ON \((k(n-1)+1)\)-SEMIGROUPS WITH \((n-2)\)-NEUTRAL OPERATIONS

Abstract

In the present paper, we define [left, right] \((n-2)\)-neutral operation \(E\;[:Q^{n-2}\to Q]\) of a \((k(n-1)+1)\)-groupoid, \((k,n)\in N\times(N\setminus\{1\})\), so that (among others) for \(n=2\) \(E(\emptyset)[a_1^{n-2}=\emptyset]\) is a neutral element of the \((k+1)\)-groupoid \((Q,A)\). The main result of the paper is the following proposition. If a \((k(n-1)+1)\)-semigroup \((Q,A)\), \(k\geq 2\), has a left [right] \((n-2)\)-neutral operation \(E\), then there is an \(n\)-semigroup \((Q,B)\) with \(\{1,n\}\)-neutral operation [:[5], 1.2.2], such that for every \(x_1^{k(n-1)+1}\in Q\), \(\displaystyle A\left(x_1^{k(n-1)+1}\right)=B^k\left(x_1^{k(n-1)+1}\right) \).

(E.g.: \(\displaystyle B^2\left(x_1^{2n-1}\right)\overset{\mathrm{def}}{=}B\left(B\left(x_1^n\right),x_{n+1}^{2n-1}\right) \). )

Moreover, if \(n\geq 3\) then \((Q,A)\) is a \((k(n-1)+1)\)-group.