42

Valentin D. Belousov and Zoran M. Stojakovic

should be replaced by the following four axioms. Let | ^4 | = a, | ^ | = ß, a, ß are the types (for infinitary case) or arities (for finitary case) of A and В pectively. We have:

1 ) \A-hB\ = \B\ + \A\, if a, ß>co,

2 ) |^ + 5| = |i5| + |^|-/ if a<co, ß>co,

3 ) |^ + Б| = |Л|, if a>co„ ß<co,

4 ) \A + B\=^\A\-i-\B\-\, if a<co, ß<6).

Let us prove the relation 1). By the definition of ( + ) we have

( 24 ) A (л'Г', В(Zß), Га) = (А + В) (х\-\ Х^, 7«),

where Х^ and Г« are well ordered sequences of the types ß and a respectively. From (24) it follows:

|^ + 5| = (/-l) + ß + a = ß + a, |у4! = (/-1)+ l+a = a,

/ i

Hence , |^ + ^| = ßfa=|^| + |^|. We note here that we cannot write \A + B\ = = \A\-b\B\ because the infinite types a and ß do not commute.

The proof of 2)—4) is similar.

We note also that the superposition on infinite place can also be considered.

At the end we formulate some problems on oo-quasigroups:

i J

1 . Solve the functional equation A-\-B=C + D when the types are rent from (0 + /:.

2 . Discuss the both notions of parastrophy given in 5°.

3 . Define the (/, /)-associativity for the oo-quasigroups of the types Фы and find its structure (of course if they exist).

4 . Find other examples of oo-quasigroups. Note that the example from §1 is not effective one.

5 . Does there exist oo-quasigroups satisfying some known identities? We note that the example of oo-loop given in §1, which has the property that all

к 00

elements are unity elements, satisfies the identities A {x, j, x) =y, for all x,y^Q, fc=l,2,... .

6 . Construct the theory of insertion algebras for infinitary case.

REFERENCES

[ 1 ] Мадевски Ж., Тр пен о веки Б., Чу по на f.. За инфинишарише асоци]а- шивни операции, Билтен ДруШт. матем. и физ. CFM, XV (1964), 19-22.

[ 2 [ M i 1 i с S., On GD-groupoids with applications to n-ary quasigroups, Publ. Inst Math. T. 13 (27), 1972, 65-76.

[ 3 ] S i e r p i n s к i W., Cardinal and ordinal numbers, Warszawa, 1958.

[ 4 ] Белоусов В. Д., п-арные квазшруппы, Кишинев, „Штиинца'% 1972.

[ 5 ] В е 1 о U s о V V. D., Balanced identities in algebras of quasigroups, Aequationes mathematicae, vol. 8, fasc. 1/2, 1972, 1-73.