Geodesic
Free rectangular $n$-tuple semigroups
Čebyševskij sbornik, Tome 20 (2019) no. 3, pp. 261-271.

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An $n$-tuple semigroup is a nonempty set $G$ equipped with $n$ binary operations $\fbox{1}\,, \fbox{2}\,, ..., \fbox{n}\,$, satisfying the axioms $(x\fbox{r} \, y) \fbox{s}\, z=x\fbox{r}\,(y\fbox{s}\,z)$ 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. 
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A. V. Zhuchok. Free rectangular $n$-tuple semigroups. Čebyševskij sbornik, Tome 20 (2019) no. 3, pp. 261-271. http://geodesic.mathdoc.fr/item/CHEB_2019_20_3_a16/

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