On a co-universal arrow in the construct of $n$-ary hyperalgebras
Josef Šlapal
2015
Miskolc Mathematical Notes
We introduce and study a new operation of product of n-ary hyperalgebras which lies, with respect to set inclusion, between their cartesian product and the cartesian product of their idempotent hulls. For every fixed n-ary hyperalgebra, the product introduced gives an endofunctor of the construct of n-ary hyperalgebras. We define a power of n-ary hyperalgebras and specify a class of n-ary hyperalgebras such that, with respect to the endofunctor, the power together with the evaluation map
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... ute a co-universal arrow for each hyperalgebra of the class. 2010 Mathematics Subject Classification: 08B25; 08C05 Keywords: hyperalgebra, diagonality, mediality, combined product, co-universal arrow with respect to a functor c 2015 Miskolc University Press 508 J.ŠLAPAL the cartesian product and a fixed idempotent hyperalgebra, the power together with the evaluation map constitute a co-universal arrow for each hyperalgebra of the class. In the present note, we restrict our considerations to n-ary hyperalgebras, i.e., hyperalgebras with just one n-ary hyperoperation. For n-ary hyperalgebras, we introduce an operation of product which is obtained as the restriction of the product of relational systems introduced and studied in [8] . The operation lies, with respect to set inclusion, between the cartesian product of n-ary hyperalgebras and the cartesian product of their idempotent hulls. For idempotent hyperalgebras, the cartesian product and the introduced one coincide. We will show that the above mentioned class has the property that, with respect to the endofunctor of the construct of n-ary hyperalgebras given by the new product and arbitrary (not only idempotent) but fixed n-ary hyperalgebra, the power together with the evaluation map constitute a co-universal arrow for each hyperalgebra of the class. We then get the first exponential law with respect to the new product which, unlike the first exponential law with respect to the cartesian product, is valid for arbitrary n-ary algebras in the exponents, not only for the idempotent ones.
doi:10.18514/mmn.2015.1029
fatcat:vxo6bq2xtnck3gqnio3kqjjhry