ON ${}^*$-BANDS AND THEIR VARIETIES

MARIO PETRICH AND PEDRO V. SILVA

Abstract:

A ${}^*$-band is a semigroup with a unary operation ${}^*$ obeying the axioms $(xy)^*=y^*x^*$, $x^{**}=x$, $x=xx^*x$, $x^2=x$. On a free involutorial semigroup $F$ on a nonempty set $X$, we define a family of operators $\delta_{t_n}$ and prove that each of them is a ${}^*$-homomorphism of $F$ onto its image with a suitable multiplication and the ${}^*$-operation of $F$. We then investigate the interplay of this operator with several others occurring in the literature as well as the relationship of the equivalence relations they induce on $F$ or on $X^+$. In particular, we obtain the structural description of all relatively free ${}^*$-bands. We conclude with a brief consideration of the problem of converting ${}^*$-identities to equivalent star-free identities.