Porubský, Štefan

2017 - English
Using an idempotent semigroup approach we describe the semigroup and group structure of the set of solutions to equation X^m = X^s in successive steps over a periodic commutative semigroup, over multiplicative semigroups of factor rings of residually finite commutative rings and finally over multiplicative semigroups of factor rings of residually finite commutative principal ideal domains. The analysis is done through the use of the maximal subsemigroups and groups corresponding to an idempotent of the corresponding semigroup and in the case of residually finite PID’s employing the available analysis of the Euler-Fermat Theorem as given in [11]. In particular the case when this set of solutions is a union of groups is handled. As a simple application we show a not yet noticed group structure of the set of solutions to x^n = x connected with the message space of RSA cryptosystems and Fermat pseudoprimes.
Keywords:
*set of solutions; idempotent; maximal semigroup corresponding to an idempotent; maximal group corresponding to an idempotent; equation X^s = X^m; finite commutative ring with identity element; residually finite commutative principal ideal domains*
Available on request at various institutes of the ASCR
Semigroup Structure of Sets of Solutions to Equation X^s = X^m

Using an idempotent semigroup approach we describe the semigroup and group structure of the set of solutions to equation X^m = X^s in successive steps over a periodic commutative semigroup, over ...

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**Idempotents, Group Membership and their Applications**

Porubský, Štefan

2017 - English
S.Schwarz in his paper [165] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought [167] into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.
Keywords:
*multiplicative semigroup; finite semigroups; power semigroups; idempotent elements; finite commutative rings; principal ideal domain; Euler-Fermat theorem; Wilson theorem; matrices over fields; maximal groups contained in a semigroup; periodic sequence; multiplicative semigroup of Zm; semigroup of circulant Boolean matrices*
Available on request at various institutes of the ASCR
Idempotents, Group Membership and their Applications

S.Schwarz in his paper [165] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought [167] into play the maximal subsemigroups and thus he embodied the ...

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**Application and Misapplication of the Czechoslovak STP Cipher During WWII - Report on an Unpublished Manuscript**

Porubský, Štefan

2017 - English
Keywords:
*STP cipher; Josef Růžek; Karol Cigáň; František Moravec; Czechoslovak military cryptography; Word War II*
Available on request at various institutes of the ASCR
Application and Misapplication of the Czechoslovak STP Cipher During WWII - Report on an Unpublished Manuscript

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**Real-valued Score Function, New Description of Continuous Random Variables and the Central Limit Theorem**

Fabián, Zdeněk

2016 - English
Keywords:
*score function; transformation-based score; generalized moment method; new descriptive characteristic of the data*
Available on request at various institutes of the ASCR
Real-valued Score Function, New Description of Continuous Random Variables and the Central Limit Theorem

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**On Nominal Automata as Models of Java-like Object-Oriented Programs**

Suzuki, Tomoyuki

2016 - English
In this paper, we proposed a model of Java-like object-oriented programs as nominal automata and a simple method invocation checker.
Available on request at various institutes of the ASCR
On Nominal Automata as Models of Java-like Object-Oriented Programs

In this paper, we proposed a model of Java-like object-oriented programs as nominal automata and a simple method invocation checker.

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**Prediction diagnostics for Uncertain Systems**

Novák, M.; Votruba, Z.; Brandejský, T.; Faber, J.; Coufal, David; Pelikán, Emil

2014 - English
Available at various institutes of the ASCR
Prediction diagnostics for Uncertain Systems

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**The use of Score Functions of Distribution for Description of Parametric Families**

Fabián, Zdeněk

2014 - English
Keywords:
*systems of distributions; Johnson transformations; score function of distribution; parametric families*
Available on request at various institutes of the ASCR
The use of Score Functions of Distribution for Description of Parametric Families

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**Description of Continuous Distributions and Data Samples by Means of Score Functions of Distribution**

Fabián, Zdeněk

2014 - English
Keywords:
*score function; score mean; score variance; generalized Fisher information; data characteristics*
Available on request at various institutes of the ASCR
Description of Continuous Distributions and Data Samples by Means of Score Functions of Distribution

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**Robust Template Matching**

Kalina, Jan

2013 - English
Keywords:
*correlation coefficient; robust statistics; image analysis*
Available on request at various institutes of the ASCR
Robust Template Matching

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**Representations of Highly-Varying Functions by One-Hidden-Layer Networks**

Kůrková, Věra

2013 - English
Keywords:
*model complexity of neural networks; one-hidden-layer networks; highly-varying functions; tractability of representations of multivariable functions by neural networks*
Available on request at various institutes of the ASCR
Representations of Highly-Varying Functions by One-Hidden-Layer Networks

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