Multi-algebra
A set in which a system of (in general, partial) multi-operations is given. A partial multi-operation on a set is a partial mapping
between Cartesian powers of
, where
. Here
means a one-element set. A homomorphism
of multi-algebras with the same system of multi-operations is a mapping
such that if
is a multi-operation mapping the
-th power into the
-th, then
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for all . The concept of a multi-algebra is a generalization of that of a universal algebra. At the same time a multi-algebra is a particular case of an algebraic system, since a mapping
can be identified with the
-ary relation
on
,
. Multi-algebras arise most naturally in connection with the functorial approach to universal algebra (see [1]). Namely, let
be a category whose objects are the natural numbers including zero, where the object
is the direct product of the objects
and
. Then a functor
from
into the category of sets that commutes with direct products is a multi-algebra on the set
with system of multi-operations
, where
in
. The homomorphisms in this case are precisely the natural transformations of functors.
References
[1] | F.W. Lawvere, "Functorial semantics of algebraic theories" Proc. Nat. Acad. Sci. USA , 50 : 5 (1963) pp. 869–872 |
[2] | V.D. Belousov, "Algebraic nets and quasi-groups" , Stiintsa , Kishinev (1971) (In Russian) |
Multi-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-algebra&oldid=13513