Note on cyclic doppelsemigroups

Gavrylkiv, Volodymyr

doi:10.12958/adm1991

Cited by 3 publications

(5 citation statements)

References 14 publications

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“…It was proved in [19] that there exist 6 pairwise non-isomorphic commutative 2-element doppelsemigroups:…”

Section: The Upfamily Functor In the Categorymentioning

confidence: 99%

“…The following Table 2 of all two-element doppelsemigroups and their automorphism groups is taken from [19].…”

Section: The Upfamily Functor In the Categorymentioning

confidence: 99%

“…In [19], the task of describing all pairwise non-isomorphic (strong) doppelsemigroups with at most three elements has been solved. We proved that there exist 8 pairwise nonisomorphic two-element doppelsemigroups among which 6 doppelsemigroups are commutative.…”

mentioning

confidence: 99%

“…Also up to isomorphism there are 65 strong doppelsemigroups of order 3, and all non-strong doppelsemigroups are not commutative. In [20], we studied cyclic doppelsemigroups…”

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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“…It was proved in [19] that there exist 6 pairwise non-isomorphic commutative 2-element doppelsemigroups:…”

Section: The Upfamily Functor In the Categorymentioning

confidence: 99%

“…The following Table 2 of all two-element doppelsemigroups and their automorphism groups is taken from [19].…”

Section: The Upfamily Functor In the Categorymentioning

confidence: 99%

mentioning

confidence: 99%

“…Also up to isomorphism there are 65 strong doppelsemigroups of order 3, and all non-strong doppelsemigroups are not commutative. In [20], we studied cyclic doppelsemigroups…”

mentioning

confidence: 99%

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On the upfamily extension of a doppelsemigroup

Gavrylkiv

2024

Mat. Stud.

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“…Thus, semigroups and doppelsemigroups can be characterized as n-tuple semigroups. The study of doppelsemigroups was initiated by the author in [28] and then it was continued in [5,6,25,30,33,35,36,41,44]. Note that doppelalgebras introduced by Richter [17] are linear analogs of doppelsemigroups.…”

Section: Introductionmentioning

confidence: 99%

Structure of relatively free n-tuple semigroups

Zhuchok

2023

ADM

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An n-tuple semigroup is an algebra defined on a set with n binary associative operations. This notion was considered by Koreshkov in the context of the theory of n-tuple algebras of associative type. The n>1 pairwise interassociative semigroups give rise to an n-tuple semigroup. This paper is a survey of recent developments in the study of free objects in the variety of n-tuple semigroups. We present the constructions of the free n-tuple semigroup, the free commutative n-tuple semigroup, the free rectangular n-tuple semigroup, the free left (right) k-nilpotent n-tuple semigroup, the free k-nilpotent n-tuple semigroup, and the free weakly k-nilpotent n-tuple semigroup. Some of these results can be applied to constructing relatively free cubical trialgebras and doppelalgebras.

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Free k -nilpotent n -tuple semigroups

Zhuchok

2023

Communications in Algebra

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Note on cyclic doppelsemigroups

Cited by 3 publications

References 14 publications

On the upfamily extension of a doppelsemigroup

On the upfamily extension of a doppelsemigroup

Structure of relatively free n-tuple semigroups

Free k -nilpotent n -tuple semigroups

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