Multi-algebra

A set in which a system of (in general, partial) multi-operations is given. A partial multi-operation on a set $ A $ is a partial mapping $ f : A ^ {n} \rightarrow A ^ {m} $ between Cartesian powers of $ A $, where $ n , m \geq 0 $. Here $ A ^ {0} $ means a one-element set. A homomorphism $ g : A \rightarrow B $ of multi-algebras with the same system of multi-operations is a mapping $ g $ such that if $ f $ is a multi-operation mapping the $ n $- th power into the $ m $- th, then

$$ g ^ {m} ( f ( x _ {1} \dots x _ {n} ) ) = \ f ( g ( x _ {1} ) \dots g ( x _ {n} ) ) $$

for all $ x _ {i} \in A $. 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 $ f : A ^ {n} \rightarrow A ^ {m} $ can be identified with the $ ( m + n ) $- ary relation $ ( x , f ( x) ) $ on $ A $, $ x \in A ^ {n} $. Multi-algebras arise most naturally in connection with the functorial approach to universal algebra (see [1]). Namely, let $ C $ be a category whose objects are the natural numbers including zero, where the object $ m + n $ is the direct product of the objects $ m $ and $ n $. Then a functor $ F $ from $ C $ into the category of sets that commutes with direct products is a multi-algebra on the set $ F ( 1) = A $ with system of multi-operations $ F ( f ) : A ^ {n} \rightarrow A ^ {m} $, where $ f : n \rightarrow m $ in $ C $. 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)