ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 20 Iranmanesh, A & Iradmusa, M. N., Tarbiat Modarres University, Iran H_{v}-structures associated with generalized P −hyperoperations Let H be a set and · : H × H ⟶ P(H) be a hyperoperation. The (·) in H is called weak associative if (xy)z ∩x(yz) ≠ ∅ for all x, y, z in H. H is called H_{v}-quasigroup if the reproduction axiom is valid, i. e. xH = Hx = H for all x in H. A hyperoperation (R, +, ·) is called H_{v}-ring if

(i) (+) and (·) are weak associative (ii) the reproduction axiom is valid for (+), and (iii) (·) is weak distributive with respect to (+), i. e. x(y + z) ∩ (xy + xz) ≠ ∅ and (x + y)z ∩ (xz + yz) ≠ ∅ for all x, y, z in R. Now, consider a ring (R, +, ·) and we take P_{1}, P_{2} ⊂ R with P_{1} ≠ ∅ ≠ P_{2}. The hyperoperations xP_{1}^{∗}y = x + y + P_{1} and xP_{2}^{∗}y = xyP_{2}, for all x, y ∈ R are called P -hyperoperations. The < R, P_{1}^{∗}, P_{2}^{∗} > becomes a H_{v}-ring called P − H_{v}-ring. In this paper we introduce generalized P - hyperoperations and find a sufficient condition so that R with these generalized P -hyperoperations become a P − H_{v}-ring. Then we introduce the concept of the homomorphism of P − H_{v}-rings and compute the number of these structures.