INDEPENDENCE OF CHARACTERIZING PROPERTIES OF (m + k,m)−RECTANGULAR BANDS - Miovska, Valentina; Dimovski, Dončo (Matematichki Bilten, 2009)
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: \[ (I)[x_1^m+k]_i = [ y_1^j-1x_i y_j+1^j+k-1x_i+ky_j+k+1^m+k ]_j, i,jN_m; \] \[ (II)[x_1^m+2k]_i = [x_1^i x_i+k+1^m+2k]_i; \] \[ (III)[^m+k\!x] = ^m\!x. \] Here, the independence between these identities is proved.