Hyperrings with n-ary composition hyperoperation

Based on the concepts of composition ring and composition hyperring, in this note, we introduce the notion of [Formula: see text]-ary composition hyperring and study the connections with composition hyperrings. Moreover, we show that special strong multiendomorphisms of hyperrings can determine the [Formula: see text]-ary composition structure of a such hyperring and also a subclass of [Formula: see text]-ary composition hyperrings. Finally, the three isomorphism theorems are studied in the framework of [Formula: see text]-ary composition hyperrings.

A note on composition (m, n)-hyperrings

Based on the concepts of composition ring and composition hyper- ring, in this note we introduce the notion of composition structure for (m, n)-hyperrings and study the connections with composition hy- perrings. Moreover we show that particular strong endomorphisms of (m, n)-hyperrings can determine the composition structure of a such (m, n)-hyperrings. Finally, the three isomorphism theorems are pre- sented in the case of composition (m, n)-hyperrings, showing that they are not a pure extension of those for composition hyperrings.

A Raman Scattering Study of the Temperature Dependence of Boroxol Ring Clusters in B2 O3-GeO2 and B2 O3-GeO2-SiO2 Glasses.

This Raman study indicates that the boroxol ring concentrations for glasses in the B2 O3-GeO2 and B2 O3-GeO2-SiO2 systems are dependent upon temperature and composition. Ring breakage and formation occur well below Tg. An Arrhenius plot of intensity data for borogermanate g glass indicates that the heat of ring rupture is ca. 7.7 Kcal/mole. The boroxol ring concentrations in various glasses were also investigated at room temperature after various heat treatments. The structural interpretation from Raman scattering data was compared with that from IR data.