Automorphism groups of semigroups of upfamilies

Gavrylkiv, Volodymyr

doi:10.1142/s1793557120500990

Cited by 1 publication

(3 citation statements)

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“…Taking into account that for any group (G, ⊣) of order |G| > 1 the semigroup (υ(G), ⊣) contains at least two right zeros: {G} and ⟨{g} : g ∈ G⟩, see [13,Prop. 15,18], we conclude that (υ(G), ⊣) is not a group, and hence the doppelsemigroup (υ(G), ⊣, ⊢) is not a group doppelsemigroup.…”

Section: The Upfamily Functor In the Categorymentioning

confidence: 81%

“…The doppelsemigroup υ(O 2 ). It was proved in [18] that the upfamily extension of a null semigroup is a null semigroup as well, and Aut (υ(O n )) = S |υ(On)|−1 for n > 1. We conclude that υ(O 2 ) ∼ = O 4 and Aut (υ(O 2 )) ∼ = S 3 .…”

Section: The Upfamily Functor In the Categorymentioning

confidence: 99%

“…In this paper, we investigate the extension (υ(D), ⊣, ⊢) of a doppelsemigroup (D, ⊣, ⊢). The thorough study of various extensions of semigroups was started in [13] and continued in [1]- [8], [14]- [18]. The largest among these extensions is the semigroup υ(S) of all upfamilies on a semigroup S. The extension υ(S) is called the upfamily extension of S. A family U of nonempty subsets of a set X is called an upfamily if for each set U ∈ U any subset F ⊃ U belongs to U (In [13] instead of the notion "upfamily" it was used the notion "inclusion hyperspace").…”

mentioning

confidence: 99%

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On the upfamily extension of a doppelsemigroup

Gavrylkiv

2024

Mat. Stud.

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A family $\mathcal{U}$ of non-empty subsets of a set $D$ is called an {\em upfamily} if for each set $U\in\mathcal{U}$ any set $F\supset U$ belongs to $\mathcal{U}$. The upfamily extension $\upsilon(D)$ of $D$ consists of all upfamilies on~$D$.Any associative binary operation $* \colon D\times D \to D$ can be extended to an associative binary operation $$*:\upsilon(D)\times \upsilon(D)\to \upsilon(D), \ \ \ \mathcal U*\mathcal V=\big\langle\bigcup_{a\inU}a*V_a:U\in\mathcal U,\;\;\{V_a\}_{a\in U}\subset\mathcal V\big\rangle.$$In the paper, we show that the upfamily extension $(\upsilon(D),\dashv,\vdash)$ of a (strong) doppelsemigroup $(D,\dashv,\vdash)$ is a (strong) doppelsemigroup as well and study some properties of this extension. Also we introduce the upfamily functor in the category $\mathbf {DSG}$ whose objects are doppelsemigroups and morphisms are doppelsemigroup homomorphisms. We prove that the automorphism group of the upfamily extension of a doppelsemigroup $(D,\dashv, \vdash)$ of cardinality $|D|\geq 2$ contains a subgroup, isomorphic to $C_2\times \mathrm{Aut\mkern 2mu}(D,\dashv, \vdash)$. Also we describe the structure of upfamily extensions of all two-element doppelsemigroups and their automorphism groups.

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Section: The Upfamily Functor In the Categorymentioning

confidence: 81%

Section: The Upfamily Functor In the Categorymentioning

confidence: 99%