Free rectangular n-tuple semigroups
https://doi.org/10.22405/2226-8383-2019-20-3-261-271
Abstract
An n-tuple semigroup is a nonempty set G equipped with n binary operations $$\fbox{1}\,, \fbox{2}\,, ..., \fbox{n}\,,$$ satisfying the axioms $$(x\fbox{r} \, y) \fbox{s}\, z=x\fbox{r}\,(y\fbox{s}\,z)$$ 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.
About the Author
Anatolii Vladimirovich Zhuchok
Luhansk Taras Shevchenko National University (Starobilsk, Ukraine)
Ukraine
Ukraine
doctor of physical and mathematical Sciences, Professor, head of the Department of algebra and system analysis
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Review
For citations:
Zhuchok A.V. Free rectangular n-tuple semigroups. Chebyshevskii Sbornik. 2019;20(3):261-271. (In Russ.) https://doi.org/10.22405/2226-8383-2019-20-3-261-271
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