Superextensions of cyclic semigroups

2017

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. An upfamily L of subsets of X is said to be linked if A ∩ B ̸ = ∅ for all A, B ∈ L. A linked upfamily M of subsets of X is maximal linked if M coincides with each linked upfamily L on X that contains M. The superextension λ(X) consists of all maximal linked upfamilies on X. Any associative binary operation * : X ×X → X can be extended to an associative binary operation * : λ(X) × λ(X) → λ(X). In the paper we study automorphisms of superextensions of semigroups and describe the automorphism groups of superextensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups. Also we find the automorphism groups of superextensions of all semigroups S of order |S| ≤ 3.

“…If φ : S → S ′ is a homomorphism of semigroups, then λφ : λ(S) → λ(S ′ ) is a homomorphism as well, see [17].…”

Section: X|mentioning

confidence: 99%

On the automorphism group of the superextension of a semigroup

2017

“…A homomorphism ρ : S → S of a semigroup S is called a homomorphic retraction if ρ • ρ = ρ. The functoriality of the superextension in the category of semigroups [15] ensures that for any homomorphic retraction ρ : S → S the map λρ : λ(S) → λ(S) is a homomorphic retraction, too.…”

Section: Semigroups Possessing a Good Shiftmentioning

confidence: 99%

“…On the other hand, the definition of the semigroup operation on Proof. If two semigroups are isomorphic, then their superextensions are isomorphic by the functoriality of the superextension in the category of semigroups [15]. Now assume that two monogenic semigroups M i,m and M j,n have isomorphic superextensions.…”

Section: Automorphisms and Characteristic Ideals Of Superextensions Omentioning

confidence: 99%

“…In this paper we investigate the automorphism group of the superextension λ(S) of a finite monogenic semigroup S. The thorough study of various extensions of semigroups was started in [14] and continued in [1]- [10] and [15]- [18]. 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%

“…If ϕ : S → S ′ is a homomorphism of semigroups, then λϕ : λ(S) → λ(S ′ ) is a homomorphism as well, see [15]. A non-empty subset I of a semigroup S is called an ideal if IS ∪ SI ⊂ I.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Automorphism groups of superextensions of groups

Банах

2017

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.

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.