All 2-potent Elements in \(\mathit{Hyp}_G(2)\)

Abstract

A generalized hypersubstitution of type \(\tau = (2)\) is a function which takes the binary operation symbol \(f\) to the term \(\sigma(f)\) which does not necessarily preserve the arity. Let \(Hyp_{G}(2)\) be the set of all these generalized hypersubstitutions of type \((2)\). The set \(Hyp_{G}(2)\) with a binary operation and the identity generalized hypersubstitution forms a monoid. The index and period of an element \(a\) of a finite semigroup are the smallest values of \(m\geq1\) and \(r\geq1\) such that \(a^{m+r}=a^m\). An element with the index \(m\) and period 1 is called an $m$-potent element. In this paper we determine all \(2\)-potent elements in \(Hyp_{G}(2)\).