Regularity in terms of hyperideals. (Q463160)

The paper deals with the notion of regularity in algebraic hypersystems. By an algebraic hypersystem \((H,f_1,\ldots,f_n)\) is meant a non-empty set \(H\) closed under a collection of \(m_i\)-ary \textit{hyperoperations} \(f_i\), i.e., \(f_i\colon\underbrace{H\times\cdots\times H}_{m_i\text{-times}}\to\mathcal P^*(H)\) is a mapping for \(1\leq i\leq n\), and often also satisfying a fixed set of laws, for instance, the associative law.NEWLINENEWLINE In classical semigroup theory it is well known that the notion of regularity can be stated in terms of ideals. For example, a commutative semigroup is regular if and only if every ideal is idempotent. In this paper the authors first define the notion of regularity in an algebraic hypersystem \((H,f)\) and then prove similar result for this hypersystem. (Note: The correct publication year of ref. [12] in the paper is 1956.)