ON \((k(n-1)+1)\)-SEMIGROUPS WITH \((n-2)\)-NEUTRAL OPERATIONS - Ušan, Janez (Matematichki Bilten, 1998)
In the present paper, we define [left, right] \((n-2)\)-neutral operation \(E [:Q^n-2→ Q]\) of a \((k(n-1)+1)\)-groupoid, \((k,n)N×(N{1})\), so that (among others) for \(n=2\) \(E()[a_1^n-2=]\) 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 \((Q,A)\), \(k≥ 2\), 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 \(x_1^k(n-1)+1Q\), \(A(x_1^k(n-1)+1)=B^k(x_1^k(n-1)+1) \). (E.g.: \(B^2(x_1^2n-1)def =B(B(x_1^n),x_n+1^2n-1) \). ) Moreover, if \(n≥ 3\) then \((Q,A)\) is a \((k(n-1)+1)\)-group.