**
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 2, pp. 569-588 (2000)
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#
Relatively Free *-Bands

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Mario Petrich, Pedro V. Silva

Centro de Matematica, Faculdade de Ciencias, Universidade do Porto, 4050 Porto, Portugal, http://www.fc.up.pt/cmup

**Abstract:** A *-band is an algebra consisting of a band (idempotent semigroup) on which an involution * is defined satisfying an extra condition; in summary

(xy)^* = y^*x^*, x^{**} = x, x = xx^*x, (xy)z = x(yz), x^2 = x.

The lattice of all *-band varieties was determined by Adair who also provided a basis for the identities of each variety. Another system of bases was devised by Petrich. Defining certain operators on the free involutorial semigroup $F$ on a nonempty set $X$, we construct a system of fully invariant congruences on $F$ which is in bijection with the set of all proper *-band varieties, with the exception of normal *-band varieties which require a different treatment. The proof of this result is based on those evoked above and is broken into a long sequence of lemmas.

**Classification (MSC2000):** 20M05, 20M07

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