INDEPENDENCE OF CHARACTERIZING PROPERTIES OF (m + k,m)−RECTANGULAR BANDS

Abstract

An \((m+k,m)\)-semigroup \((Q;[\,])\) which is a direct product of a left-zero \((m+k,m)\)-semigroup and a right-zero \((m+k,m)\)-semigroup is called an \((m+k,m)\)-rectangular band. The class of \((m+k,m)\)-rectangular bands is characterized by the following identities:

\[
\text{(I)}\quad
\left[x_1^{m+k}\right]_i
=
\left[
y_1^{j-1}x_i y_{j+1}^{j+k-1}x_{i+k}y_{j+k+1}^{m+k}
\right]_j,
\qquad i,j\in N_m;
\]

\[
\text{(II)}\quad
\left[x_1^{m+2k}\right]_i
=
\left[x_1^i x_{i+k+1}^{m+2k}\right]_i;
\]

\[
\text{(III)}\quad
\left[{}^{m+k}\!x\right] = {}^m\!x.
\]

Here, the independence between these identities is proved.