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Chebyshevskii Sb., 2019, Volume 20, Issue 3, Pages 261–271 (Mi cheb810)  

Free rectangular $n$-tuple semigroups

A. V. Zhuchok

Luhansk Taras Shevchenko National University (Starobilsk, Ukraine)

Abstract: An $n$-tuple semigroup is a nonempty set $G$ equipped with $n$ binary operations $\fbox{1} , \fbox{2} , ..., \fbox{n} $, satisfying the axioms $(x\fbox{r}   y) \fbox{s}  z=x\fbox{r} (y\fbox{s}ż)$ for all $x,y,z \in G$ and $r,s\in \{1,2,...,n\}$. This notion was considered by Koreshkov in the context of the theory of $n$-tuple algebras of associative type. Doppelsemigroups are $2$-tuple semigroups. The $n$-tuple semigroups are related to interassociative semigroups, dimonoids, trioids, doppelalgebras, duplexes, $g$-dimonoids, and restrictive bisemigroups. If operations of an $n$-tuple semigroup coincide, the $n$-tuple semigroup becomes a semigroup. So, $n$-tuple semigroups are a generalization of semigroups.
The class of all $n$-tuple semigroups forms a variety. Recently, the constructions of the free $n$-tuple semigroup, of the free commutative $n$-tuple semigroup, of the free $k$-nilpotent $n$-tuple semigroup and of the free product of arbitrary $n$-tuple semigroups were given. The class of all rectangular $n$-tuple semigroups, that is, $n$-tuple semigroups with $n$ rectangular semigroups, forms a subvariety of the variety of $n$-tuple semigroups.
In this paper, we construct the free rectangular $n$-tuple semigroup and characterize the least rectangular congruence on the free $n$-tuple semigroup.

Keywords: $n$-tuple semigroup, free rectangular $n$-tuple semigroup, free $n$-tuple semigroup, semigroup, congruence.

DOI: https://doi.org/10.22405/2226-8383-2018-20-3-261-271

Full text: PDF file (566 kB)

UDC: 512.57, 512.579
Received: 08.10.2019
Accepted:12.11.2019
Language:

Citation: A. V. Zhuchok, “Free rectangular $n$-tuple semigroups”, Chebyshevskii Sb., 20:3 (2019), 261–271

Citation in format AMSBIB
\Bibitem{Zhu19}
\by A.~V.~Zhuchok
\paper Free rectangular $n$-tuple semigroups
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 3
\pages 261--271
\mathnet{http://mi.mathnet.ru/cheb810}
\crossref{https://doi.org/10.22405/2226-8383-2018-20-3-261-271}


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