arXiv subjects

Georg Klein

Publications and source records attributed to Georg Klein.

8 recordsLinked to original sources

arXiv2022

Combinatorial results for order-preserving partial injective contraction mappings

Let $ \mathcal{I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots , n\}$. Let $\mathcal{OCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving injective partial contraction mappings, and let $\mathcal{ODCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving and order-decreasing injective partial contraction mappings of $X_n$. In this paper, we investigate the cardinalities of some equivalences on $\mathcal{OCI}_n$ and $\mathcal{ODCI}_n$ which lead naturally to obtaining the order of these semigroups. Then, we relate the formulae obtained to Fibonacci numbers. Similar results about $\mathcal{ORCI}_n$, the semigroup of order-preserving or order-reversing injective partial contraction mappings, are deduced.

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arXiv2022

Operadic approach to HNN-extensions of Leibniz algebras

We construct HNN-extensions of Lie di-algebras in the variety of di-algebras and provide a presentation for the replicated HNN-extension of a Lie di-algebras. Then, by applying the method of Gr\"obner-Shirshov bases for replicated algebras, we obtain a linear basis. As an application of HNN-extensions, we prove that Lie di-algebras are embedded in their HNN-extension.

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arXiv2017

A note on identities in plactic monoids and monoids of upper-triangular tropical matrices

This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank $n$ cannot satisfy a non-trivial identity of length less than or equal to $n$. A new identity is then proven to hold for the monoid of $n \times n$ upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank $3$ into the direct product of two copies of the monoid of $3\times 3$ upper-triangular tropical matrices, giving a new proof that the plactic monoid of rank $3$ satisfies a non-trivial identity.

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arXiv2014

Construction of a two unique product semigroup defined by permutation relations of quaternion type

For a regular representation $H \subseteq \text{Sym}_n$ of the generalized quaternion group of order $n=4k$, with $k\geq 2$, the monoid $S_n(H)$ presented with generators $a_1,a_2,\dots ,a_n$ and with relations $a_1a_2\cdots a_n=a_{\sigma(1)}a_{\sigma(2)}\cdots a_{\sigma(n)}$, for all $\sigma\in H$, is investigated. It is shown that $S_n(H)$ has the two unique product property. As a consequence, for any field $K$, the monoid algebra $K[S_n(H)]$ is a domain with trivial units which is semiprimitive.

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arXiv2014

Finitely presented algebras defined by permutation relations of dihedral type

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},\ldots , a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}\cdots a_{n} =a_{\sigma (1)} a_{\sigma (2)} \cdots a_{\sigma (n)}$, where $\sigma$ runs through a subset $H$ of the symmetric group $\text{Sym}_{n}$ of degree $n$, is investigated. Groups $H$ in which the cyclic group $\langle (1,2, \ldots ,n) \rangle$ is a normal subgroup of index $2$ are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid $S_n(H)$ with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovski\u{\i}. The universal group $G_n$ of $S_n(H)$ is a unique product group, and it is the central localization of a cancellative subsemigroup of $S_n(H)$. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra $K[S_n(H)]$ is semiprimitive.

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arXiv2014

Group algebras and semigroup algebras defined by permutation relations of fixed length

Let $H$ be a subgroup of $\text{Sym}_n$, the symmetric group of degree $n$. For a fixed integer $l \geq 2$, the group $G$ presented with generators $x_1, x_2, \ldots ,x_n$ and with relations $x_{i_1}x_{i_2}\cdots x_{i_l} =x_{\sigma (i_1)} x_{\sigma (i_2)} \cdots x_{\sigma (i_l)}$, where $\sigma$ runs through $H$, is considered. It is shown that $G$ has a free subgroup of finite index. For a field $K$, properties of the algebra $K[G]$ are derived. In particular, the Jacobson radical $\mathcal{J}(K[G])$ is always nilpotent, and in many cases the algebra $K[G]$ is semiprimitive. Results on the growth and the Gelfand-Kirillov dimension of $K[G]$ are given. Further properties of the semigroup $S$ and the semigroup algebra $K[S]$ with the same presentation are obtained, in case $S$ is cancellative. The Jacobson radical is nilpotent in this case as well, and sufficient conditions for the algebra to be semiprimitive are given.

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arXiv2013

Finitely Presented Monoids and Algebras defined by Permutation Relations of Abelian Type, II

The class of finitely presented algebras A over a field K with a set of generators x_{1},...,x_{n} and defined by homogeneous relations of the form x_{i_1}x_{i_2}...x_{i_l}=x_{sigma(i_1)}x_{sigma(i_2)}...x_{sigma(i_l)}, where l geq 2 is a given integer and sigma runs through a subgroup H of Sym_n, is considered. It is shown that the underlying monoid S_{n,l}(H)= is cancellative if and only if H is semiregular and abelian. In this case S_{n,l}(H) is a submonoid of its universal group G. If, furthermore, H is transitive then the periodic elements T(G) of G form a finite abelian subgroup, G is periodic-by-cyclic and it is a central localization of S_{n,l}(H), and the Jacobson radical of the algebra A is determined by the Jacobson radical of the group algebra K[T(G)]. Finally, it is shown that if H is an arbitrary group that is transitive then K[S_{n,l}(H)] is a Noetherian PI-algebra of Gelfand-Kirillov dimension one; if furthermore H is abelian then often K[G] is a principal ideal ring. In case H is not transitive then K[S_{n,l}(H)] is of exponential growth.

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arXiv2010

Finitely Presented Monoids and Algebras defined by Permutation Relations of Abelian Type

The class of finitely presented algebras over a field K with a set of generators a_1,...,a_n and defined by homogeneous relations of the form a_1a_2...a_n = a_{sigma(1)}a_{sigma(2)}...a_{sigma(n)}, where sigma runs through an abelian subgroup H of Sym_{n}, the symmetric group, is considered. It is proved that the Jacobson radical of such algebras is zero. Also, it is characterized when the monoid S_n(H), with the "same" presentation as the algebra, is cancellative in terms of the stabilizer of 1 and the stabilizer of n in H. This work is a continuation of earlier work of Cedo, Jespers and Okninski.

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