On the automorphism group of the superextension of a semigroup

2021

Carpathian Math. Publ.

A subset $B$ of a group $G$ is called a basis of $G$ if each element $g\in G$ can be written as $g=ab$ for some elements $a,b\in B$. The smallest cardinality $|B|$ of a basis $B\subseteq G$ is called the basis size of $G$ and is denoted by $r[G]$. We prove that each finite group $G$ has $r[G]>\sqrt{|G|}$. If $G$ is Abelian, then $r[G]\ge \sqrt{2|G|-|G|/|G_2|}$, where $G_2=\{g\in G:g^{-1} = g\}$. Also we calculate the basis sizes of all Abelian groups of order $\le 60$ and all non-Abelian groups of order $\le 40$.

Section: Theorem 4 ([13]mentioning

confidence: 99%

Bases in finite groups of small order

Банах

2021

Carpathian Math. Publ.

“…In this paper, we investigate the automorphism groups of the extensions N k (S) of a semigroup S. The thorough study of various extensions of semigroups was started in [13] and continued in [1][2][3][4][5][6][7][8][9][10][14][15][16][17][18][19]. The largest among these extensions is the semigroup υ(S) of all upfamilies on S. A family A of non-empty subsets of a set X is called an upfamily if for each set A ∈ A any subset B ⊃ A of X belongs to A.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of semigroups of k-linked upfamilies

2018

J Math Sci

A family A of non-empty subsets of a set X is called an upfamily if for each set A ∈ A any set B ⊃ A belongs to A.Any associative binary operation * : X × X → X can be extended to an associative binary operation * : N k (X)×N k (X) → N k (X). In the paper, we study automorphisms of the extensions of groups, finite monogenic semigroups and describe the automorphism groups of extensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups.

“…The thorough study of automorphism groups of superextensions of semigroups was started in [19] and continued in [9]. In [19] it was shown that each automorphism of a semigroup S can be extended to an automorphism of its superextension λ(S) and the automorphism group Aut (λ(S)) of the superextension of a semigroup S contains a subgroup, isomorphic to the automorphism group Aut (S) of S. Also the automorphism groups of superextensions of null semigroups, almost null semigroups, right zero semigroups, left zero semigroups and all three-element semigroups were described. In [9] we studied the automorphism groups of superextensions of groups and described the automorphism groups Aut (λ(G)) of the superextensions of all groups G of cardinality |G| ≤ 5.…”

Section: Introductionmentioning

confidence: 99%

Automorphism groups of superextensions of groups

Банах

2017

Self Cite

The superextension λ(X) consists of all maximal linked families on X. Any associative binary operation * : X × X → X can be extended to an associative binary operation * colonλ(X) × λ(X) → λ(X). In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of order ≤ 5.